Download Abstract Book - the ICAAM 2012 Conference in Gumushane

TABLE OF CONTENTS 

P a g e | i 

Shadowing in the parabolic equations 

S. Piskarev________________________________________________________________1 

Fractals arising from Newton's method 

F. Çilingir _________________________________________________________________2 

On the Classifications of C-Algebras Using Unitary Groups 

A. Al-Rawashdeh___________________________________________________________3 

Solution of One Problem for the Equation of Parabolic type with Involution Perturbation 

A. A.Sarsenbi, A.A. Tengaeva ________________________________________________4 

A Note on the Numerical Solution of Fractional Schrödinger Differential Equations 

A. Ashyralyev, B. Hicdurmaz _________________________________________________5 

On Bitsadze-Samarskii type nonlocal boundary value problems for semilinear elliptic equations 

A. Ashyralyev, E. Ozturk_____________________________________________________6 

A Third-Order of Accuracy Difference Scheme for the Bitsadze-Samarskii Type Nonlocal 

Boundary Value Problem 

A. Ashyralyev, F. S. Ozesenli Tetikoglu_________________________________________7 

NBVP for Hyperbolic Equations Involving Multi-point and Integral Conditions 

A. Ashyralyev, N. Aggez _____________________________________________________8 

Boundary Value Problem for a Third Order Partial Differential Equation 

A. Ashyralyev, N. Aggez, F. Hezenci___________________________________________9 

Fractional Parabolic Differential and Difference Equations with the Dirichlet-Neumann 

Condition 

A. Ashyralyev, N. Emirov, Z. Cakir _________________________________________10-11 

High Order of Accuracy Stable Difference Schemes for Numerical Solutions of NBVP for 

Hyperbolic Equations 

A. Ashyralyev, O. Yildirim___________________________________________________12 

Positivity of Two-dimensional Elliptic Differential Operators in Hölder Spaces 

A. Ashyralyev, S. Akturk, Y. Sozen_________________________________________13-14 

On the Numerical Solution of Ultra Parabolic Equations with Neumann Condition 

A. Ashyralyev, S. Yılmaz____________________________________________________15 

Existence and Uniqueness of Solutions for Nonlinear Impulsive Differential Equations with 

Two-point and Integral Boundary Conditions 

A. Ashyralyev, Y.A. Sharifov_________________________________________________16 

Optimal Control Problem for Impulsive Systems with Integral Boundary Conditions 

A. Ashyralyev, Y.A. Sharifov______________________________________________17-18 

On Stability of Hyperbolic- Elliptic Differential Equations With Nonlocal Integral Condition 

A. Ashyralyev, Z. Odemis Ozger, F. Ozger___________________________________19-20 

Fuzzy Continuous Dynamical System: A Multivariate Optimization Technique 

A. Bandyopadhyay, Samarjit Kar_____________________________________________21 

Analysis of Dynamical Complex Network of Ecological Stability Diversity and Persistence 

A. Bandyopadhyay, Samarjit Kar__________________________________________22-23 

Analytical solution for the recovery tests after constant-discharge tests in confined aquifers 

A. Atangana______________________________________________________________24 

Bright and dark soliton solutions for the variable coefficient generalizations of the KP equation 

A. Bekir, Ö. Güner, A.C. Çevikel___________________________________________25-27 

A Characterization of Compactness in Banach Spaces with Continuous Linear 

Representations of the Rotation Group of a Circle 

A. Cavus, M. Kunt_______________________________________________________28-29 

The approximate solutions of linear Goursat Problems via Homotopy Analysis Method 

A. Eryılmaz, M. Basbük, H.Tuna______________________________________________30 

Paths of Minimal Length on Suborbital Graphs with Recurrence Relations 

A.H. Deger, M. Besenk, B.O. Guler____________________________________________31 

Riesz Basis Property of Eigenfunctions of One Boundary-Value Transmission Problem 

A. H. Olgar, O. Sh. Mukhtarov________________________________________________32

P a g e | ii 

On a Subclass of Univalent Functions with Negative Coefficients 

A. S. Juma, H. Zırar________________________________________________________33 

Fine spectra of upper triangular triple-band matrices over the sequence space lp,(0 

A. Karaisa________________________________________________________________34 

Approximation by Certain Linear Positive Operators of Two Variables 

A.K. Gazanfer, I. Büyükyazıcı________________________________________________35 

Impulsive differential equations with variable times 

A.Lakmeche, F.Berrabah____________________________________________________36 

Parabolic Problems with Parameter Occuring in Environmental Engineering 

A. Sahmurova, V.B. Shakhmurov_____________________________________________37 

On Darboux Helices in Minkowski Space R1 3 

A. Senol, E. Zıplar, Y. Yaylı__________________________________________________38 

On the Numerical Solution of a Diffusion Equation Arising in Two-phase Fluid Flow 

A.S. Erdogan, A.U. Sazaklioglu_______________________________________________39 

On the Solution of a Three Dimensional Convection Diffusion Problem 

A. S. Erdogan, M. Alp____________________________________________________40-41 

A Fuzzy Max-Min Approach to Multi Objective, Multi Echelon Closed Loop Supply Chain 

B. Ahlatcioglu Ozkok, E. Budak, S. Ercan______________________________________42 

A Fuzzy Approach to Multi Objective Multi Echelon Supply Chain 

B. Ahlatcioglu Ozkok, S. Ercan, E. Budak______________________________________43 

Modeling Voting Behavior in the Eurovision Song Contest 

B. Dogru_________________________________________________________________44 

Derivation and numerical study of relativistic Burgers equations posed on Schwarzschild 

spacetime 

B. Okutmustur____________________________________________________________45 

Newton-Pade Approximations for Univariate and Multivariate Functions 

C. Akal, A. Lukashov____________________________________________________46-47 

Finite Difference Method for The Reverse Parabolic Problem with Neumann Condition 

C. Ashyralyyev, A. Dural, Y. Sozen____________________________________________48 

Three Models based Fusion Approach in the Fuzzy Logic Context for the Segmentation of MR 

Images : A Study and an Evaluation 

C. Lamiche, A. Moussaoui___________________________________________________49 

Approximate Solutions of The Cauchy Problem for The Heat Equations 

D. Agirseven______________________________________________________________50 

The normal inverse Gaussian distribution: exposition and applications to modeling asset,index 

and foreign exchange closing prices 

D. Teneng, K. Parna________________________________________________________51 

Radial basis functions method for determining of unknown coefficient in parabolic equation 

E. Can___________________________________________________________________52 

Compact and Fredholm Operators on Matrix Domains of Triangles in the Space of Null 

Sequences 

E. Malkowsky_____________________________________________________________53 

Compact Operators on Spaces of Sequences of Weighted Means 

E. Malkowsky, F. Özger_____________________________________________________54 

Extended Eigenvalues of Direct Integral of Operators 

E. Otkun Cevik, Z.I. Ismailov_________________________________________________55 

Exponential decay and blow up of a solution for a system of nonlinear higher-order wave 

equations 

E. Piskin, N. Polat__________________________________________________________56 

A New General Inequality for double integrals 

E. Set, M. Z. Sarıkaya, A. O. Akdemir_______________________________________57-58 

Exact Solutions of the Schrödinger Equation with Position Dependent Mass for the solvable 

Potentials 

F. Aricak, M.Sezgin________________________________________________________59

P a g e | iii 

Sturm Liouville Problem with Discontinuity Conditions at Several Points 

F.Hıra, N. Altınısık_______________________________________________________60-61 

Characterization of Three Dimensional Cellular Automata over Zm 

F. Sah, I. Siap, H. Akın______________________________________________________62 

Positivity of Elliptic Difference Operators and its Applications 

G.E. Semenova____________________________________________________________63 

On (α, β)-derivations in BCI-algebras 

G. Muhiuddin _____________________________________________________________64 

Transient and Cycle Structure of Elementary Rule 150 with Reflective Boundary 

H. Akın, I. Siap, M.E. Koroglu________________________________________________65 

Numerical solution of a laminar viscous .ow boundary layer equation using Haar Wavelet 

Quasilinearization Method 

H. Kaur, R.C. Mittal, V. Mishra________________________________________________66 

Characterizations of Slant Helices According to Quaternionic Frame 

H. Kocayigit, M. Önder, B. B. Pekacar _________________________________________67 

Some Characterizations of Constant Breadth Tımelıke Curves in Minkowskı 4-space E 4 

H. Kocayiğit, M. Önder, Z. Çiçek ___________________________________________68-69 

Using Inverse Laplace Transform for the solution of a Flood Routing Problem 

H. Saboorkazeran, M.F. Maghrebi_____________________________________________70 

Applied Mathematics Analysis of the Multibody Systems 

H. Sahin, A. K. Kar, E. Tacgin_____________________________________________71-72 

Multibody Railway Vehicle Dynamics Using Symbolic Mathematics 

H. Sahin, A. K. Kar, E. Tacgin_____________________________________________73-74 

Existence of Global Solutions for a Multidimensional Boussinesq-Type Equation with 

Supercritical Initial Energy 

H. Taskesen, N. Polat_______________________________________________________75 

Dissipative Extensions of Fourth Order Differential Operators in the Lim- 3 case 

H. Tuna __________________________________________________________________76 

On the stability of the steady-state solutions of cell equations in a tumor growth model 

I. Atac, S. Pamuk __________________________________________________________77 

The Cuttings Transport Modelling with Couette Flow 

I. Cumhur ________________________________________________________________78 

On the Numerical Solution of Diffusion Problem with Singular Source Terms 

I. Turk, M. Ashyraliyev______________________________________________________79 

One Boundary-Value Problem Perturbed by Abstract Linear Operator 

K. Aydemir, O. Sh. Mukhtarov________________________________________________80 

Using expanding method of (G′/G) to find the travelling wave solutions of nonlinear 

partial differential equations and solving mkdv equation by this method 

K. Nojoomi, M. Mahmoudi, A. Rahmani _____________________________________81-82 

Weighted Bernstein Inequality for Trigonometric Polynomials on a Part of The Period 

M. A. Akturk, A. Lukashov___________________________________________________83 

Mixed problem for a differential equation with involution under boundary conditions of general 

form 

M.A. Sadybekov, A.M.Sarsenbi_______________________________________________84 

An Application on Suborbital Graphs 

M. Besenk, A.H. Deger, B.O. Guler____________________________________________85 

Cellular Automata Based Byte Error Correcting Codes over Finite Fields 

M. E. Koroglu, I. Siap, H. Akin________________________________________________86 

On The First Fundamental Theorem for Special Dual Orthogonal Group SO(2;D) and its 

Application to Dual Bezier Curves 

M.Incesu, O. Gursoy _______________________________________________________87 

On Euler’s differential method for continued fractions 

M. J. Shah Belaghi, A. Bashirov______________________________________________88 

Almost Convergence and Generalized Weighted 

M. Kirişçi_________________________________________________________________89

P a g e | iv 

Wavelet-based prediction of crude oil prices 

M. Mahmoudi, K. Nojoomi, A. Rahmani_____________________________________90-91 

Numerical solution of a time-fractional Navier–Stokes Equation with modified Riemann- 

Liouville derivative 

M. Merdan, A. Gökdoğan_________________________________________________92-93 

The Modified Simple Equation Method for Solving Some Nonlinear Evolution Equations 

M.Mızrak, A.Ertaş_______________________________________________________94-95 

Application of Cross Efficiency in Stock Exchange 

M. M. Kaleibara, S. Daneshvar_______________________________________________96 

Application of the Trial Equation Method for some Nonlinear Evolution Equations 

M. Odabasi, E. Misirli ______________________________________________________97 

Paranormality of Some Class Differential Operators for First Order 

M. Sertbas, L. Cona________________________________________________________98 

Oscillation Theorems for Second-Order Damped Dynamic Equation on Time Scales 

M. T. Senel_______________________________________________________________99 

On the fine spectrum of the $\Lambda$ operator defined by a lambda matrix over the 

sequence space $c_{0}$ and $c$ 

M. Yeşilkayagil, F. Başar___________________________________________________100 

On Hadamard Type Integral Inequalities For Nonconvex Functions 

M. Z. Sarikaya, H. Bozkurt, N. Alp________________________________________101-102 

A geometrical approach of an optimal control problem governed by EDO 

N. Driai__________________________________________________________________103 

Existence of Local Solution for a Double Dispersive Bad Boussinesq-Type Equation 

N. Dündar, N. Polat________________________________________________________104 

A Perturbation Solution Procedure for a Boundary Layer Problem 

N. Elmas, A. Ashyralyev, H. Boyaci_______________________________________105-106 

Solution of Differential Equations by Perturbation Technique Using any Time Transformation 

N. Elmas, H. Boyaci ___________________________________________________107-108 

Aproximation Properties of a Generalization of Linear Positive Operators in C[0,A] 

N.Gonul_________________________________________________________________109 

Three-term Asymptotic Expansion for the Moments of the Ergodic Distribution of a Renewalreward 

Process with Gamma Distributed Interference of Chance 

N. Okur Bekar, R. Aliyev, T. Khaniyev________________________________________110 

Blow up of a solution for a system of nonlinear higher-order wave equations with strong 

damping 

N. Polat, E. Piskin_________________________________________________________111 

Reduction of spectral problem of Cauchy-Riemann operator with homogeneous boundary 

conditions to an integral equation 

N.S.Imanbayev___________________________________________________________112 

A Note on Some Elementary Geometric Inequalities 

O. Gercek, D. Caliskan, A. Sobucova, F. Cekic_____________________________113-114 

Solving Crossmatching Puzzles Using Multi-Layer Genetic Algorithms 

O. Kesemen, E. Ozkul _____________________________________________________115 

Generate Adaptive Quasi-Random Numbers 

O. Kesemen, N. Jabbari____________________________________________________116 

Polygonal Approximation of Digital Curve Using Artificial Bee Colony Optimization Algorithms 

O. Kesemen, S. Vafaei_____________________________________________________117 

Generating Random Points from Arbitrary Distribution In Polygonal Areas 

O. Kesemen, U. Unsal _____________________________________________________118 

Panoramic Image Mosaicing Using Multi-Object Artificial Bee Colony Optimization Algorithm 

O. Kesemen, Y. Yeginoglu__________________________________________________119 

Some Properties of a Sturm-Liouville-Type Problem and The Green Function 

O. Kuzu, Y. Kuzu, M. Kadakal ___________________________________________120-121

P a g e | v 

Real Time 3D Palmprint Pose Estimation and Feature Extraction Using Multiple View 

Geometry Techniques 

Ö. Bingöl, M. Ekinci_______________________________________________________122 

Chaos in cubic-quintic nonlinear oscillator 

P. Sharma, V. G. Gupta____________________________________________________123 

The Numerical Solution of Boundary Value Problems by using Galerkin Method 

S. Alkan, T. Yeloğlu, D. Yılmaz______________________________________________124 

Semismooth Newton method for gradient constrained minimization problem 

S.Anyyeva, K.Kunisch_____________________________________________________125 

The Finite Element Method Solution of Variable Diffusion Coefficient Convection-Diffusion 

Equations 

S.H Aydın, C. Çiftçi________________________________________________________126 

Multiple solutions for quasilinear equations depending on a parameter 

S. Heidarkhani________________________________________________________127-128 

The Modified Bi-quintic B-spline base functions: An Application to Diffusion Equation 

S. Kutluay, N.M. Yagmurlu _________________________________________________129 

Numerical Solutions of the Modified Burgers' Equation by Cubic B-spline Collocation Method 

S. Kutluay, Y. Ucar, N.M. Yagmurlu__________________________________________130 

The Modified Kudryashov Method for Solving Some Evolution Equations 

S.M. Ege, E. Misirli____________________________________________________131-132 

Study of an inverse problem that models the detection of corrosion in metalic plate whose 

lower part is embedded 

S. M. Said _______________________________________________________________133 

Commuting nilpotent operators with maximal rank 

S. Ozturk Kaptanoglu______________________________________________________134 

Finding Global minima with a new class of filled function 

T. Hamaizia______________________________________________________________135 

Weak Convergence Theorem For A Semi-Markovian Random Walk With Delay And Pareto 

Distributed Interference Of Chance 

T. Kesemen, F. Yetim______________________________________________________136 

Parameter dependent Navier-Stokes like problems 

V. B. Shakhmurov________________________________________________________137 

A New Spline Approximation for the Solution of One-space Dimensional Second Order Nonlinear 

Wave Equations With Variable Coefficients 

V. Gopal, R. K. Mohanty________________________________________________138-139 

An error correction method for solving stiff initial value problems based on a cubic C1-spline 

collocation method 

Xi. Piaoa, S. Dong Kima, P. Kim_________________________________________140-141 

On Numerical Solution of Multipoint NBVP for Hyperbolic-Parabolic Equations with Neumann 

Condition 

A. Ashyralyev, Y. Ozdemir__________________________________________________142 

Classification of exact solutions for the Pochhammer-Chree equations 

Y. Gurefe, Y. Pandir, E. Misirli_______________________________________________143 

On the Density of Regular Functions in Variable Exponent Sobolev Spaces 

Y. Kaya _________________________________________________________________144 

Numerical Solution of a Hyperbolic-Schrödinger Equation with Nonlocal Boundary Conditions 

Y. Ozdemir, M. Kucukunal__________________________________________________145 

New generalized hyperbolic functions to find exact solution of the nonlinear partial differential 

equation 

Y. Pandir, H. Ulusoy___________________________________________________146-148 

Equivalence of afine curves 

Y. Sağıroğlu__________________________________________________________149-150 

Modified trial equation method for nonlinear differential equations 

Y. A Tandogan, Y. Pandir, Y. Gurefe_________________________________________151

P a g e | vi 

Finite Difference Method for the Integral-Differential Equation of the Hyperbolic Type 

Z. Direk, M. Ashyraliyev____________________________________________________152 

Normal Extensions of a Singular Differential Operator For First Order 

Z.I. Ismailov, R. Ozturk Mert ________________________________________________153 

Reproducing Kernel Hilbert Space Method for Solving the Pollution Problem of Lakes 

Z. Karabulut, V. S. Ertürk___________________________________________________154

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Page 1

F˙IGEN Ç˙IL˙ING˙IR 

Çankaya University, Turkey 

cilingirfigen@gmail.com 

Fractals arising from Newton’s method 

Abstract. The aim of this talk is to introduce the concept of fractals arising from Newton’s 

method. We consider the dynamics as a special class of rational functions that are obtained 

from Newton’s method when applied to a polynomial equation. Finding solutions of these 

equations leads to some beautiful images in complex functions. These images represent the 

basins of attraction of roots of complex functions. If z0 is an attracting periodic point of 

some rational function of degree larger that one, its basin of attraction is as follows: 

B(z0) := {z ∈ C |Nf n (z0) converges to z0, n → ∞}. 

The basin of attraction B(z0) is a union of components of the Fatou set, and the boundary 

of B(z0) coincides with the Julia sets of a rational function Nf. In this presentation, we seek 

will the answer of the following question: 

For example, 

“What is the dynamics near the chosen parabolic fixed points?” 

f(z) = (z 2 + 4)e z the Newton function of f is Nf(z) = z3 +z 2 +4z−z 

z 2 +2z+4 

and the fractal image of that function on Riemann sphere is presented. 

References 

Page 2 

[1] Ahlfors,L.V.[1979] Complex Analysis, McGraw-Hill. 

[2] Alexander,D.[1992] The Historical background to the works of Pierre Fatou and Gaston Julia in 

Complex Dynamics, Thesis, Boston University. 

[3] Beardon,A.[1991] Iteration of RationalFunctions, Springer-Verlag. 

[4] Çilingir, F. [2004] ”Finiteness of the Area of Basins of Attraction of Relaxed Newton Method for 

Certain Holomorphic Functions”, IJBC, Vol14, No.12 (2004) 4177 − 4190. 

[5] Devaney,R.L.[1989] An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City,Calif. 

[6] Keen,L.[1989] Julia sets, Chaos and Fractals, the Mathematics behind the Compuer Graphics, ed. 

R.L.Devaney & L.Keen,Proc.Symp.Appl.Math.39,Amer.Math.Soc., pp.57-75.

Abstract 

On the Classifications of C ∗ -Algebras Using Unitary Groups 

Ahmed Al-Rawashdeh 

Department of Mathematical Sciences, UAEU, Al Ain 

United Arab Emirates 

In 1955, Dye proved that the discrete unitary group in a factor determines the algebraic type of the 

factor. Using Dye’s approach, we prove similar results to a larger class of amenable unital C ∗ -algebras 

including simple unital AH-algebras (of SDG) with real rank zero. If φ is an isomorphism between the 

unitary groups of two unital C ∗ -algebras, it induces a bijective map θφ between the sets of projections of 

the algebras. For some UHF-algebras, we construct an automorphism φ of their unitary group, such that 

θφ does not preserve the orthogonality of projections. For a large class of unital C ∗ -algebras, we show that 

θφ is always an orthoisomorphism. This class includes in particular the Cuntz algebras On, 2 ≤ n ≤ ∞, 

and the simple unital AF-algebras having 2-divisible K0-group. If φ is a continuous automorphism of 

the unitary group of a UHF-algebra A, we show that φ is implemented by a linear or a conjugate linear 

∗-automorphism of A. 

References 

[1] K.R. Davidson, C ∗ -Algebras by Example, Fields Institute Monographs, 6, Amer. Math. Soc., 

Providencs, RI (1996). 

[2] H. Dye, On the Geometry of Projections in Certain Operator Algebras, Ann. of Math., 61 (1955), 

p.73-89. 

[3] G. Elliott and G. Gong, On the Classification of C ∗ -algebras of Real Rank Zero,II, Ann. of Math., 

144 (1996), p.497-610. 

[4] M. Rørdam, Classification of Nuclear C ∗ -Algebras, in Encyclopedia of Math. Sci., Operator Alge- 

bras and Non-commutative Geometry VII, Springer-Verlag Berlin Heidelberg-New York (2000). 

Page 3

Solution of One Problem for the Equation of Parabolic type 

with Involution Perturbation 

Abdisalam A.Sarsenbi ∗ and A.A. Tengaeva † 

∗ M.Auezov South-Kazakhstan State University, Shymkent, Kazakhstan, abzhahan@mail.ru 

† M.Auezov South-Kazakhstan State University, Shymkent, Kazakhstan, aijan0973@mail.ru 

Abstract. In the domain D = {(x,t) : −1 < x < 1,0 < t < T } the following problem is considered: Obtain the solution 

u ∈ C 2,1 (D) ∩C(D) of the equation 

ut(x,t) = uxx(−x,t) − αuxx(x,t), (1) 

which satisfies the conditions 

u(x,0) = ϕ(x),−1 ≤ x ≤ 1;u(−1,t) = 0,u(1,t) = 0,0 ≤ t ≤ T,ϕ(−1) = 0,ϕ(1) = 0. (2) 

Application of the Fourier method gives the spectral problem of the form 

−X ′′ (−x) + αX ′′ (x) = λX(x),−1 < x < 1,X(−1) = 0,X(1) = 0. (3) 

The system of eigenfunctions of (3) is generated Riesz basis in L2(−1,1). 

Theorem 1. If α is real number α and |α| > 1, then the problem (1)-(2) has a unique solution. 

Theorem 2. If α is a complex number and |Reα| ≥ 1, then the problem (1)-(2) has a unique solution. 

Moreover, we have the following formula 

∞ 

u(x,t) = ∑ 

k=0 

π (1−α)( 

cke 2 +kπ)2t π 

cos( 

2 

∞ 

+ kπ)x + ∑ dke 

k=1 

−(1+α)(kπ)2 t 

sinkπx. 

Many papers are devoted to investigation of partial differential equations and spectral problems of differential operators 

with involution see, for example, [1-4]. 

Keywords: Spectral of Problems, Differential Operators with involutions, Exact Solutions, Fourier Series 

PACS: 87.10.Ed 

REFERENCES 

Page 4 

1. J. Wiener, Generalized solutions of functional differential equations, 1993. 

2. A. B. Lin’kov, The substantiation of a method of Fourier for bounders value provlems with involution deviation, The Bulletin 

Sam. GU., 2, 60-66, 1999. 

3. A.M. Sarsenbi, Unconditional’s the bases connected with the nonclassical differential operator of the second order, Dif. 

Equation, 46(4), 506-511, 2010. 

4. A.A. Tengaeva, A.M. Sarsenbi, About basic properties of root functions of two generalized spectral problems, Dif.Equation, 

48(2), 294-296, 2012.

A Note on the Numerical Solution of Fractional Schrödinger Di¤erential Equations 

A. Ashyralyev 1;2 , B. Hicdurmaz 3;4 

1 Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey 2 International 

Turkmen-Turkish University, Ashgabat, Turkmenistan 3 Department of Mathematics, Faculty of 

Sciences, Istanbul Medeniyet University, 34720 Istanbul, Turkey 4 Department of Mathematics, Gebze 


Institute of Technology, Kocaeli, Turkey 

Many di¤erent equations are called by fractional Schrödinger di¤erential equation (FSDE) until today. 

In recent years, the FSDE which is derived from classical Schrödinger di¤erential equation has received 

more attention. This problem is solved by some numerical methods (see [2]-[5]). However, …nite di¤erence 

method which is a useful tool for investigation of fractional di¤erential equations has not been applied 

to a FSDE yet. The present paper …lls a gap by applying …nite di¤erence method to the following 

multi-dimensional linear FSDE 

8 

>< 

@ u(t;x) 

i @t 

mP 

(ar(x)uxr )xr + u(t; x) = f(t; x); 

r=1 

0 < t < 1; x = (x1; ; xm) 2 ; 

u(0; x) = 0; x 2 ; 

>: u(t; x) = 0; x 2 S 

where 0 < < 1. Here ar(x); x 2 and f(t; x) (t 2 [0; 1]; x 2 ) are given smooth functions and 

ar(x) a 0: First and second orders of accuracy di¤erence schemes are constructed for problem (1). 

Numerical experiment on a one-dimensional FSDE shows the e¤ectiveness of the di¤erence schemes. 

References 

[1] Ashyralyev A., A note on fractional derivatives and fractional powers of operators, Journal of 

Mathematical Analysis and Applications, 357(1), 232-236, 2009. 

[2] Ashyralyev A., Hicdurmaz B., A note on the fractional Schrödinger di¤erential equations, Kyber- 

netes, 40(5/6), 736-750, 2011. 

[3] Naber M., Time fractional Schrodinger equation, Journal of Mathematical Physics, 45(8), 18, 2004. 

[4] Odibat Z., Moman S. and Alawneh A., Analytic Study on Time-Fractional Schrödinger Equations: 

Exact Solutions by GDTMS, J. Phys.: Conf. Ser. 96 012066, 2008. 

[5] Rida S. Z., El-Sherbiny H. M. and Arafa A. A. M., On the solution of the fractional nonlinear 

Schrödinger equation, Physics Letters A, 372(5), 553–558, 2008. 

Page 5 

(1)

On Bitsadze-Samarskii type nonlocal boundary value problems for semilinear elliptic 

equations 

A. Ashyralyev 1 , E. Ozturk 2 

1 Department of Mathematics, Fatih University, Istanbul, Turkey 2 Department of Mathematics, Uludag 


University, Bursa, Turkey 

In the literature, the problem of Bitsadze-Samarskii type is often referred to as the boundary value 

problem with Bitsadze-Samarskii condition (see [2], [4] and [7]). Previously, the Bitsadze-Samarskii type 

nonlocal boundary value problems for linear elliptic equations were studied ([5]). In this paper, the 

Bitsadze-Samarskii type nonlocal boundary value problems for semilinear elliptic equations 

8 

>< 

>: 

d 2 u(t) 

dt 2 

+ Au(t) = f(t; u(t)); 0 < t < 1; 

u(0) = '; u(1) = JP 

ju( j) + ; 

j=1 

0 < 1 < < J < 1; JP 

j jj 1 

in a Hilbert space H with the self-adjoint positive de…nite operator A is considered. The …rst and second 

orders of accuracy di¤erence schemes approximately solving these problems are studied. A procedure of 

modi…ed Gauss elimination method is used for solving these di¤erence schemes for the two-dimensional 

elliptic di¤erential equation. The method is illustrated by numerical examples. The converge estimates 

for the solution of these di¤erence schemes are obtained. 

References 

[1] Ashyralyev A. and Yurtsever A.,On a nonlocal boundary value problem for semilinear hyperbolic- 

j=1 

parabolic equations, Nonlinear Analysis, 47, 3585-3592, 2001. 

[2] Skubaczewski A.L., Solvability of ellipic problems wih Bitsadze-Samarskii boundary conditions, 

Di¤erencial’nye Uravneniya, 21, 701-706, 1985. 

[3] Ashyralyev A. and S¬rma A., A note on the numerical solution of the semilinear Schrödinger 

equation, Nonlinear Analysis, 71, e507-e2516, 2009. 

[4] Bitsadze A.V. and Samarskii A.A., On some simple generalizations of linear elliptic boundary 

problems, Soviet Mat. Dokl., 10 (2), 398- 400, 1969. 

[5] Ashyralyev A. and Ozturk E., The numerical solution of Bitsadze-Samarskii nonlocal boundary 

value problems with the dirichlet-Neumann condition, Abstract and Applied Analysis, 730804, 2012. 

[6] Chabrowski J., Multiple solutions for a class of non-local problems for semilinear elliptic equations, 

RIMS. Kyoto Uni., 28, 1-11, 1992. 

[7] Samarskii A.A., Some problems in di¤erential equation theory, Di¤erencial’nye Uravneniya, 16, 

1925-1935, 1980. 

Page 6


A Third-Order of Accuracy Difference Scheme 

for the Bitsadze-Samarskii Type Nonlocal Boundary Value Problem 

A. Ashyralyev 1,2 , F. S. Ozesenli Tetikoglu 1 

1 Department of Mathematics, Fatih University, Istanbul, Turkey 

2 Department of Mathematics, ITTU, Ashgabat, Turkmenistan 

The role played by coercive inequalities in the study of local boundary-value problems for elliptic and 

parabolic differential equations is well-known ([1], [2]). Theory, applications and methods of solutions of 

Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied 

extensively by many researchers ([3]-[5]). The Bitsadze-Samarskii type nonlocal boundary value problem 

⎧ 

⎪⎨ 

− d2 u(t) 

dt 2 

+ Au(t) = f(t), 0 < t < 1, 

ut(0) = ϕ, ut(1) = βut(λ) + ψ, 

⎪⎩ 0 ≤ λ < 1, |β| ≤ 1 

for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A is 

considered. The third order of accuracy difference scheme for the approximate solution of this problem 

is presented. The well-posedness of this difference scheme in difference analogue of Hölder spaces is 

established. In applications, the stability, the almost coercivity and the coercivity estimates for solution 

of difference scheme for elliptic equations are obtained. 

References 

[1] V. L. Gorbachuk, M.L. Gorbachuk, Boundary Value Problems for Dierential - Operator Equations, 

Naukova Dumka, Kiev, 1984 (in Russian). 

[2] G. Berikelashvili, ”On a nonlocal boundary value problem for a two-dimensional elliptic equation”, 

Comput. Methods Appl. Math. 3, no.1, pp. 35-44, 2003. 

[3] A.V. Bitsadze, A. A. Samarskii, ”On some simplest generalizations of linear elliptic problems”, 

Dokl. Akad. Nauk SSSR 185, 1969. 

[4] A. Ashyralyev, ”Nonlocal boundary-value problems for elliptic equations: Well- posedness in 

Bochner spaces”, Conference Proceedings, ICMS International Con- ference on Mathematical Science, 

vol. 1309, pp. 66-85, 2010. 

[5] A. Ashyralyev, ”On well-posedness of the nonlocal boundary value problem for elliptic equations”, 

Numerical Functional Analysis and Optimization, vol. 24, no.1-2, pp. 1-15, 2009. 

Page 7

NBVP for Hyperbolic Equations Involving Multi-point and Integral Conditions 

A. Ashyralyev 1 ;N. Aggez 1 


1 Department of Mathematics, Fatih University, 34500 Istanbul, Turkey 

Nonlocal boundary value problems involving multi-point and integral conditions for a hyperbolic 

equation in a Hilbert space are investigated. The stability estimates for the solution of these 

multi-point NBVP are established. In applications, the stability estimates for the solution of these 

problems are obtained. 

The authors of [3] developed a numerical procedure for the NBVP with a integral conditions for hyper- 

bolic equations. In the paper [4], instead of nonlocal integral conditions multi-point nonlocal conditions 

used. In the present work, we consider the NBVP with multi-point and integral conditions 

8 

>< 

>: 

d 2 u(t) 

dt2 + Au(t) = f(t) (0 t 1); 

1R 

u(0) = ( ) u( )d + 

0 

nP 

aiu( i) + '; 

1R 

ut(0) = 

i=1 

( ) ut( )d + nP 

biut( i) + 

0 

for the di¤erential equation in a Hilbert space H with a self-adjoint positive de…nite operator A. We are 

interested in studying the stability of solutions of problem (1) under the assumption 

> 

Z 

1 + 

Z1 

0 

0 

1 

(s) (s) ds + 

(j (s)j + j (s)j) ds + 

nX 

k=1 

akbk + 

nX 

k=1 

nX 

jak + bkj : 

k=1 

i=1 

ak 

Z1 

0 

(s) ds + 

A function u(t) is a solution of problem (1) if the following conditions are satis…ed: 

nX 

k=1 

bk 

Z1 

0 

(1) 

(s)ds (2) 

i) u(t) is twice continuously di¤erentiable on the interval (0; 1) and continuously di¤erentiable on 

the segment [0; 1]. The derivatives at the endpoints of the segment are understood as the appropriate 

unilateral derivatives. 

[0; 1]. 

ii) The element u(t) belongs to D(A) for all t 2 [0; 1], and function Au(t) is continuous on the segment 

iii) u(t) satis…es the equation and nonlocal boundary conditions (1). 

References 

[1] Ashyralyev A. and Sobolevskii P.E., A note on the di¤erence schemes for hyperbolic equations, 

Abstract and Applied Analysis, 6 (2), 63-70, 2001. 

[2] Fattorini H. O., Second Order Linear Di¤erential Equations in Banach Space,Notas de Matematica. 

North-Holland, 1985. 

[3] Ashyralyev A. and Aggez N., Finite Di¤erence Method for Hyperbolic Equations with the Nonlocal 

Integral Condition, Discrete Dynamics in Nature and Society, 2011, 1-15, 2011. 

[4]Ashyralyev A., Yildirim O., On multipoint nonlocal boundary value problems for hyperbolic di¤er- 

ential and di¤erence equations, Taiwanese Journal of Mathematics, 14, 165-194, 2010. 

Page 8

Boundary Value Problem for a Third Order Partial Di¤erential Equation 

A. Ashyralyev 1 ; N. Aggez 1 ; F. Hezenci 1 

1 Department of Mathematics, Fatih University, 34500 Istanbul, Turkey 

Abstract. Boundary value problems for third order partial di¤erential equations in a Hilbert space 

are investigated. The stability estimates for the solution of the boundary value problem is established. 

To validate the main result, some stability estimates for solutions of the boundary value problems for 

third order equations are given. Here, the boundary value problem 

8 

< 

: 

d 3 u(t) 

dt 3 

Au(t) = f(t); 0 < t < 1; 

u(0) = '; ut(0) = ; utt(1) = ; 

for a third order partial di¤erential equation in a Hilbert space H with a self-adjoint positive de…nite 

operator A is considered. We are interested in studying the stability of solutions of problem (1). 

A function u(t) is a solution of problem (1) if the following conditions are satis…ed: 

i) u(t) is three times continuously di¤erentiable on the interval (0; 1) and continuously di¤erentiable 

on the segment [0; 1]. The derivatives at the endpoints of the segment are understood as the appropriate 

unilateral derivatives. 

[0; 1]. 

ii) The element u(t) belongs to D(A) for all t 2 [0; 1], and function Au(t) is continuous on the segment 

iii) u(t) satis…es the equation and boundary conditions (1). 

References 

[1] A. Ashyralyev and Sobolevskii P.E, Abstr. Appl. Anal., 6(2), 63-70 (2001). 

[2] A. Ashyralyev and Aggez N., Discrete Dyn. Nat. Soc., 2011, 1-15 (2011). 

[3] A. Guezane-Lakoud, N. Hamidane and R. KhaldiInt., Int. J. Math. Math. Sci., 2012, (2012). 

[4] K. Schrader, Proc. Am. Math. Soc., 32(1), 247-252 (2012). 

[5] B. Ahmad, Electron. J. Di¤er. Equ., 2011(94), 1-7 (2011). 

[6] S. Simirnov, Nonlinear Anal., 16(2), 231-241 (2011). 

[7] Yu.P. Apakov and S. Rutkauskas, Nonlinear Analysis, 16(3), 255-269 (2011). 

[8] A. P. Palamides and A. N. Veloni, Electron. J. Di¤er. Equ, 2007(151), 1-13 (2007). 

[9] M. Denche and A. Memou, J. Appl. Math., 2003(11), 553-567(2003). 

Page 9 

(1)

Fractional Parabolic Differential and Difference Equations with the Dirichlet-Neumann 

Condition 

A. Ashyralyev 1 , N. Emirov 1 and Z. Cakir 2 

1 Department of Mathematics, Fatih University, Istanbul, Turkey 2 Department of Mathematical 


Engineering, Gumushane University,Gumushane, Turkey 

The multidimensional fractional parabolic equation with the Dirichlet-Neumann condition is studied. 

Stability estimates for the solution of the initial-boundary value problem for this fractional parabolic 

equation are established. The stable difference schemes for this problem are presented. Stability estimates 

for the solution of the first order of accuracy difference scheme are obtained. A procedure of modified 

Gauss elimination method is applied for the solution of first and second order of accuracy difference 

schemes of one-dimensional fractional parabolic differential equations. 

References [1] I. Podlubny, Fractional differential Equations,vol. 198 of Mahematics in Science and 

Engineering, Academic Press, San Diego, California, USA, 1999. 

[2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and 

Breach, Yverdon, Switzerland, 1993. 

[3] A. A. Kilbas, H. M. Sristava, and J. J. Trujillo, Theory and Applications of Fractional Differential 

Equations, North-Holland Mathematics Studies, 2006. 

[4] J. L. Lavoie, T. J. Osler, and R. Tremblay, SIAM Review 18(2), 240–268 (1976). 

[5] V. E. Tarasov, Internatioanl Journal of Mathematics 18(3), 281–299 (2007). 

[6] M. De la Sen, R. P. Agarwal, A. Ibeas, et al. Advances in Difference Equations, 2011, Article ID 

748608, 32 pages (2011). 

[7] R. P. Agarwal, B. de Andrade, C. Cuevas, Nonlinear Analysis Series B: Real World Applications 

11, pp 3532–3554 (2010). 

[8] R. Gorenflo, and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Frac- 

tional Order, in Fractals and Fractional Calculus in Continuum Mechanics, Edited by A.Carpinteri and 

F.Mainardi, 378 of CISM Courses and Lectures, Springer, Vienna, Austria 1997, pp. 223–276 

[9] A. S. Berdyshev, A. Cabada, E. T. Karimov, Nonlinear Anal.,75(6) 3268–3273 (2011). 

[10] A. Ashyralyev, D. Amanov, Abstract and Applied Analysis , 2012, Article ID 594802, 14 pages 

(2012). 

[11] A. Ashyralyev, B. Hicdurmaz, Kybernetes 40(5-6), 736–750 (2011). 

[12] F. Mainardi, Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics, 

in Fractals and Fractional Calculus in Continuum Mechanics , Edited by A. Carpinteri and F.Mainardi, 

Springer-Verlag, New-York,USA, 1997, pp. 291–348. 

[13] M. Kirane, Y. Laskri, Applied Mathematics and Computation, 167(2), 1304–1310 (2005). 

[14] A. Ashyralyev, F. Dal, and Z. Pınar, Appl. Math. Comput. 217(9), 4654–4664 (2011). 

[15] V. Lakshimikantham, A. Vatsala, Appl.Anal., 11(3-4), 395–402 (2007). 

[16] A. Ashyralyev, Applied Mathematics Letters 24, 1176–1180 (2011). 

Page 10 

[17] A. Ashyralyev, Journal of Mathematical Analysis and Applications357(1), 232–236 (2009).

[18] S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow, 1966(Russian). 

[19] G. Da Prato, P. Grisvard, J. Math. Pures et Appl., 54, 305–387 (1975). 

[20] P. E. Sobolevskii, Dokl. Akad. Nauk, 225(6), 1638–1641 (1975). 

[21] Ph. Clement, On (L p -L a ) coerciveness for a class of integrodifferential equation on the line p 

Preprint 5-4-90, VGU, Voronezh, 1990. 

[22] Z. Cakir Abstract and Applied Analysis , 2012, Article ID 463746, 17 pages (2012). 

Page 11

High Order of Accuracy Stable Di¤erence Schemes for Numerical 

Solutions of NBVP for Hyperbolic Equations 

A. Ashyralyev 1 , O. Yildirim 2 

1 Department of Mathematics, Fatih University, Istanbul, Turkey 2 Department of Mathematics, Yildiz 


Technical University, Istanbul, Turkey 

The abstract nonlocal boundary value problem for the hyperbolic equation 

8 

< u 

: 

00 

(t) + Au (t) = f (t) ; 0 < t < T; 

u(0) = u(1) + '; u 0 (0) = u 0 (1) + 

in a Hilbert space H with the self -adjoint positive de…nite operator A is considered. The third and 

fourth order of accuracy di¤erence schemes for the approximate solutions of this problem are presented. 

The stability estimates for the solutions of these di¤erence schemes are obtained and numerical results 

are presented in order to verify theoretical statements. 

References 

[1] A. Ashyralyev and P. E. Sobolevskii, Two new approaches for construction of the high order of 

accuracy di¤erence schemes for hyperbolic di¤erential equations, Discrete Dynamics Nature and Society, 

vol. 2, no. 1, pp. 183-213, 2005. 

[2] A. Ashyralyev and P. E. Sobolevskii, A note on the di¤erence schemes for hyperbolic equations, 

Abstract and Applied Analysis, vol. 6, no. 2, pp. 63-70, 2001. 

[3] A. Ashyralyev and O. Yildirim, On multipoint nonlocal boundary value problems for hyperbolic 

di¤erential and di¤erence equations,Taiwanese Journal of Mathematics, vol. 14, no.1, pp. 165-194, 2010. 

[4] S. Piskarev and Y. Shaw, On certain operator families related to cosine operator function, Tai- 

wanese Journal of Mathematics, vol. 1, no. 4, pp. 3585-3592, 1997. 

Page 12

Positivity of Two-dimensional Elliptic Differential Operators in Hölder Spaces 


A. Ashyralyev 1 , S. Akturk 1 and Y. Sozen 1 


This paper considers the following operator 

Au(t, x) = −a11(t, x)utt(t, x) − a22(t, x)uxx(t, x) + σu(t, x), 

defined over the region R + × R with the boundary condition u(0, x) = 0, x ∈ R. Here, the coefficients 

aii(t, x), i = 1, 2 are continuously differentiable and satisfy the uniform ellipticity 

a 2 11(t, x) + a 2 22(t, x) ≥ δ > 0, 

and σ > 0. It investigates the structure of the fractional spaces generated by this operator. Moreover, 

the positivity of the operator in Hölder spaces is proved. 

1968. 

1984. 

References 

[1] Krein S.G., Linear Differential Equations in a Banach Space, Amer. Math. Soc., Providence RI, 

[2] Grisvard P., Elliptic Problems in Nonsmooth Domains, Patman Adv. Publ. Program, London, 

[3] Fattorini H.O., Second Order Linear Differential Equations in Banach Spaces, North-Holland: 

Mathematics Studies, 1985. 

[4] Solomyak M.Z., Analytic semigroups generated by elliptic operators in Lp spaces, Dokl. Acad. 

Nauk. SSSR, 127(1) 37–39, 1959 (Russian). 

[5] Solomyak M.Z., Estimation of norm of the reseolvent of elliptic operator in Lp spaces, Usp. Mat. 

Nauk., 15(6) 141–148, 1960 (Russian). 

[6] Krasnosel’skii M.A., Zabreiko P.P., Pustyl’nik E.I. and Sobolevskii P.E., Integral Operators in 

Spaces of Summable Functions, Nauka, Moscow, 1966 (Russian). English transl.: Integral Operators in 

Spaces of Summable Functions, Noordhoff, Leiden, 1976. 

[7] Stewart H.B., Generation of analytic semigroups by strongly elliptic operators under general bound- 

ary conditions, Trans. Amer. Math. Soc. 259 299–310, 1980. 

[8] Ashyralyev A. and Sobolevskii P.E., Well-Posedness of Parabolic Difference Equations, Birkhauser 

Verlag, Basel, Boston, Berlin, 1994. 

[9] Ashyralyev A. and Sobolevskii P.E., New difference schemes for partial differential equations, 

Birkhauser Verlag, Basel, Boston, Berlin, 2004. 

[10] Sobolevskii P.E., The coercive solvability of difference equations, Dokl. Acad. Nauk. SSSR 201(5) 

1063–1066, 1980 (Russian). 

[11] Alibekov Kh.A., Investigations in C and Lp of Difference Schemes of High Order Accuracy for 

Apporoximate Solutions of Multidimensional Parabolic boundary value problems, PhD Thesis, Voronezh 

State University, Voronezh, 1978 (Russian). 

[12] Alibekov Kh.A. and Sobolevskii P.E., Stability of difference schemes for parabolic equations, Dokl. 

Acad. Nauk SSSR 232(4) 737–740, 1977 (Russian). 

Page 13

[13] Alibekov Kh.A. and Sobolevskii P.E., Stability and convergence of difference schemes of a high 

order for parabolic differential equations, Ukrain. Math. Zh. 31(6) 627–634, 1979 (Russian). 

[14] Ashyralyev A. and Sobolevskii P.E., The linear operator interpolation theory and the stability of 

the difference schemes, Dokl. Acad. Nauk SSSR 275(6) 1289–1291, 1984 (Russian). 

[15] Ashyralyev A., Method of Positive Operators of Investigations of the High Order of Accuracy 

Difference Schemes for Parabolic and Elliptic Equations, Doctor of Sciences Thesis, Kiev: Inst. of Math. 

of Acad. Sci. Kiev, 1992 (Russian). 

[16] Simirnitskii Yu.A. and Sobolevskii P.E., Positivity of multidimensional difference operators in the 

C−norm, Usp. Mat. Nauk. 36(4) 202–203, 1981 (Russian). 

[17] Danelich S.I., Fractional Powers of Positive Difference Operators, PhD Thesis, Voronezh: Voronezh 

State University 1989 (Russian). 

[18] Ashyralyev A. and N. Yaz, On Structure of Fractional Spaces Generated by Positive Operators 

with the Nonlocal Boundary Value Conditions, in Proceedings of the Conference Differential and Differ- 

ence Equations and Applications,, Hindawi Publishing Corporation, New York, edited by R.F. Agarwal 

and K. Perera, 91–101, 2006. 

Page 14

On the Numerical Solution of Ultra Parabolic Equations with Neumann Condition 

A. Ashyralyev and S. Yılmaz 


Department of Mathematics, Fatih University, Istanbul, Turkey 

In this paper, our interest is studying the stability of first order difference scheme for the approximate 

solution of the initial boundary value problem for ultra parabolic equations 

⎧ 

∂u(t,s) ∂u(t,s) 

⎪⎨ ∂t + ∂s + Au(t, s) = f(t, s), 0 < t, s < T, 

u(0, s) = ψ(s), 0 ≤ s ≤ T, 

(1) 

⎪⎩ u(t, 0) = ϕ(t), 0 ≤ t ≤ T 

in an arbitrary Banach space E with a strongly positive operator A.We refer to [1, 2] and the references 

therein for a series of papers by the authors, dealing with ultra parabolic equations , arising in diffusion 

theory, probability and finance. Some new results about numerical methods for ultra-parabolic equations 

are also announced, see [3-5]. For approximately solving problem (1), the first-order of accuracy difference 

scheme 

⎧ 

uk,m−uk−1,m 

τ 

+ uk−1,m−uk−1,m−1 

τ 

+Auk,m = fk,m , 

⎪⎨ fk,m = f(tk, sm), tk = kτ, sm = mτ, 1 ≤ k, m ≤ N, Nτ = 1, 

(2) 

u0,m = ψm, ψm = ψ(sm), 0 ≤ m ≤ N, 

⎪⎩ uk,0 = ϕk, ϕk = ϕ(tk), 0 ≤ k ≤ N 

is presented. The stability estimates and almost coercive stability estimates for the solution of difference 

schemes (2) is established. In applications, the stability in maximum norm of difference shemes for mul- 

tidimensional ultra parabolic equations with Neumann condition is established. Applying the difference 

schemes, the numerical methods are proposed for solving one dimensional ultra parabolic equations. 

References 

[1] Ashyralyev A. and Yılmaz S., An approximation of ultra-parabolic equations, Abstract and Applied 

Analysis, Article ID 840621, 13 pages, 2012.(in press) 

[2] Lanconelli E., Pascucci A. and Polidoro S, Linear and nonlinear ultraparabolic equations of Kol- 

mogorov type arising in diffusion theory and in finance, in Proceedings of the International Mathematical 

Series Conference, Nonlinear Problems in Mathematical Physics and Related Topics VOL. II in Honor of 

Professor O.A. Ladyzhenskya, 243-265, 2002. 

[3] Akrivis G., Crouzeix M. and Thom˙ee V., Numerical methods for ultraparabolic equations,Calcolo 

31 , 179-190, 1994. 

[4] Ashyralyev A. and Yılmaz S., Second Order of Accuracy Difference Schemes for Ultra Parabolic 

Equations, in Proceedings of the International Conference on Numerical Analysis and Applied Mathe- 

matics, AIP Conference Proceedings, Volume 1389, 601-604, 2011. 

[5] Ayati B. P., A variable time step method for an age-dependent population model with nonlinear 

diffusion, Sıam J. Numer. Anal., Volume 37(5), 1571-1589, 2000. 

Page 15

Existence and Uniqueness of Solutions for Nonlinear Impulsive Differential Equations with Two-point 


and Integral Boundary Conditions 

A. Ashyralyev 1 and Y.A. Sharifov 2 


2 Baku State University, Institute of Cybernetics of ANAS, Baku, Azerbaijan 

The theory of impulsive differential equations is an important branch of differential equations, which 

has an extensive physical background. Impulsive differential equations arise frequently in the modeling 

many physical systems whose states are subjects to sudden change at certain moments. There has a 

significant development in impulsive theory especially in the area of impulsive differential equations with 

fixed moments; see for instance the monographs [1-4] the references therein. 

In this paper, the suffi cient conditions are established for the existence of solutions for a class of 

two-point and integral boundary value problems for impulsive differential equations. 

References 

[1] M. Benchohra, J. Henderson, S.K. Ntouyas. Impulsive differential equations and inclusions. Hin- 

dawi Publishing Corparation, Vol. 2, New York, 2006. 

[2] D. D. Bainov, P. S. Simeonov. Systems with impulsive effect, Horwood. Chichister, 1989. 

[3] V. Lakshmikantham, D. D. Bainov, P. S. Semeonov, Theory of impulsive differential equations. 

Worlds Scientific, Singapore, 1989. 

1995. 

Page 16 

[4] A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations. Worlds Scientific, Singapore,


Optimal Control Problem for Impulsive Systems with Integral Boundary Conditions 

A. Ashyralyev 1 and Y.A. Sharifov 2 


2 Baku State University, Institute of Cybernetics of ANAS, Baku, Azerbaijan 

Impulsive differential equations have become important in recent years as mathematical models of 

phenomena in both physical and social sciences. There is a significant development in impulsive the- 

ory especially in the area of impulsive differential equations with fixed moments; see for instance the 

monographs [1-4] and the references therein. 

Many of the physical systems can be described better by integral boundary conditions. Integral 

boundary conditions are encountered in various applications such as population dynamics, blood flow 

models, chemical engineering and cellular systems. Moreover, boundary value problems with integral 

conditions constitute a very interesting and important class of problems. They include two, three, multi 

and nonlocal boundary value problems as special cases, (see [5-7]). For boundary value problems with 

nonlocal boundary conditions and comments on their importance, we refer the reader to the papers [8-10] 

and the references therein. 

In this paper the optimal control problem is considered, when the state of the system is described 

by the impulsive differential equations with integral boundary conditions. By the help of the Banach 

contraction principle the existence and uniqueness of solution is proved for the corresponding boundary 

problem by the fixed admissible control. The first and the second variation of the functional is calculated. 

Various necessary conditions of optimality of the first and the second order are obtained by the help of 

the variation of the controls. 

References 

[1] M. Benchohra, J. Henderson, S.K. Ntouyas. Impulsive differential equations and inclusions. Hin- 

dawi Publishing Corparation, Vol. 2, New York, 2006. 

[2] D. D. Bainov, P. S. Simeonov. Systems with impulsive effect, Horwood. Chichister, 1989. 

[3] V. Lakshmikantham, D. D. Bainov, P. S. Semeonov, Theory of impulsive differential equations. 

Worlds Scientific, Singapore, 1989. 

1995. 

[4] A. M. Samoilenko, N. A. Perestuk, Impulsive differential equations. Worlds Scientific, Singapore, 

[5] N.A.Perestyk, V.A. Plotnikov, A.M. Samoilenko, N.V. Skripnik Differential Equations with impulse 

Effect: Multivalued Right-hand Sides with Discontinuities, DeGruyter Studies in Mathematics 40, Walter 

de Gruter Co, Berlin, 2011. 

[6] M. Benchohra, J.J. Nieto, A. Quahab, Second-order boundary value problem with integral bound- 

ary conditions. Boundary Value Problems, vol. 2011, Article ID 260309, 9 pages, 2011. 

[7] B. Ahmad, J. Nieto, Existence results for nonlinear boundary value problems of fractional integro- 

differential equations with integral boundary conditions. Boundary Value Problems, vol. 2009 (2009), 

Article ID 708576, 11 pages. 

[8] A. Bouncherif, Second order boundary value problems with integral boundary conditions, Nonlinear 

Analysis, 70, 1 (2009), pp. 368-379. 

Page 17

[9] R. A. Khan, Existence and approximation of solutions of nonlinear problems with integral boundary 

conditions, Dynamic Systems and Applications, 14, (2005), pp. 281-296. 

[10] A. Belarbi, M. Benchohra, A. Quahab, Multiple positive solutions for nonlinear boundary value 

problems with integral boundary conditions, Archivum Mathematicum, vol. 44, no. 1, pp. 1-7, 2008. 

Page 18

On Stability Of Hyperbolic- Elliptic Differential Equations With Nonlocal Integral 

Condition 

A. Ashyralyev 1,2 , Z. Ödemi¸s Özger 1 and F. Özger1 

1 Department of Mathematics, Fatih University, Istanbul, Turkey 2 Department of Mathematics, ITT 


University 74400, Ashgabat, Turkmenistan 

The nonlocal boundary value problem for a hyperbolic-elliptic equation 

⎧ 

utt(t) + Au(t) = f(t), 0 ≤ t ≤ 1, 

⎪⎨ 

−utt(t) + Au(t) = g(t), − 1 ≤ t ≤ 0, 

1� 

⎪⎩ u(−1) = α(s)u(s)ds + ψ, u(0) = ϕ. 

0 

in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability 

estimates for the solution of this problem are established. 

References 

[1] Ashyralyev A., Judakova G. and Sobolevskii PE., A note on the difference schemes for hyperbolic- 

elliptic equations, Abstr. Appl. Anal., 1486, 1–13, 2006. 

[2] Ashyralyev A. and Sobolevskii P.E., A note on the difference schemes for hyperbolic equations, 

Abstr. Appl. Anal.,16(2), 63–70, 2001. 

[3] Sobolevskii P.E. and Chebotaryeva L.M., Approximate solution by method of lines of the Cauchy 

problem for abstract hyperbolic equations, Izv. Vyssh. Uchebn. Zaved. Mat., 5, 103–116, 1977 (Russian). 

[4] Berdyshev A.S. and Karimov E.T., Some non-local problems for the parabolic-hyperbolic type 

equation with non-characteristic line of changing type, Cent. Eur. J. Math., 4(2), 183–193, 2006. 

[5] Ashyralyev A. and Ozdemir Y., On nonlocal boundary value problems for hyperbolic-parabolic 

equations, Taiwanese Journal of Mathematics, 11(4), 1075–1089, 2007. 

[6] Yamazaki T., Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff 

type with weak dissipation, Mathematical Methods in the Applied Sciences,32(15),1893–1918, 2009. 

[7] Djuraev T.D., ,Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, 

FAN, Tashkent, 1979 (Russian). 

[8] Salakhitdinov M.S., Equations of Mixed-Composite Type, FAN, Tashkent, 1974. 

[9] Bazarov V. and Soltanov H., Some Local and Nonlocal Boundary Value Problems for Equations 

of Mixed and Mixed-Composite Types, Ylym, Ashgabat, 1995 (Russian). 

[10] Vragov V.N., Boundary Value Problems for Nonclassical Equations of Mathematical Physics, 

Textbook for Universities, NGU, Novosibirsk, 1983 (Russian). 

1968. 

[11] Krein S.G., Linear Differential Equations in a Banach Space, Amer. Math. Soc, Providence RI, 

[12] Samarskii A.A. and Nikolaev E.S., Numerical Methods for Grid Equations 2, Iterative Methods, 

Birkhauser, Basel, Switzerland, 1989. 

[13] Ashyralyev A. and Muradov I., On one difference scheme of a second order of accuracy for 

hyperbolic equations, Trudy Instituta Matematiki i Mechaniki Akad.Nauk Turkmenistana Ashgabat, 1, 

58–63, 1995(Russian). 

Page 19

[14] Ashyralyev A. and Aggez N., A note on the difference schemes of the nonlocal boundary value 

problems for hyperbolic equations, Numer. Funct. Anal. Optim., 25, 439–462, 2004. 

[15] Ashyralyev A. and Yildirim O., On multipoint nonlocal boundary value problems for hyperbolic 

differential and difference equations, Taiwanese J. Math., 14,165–194, 2010. 

[16] Ashyralyev A. and Yurtsever H.A., The stability of difference schemes of second-order of accuracy 

for hyperbolic–parabolic equations, Comput. Math. Appl., 52, 259–268, 2006. 

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Page 20


FUZZY CONTINUOUS DYNAMICAL SYSTEM: A 

MULTIVARIATE OPTIMIZATION TECHNIQUE 

Abhirup Bandyopadhyay 1 and Samarjit Kar 1 

1 Department of Mathematics, National Institute of Technology, Durgapur, India 

This paper presents a multivariate optimization technique for the numerical simulation of contin- 

uous dynamical systems whose parameters, functional forms and/or initial conditions are modeled by 

fuzzy distributions. Fuzzy differential equation (FDE) is interpreted by using the strongly generalized 

differentiability concept and is shown that by this concept any FDE can be transformed to a system of 

ordinary differential equations (ODEs). By solving the associate ODEs one can find solutions for FDE. 

This approach has an inherited drawback of increasing uncertainty at each instance of time generally with 

nonlinear functional forms. Here we present a methodology to numerically simulate interval calculus and 

implements a new approach to the numerical integration of fuzzy dynamical systems, where the propa- 

gation of imprecision as a fuzzy distribution in the phase space is solved by a constrained multivariate 

optimization technique. Numerical simulations of some fuzzy dynamical systems (viz. Lotka Volterra 

model, Lorenz model) are also reported. Finally ecological degradation in wetlands of India is modeled 

by fuzzy initial value problem and some sustainable solution is proposed. 

References 

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method, Journal of Computational Methods in applied Mathematics 2, 113-124, 2002. 

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8, 583-590, 2000. 

Page 21


Analysis of Dynamical Complex Network of Ecological 

Stability Diversity and Persistence 

Abhirup Bandyopadhyay 1 and Samarjit Kar 1 

1 Department of Mathematics, National Institute of Technology, Durgapur, India 

Explorations of ecological networks have led a long line of scientists to debate the influence of diversity 

(number of nodes) in terms of species richness and complexity in terms of the number and structure of 

interactions. This research on how vast numbers of interacting species manage to coexist in nature 

reveals a deep disparity between the ubiquity of complex ecosystems in nature and their mathematical 

improbability in theory. In this paper ecological networks are assumed to be complex dynamical network. 

Population dynamics is simulated over ecological complex network and species migration and changing 

food habits are found to be two keystones to species persistence on the earth. Also a comparative study 

on stability, complexity and persistence over complex dynamical network is shown. Here, we show how 

integrating models of food-web structure and nonlinear bioenergetic dynamics bridges this disparity and 

helps elucidate the mechanics of ecological complexity. Structural constraints of these networks including 

the trophic hierarchy, contiguity, and looping formalized by the “niche model” are shown to greatly 

increase persistence in complex model ecosystems. We explore the interplay of structure and nonlinear 

dynamics by systematically varying diversity, complexity, and function in order to “elucidate the devious 

strategies which make for stability in enduring natural systems.” ([19]). Our exploration expands on 

previously proposed strategies and shows how recently discovered structural and functional properties of 

ecological networks appear to promote stability and persistence in large complex ecosystems. 

References 

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Springer-Verlag, Berlin. 

[2] A.R. Solow, and A.R. Beet, (1998). On lumping species in food webs. Ecology. 79, 2013-2018. 

[3] R.J. Williams, and N.D. Martinez, (2000). Simple rules yield complex food webs. Nature. 404, 

180-183. 

[4] P. Yodzis, and S. Innes, (1992). Body-size and consumer-resource dynamics. American Naturalist. 

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webs. Proc R Soc Lond. B. 264, 1249-1254. 

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nature. Nature. 395, 794-798. 

[8] G.F. Fussman, and G. Heber, (2002). Food web complexity and chaotic population dynamics. 

Ecology Letters. 5, 394-401. 

[9] C.S. Holling, (1959b). Some characteristics of simple types of predation and parasitism. Can. 

Entom. 91, 385-399. 

Page 22

[10] D.L. DeAngelis, R.A. Goldstein,and R.V. O’Neill, (1975). A model for trophic interaction. Ecol- 

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[11] G.T. Skalski, and J.F. Gilliam, (2001). Functional responses with predator interference: viable 

alternatives to the Holling type II model. Ecology. 82, 3083- 3092. 

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predation of the European pine sawfly. Can. Entom. 91, 293-320. 

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stability of linked food chains. Ecology. 81, 8-14. 

1-131. 

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Press, Princeton. 

[18] R.J. Williams, and N.D. Martinez, (2004b). Stabilization of chaotic and nonpermanent food web 

dynamics. European Physics Journal B. 

ton. 

Page 23 

[19] R.M. May, (1973). Stability and Complexity in Model Ecosystems. Princeton Univ Press, Prince

Analytical solution for the recovery tests after constant-discharge 

tests in confined aquifers 


ABDON ATANGANA 

Institute for Groundwater Study, Faculty of Natural and Agricultural Sciences, 

University of The Free State, Bloemfontein, 9301, South Africa 

abdonatangana@yahoo.fr 

In this paper we provide a new analytical solution for residual drawdown during the recovery 

period after constant rate pumping test. We first compare the proposed solution with the existing 

solution, secondary we compare the solution with experimental data from field observation. The 

analytical solution is in perfect agreement with the experimental data for than Cooper 

Jacob solution. We derive a new analytical solution for determination of the skin factor without 

any restriction on the variables t and t , . We present an analytical solution for the drawdown 

response in a confined aquifer that is pumped step-wise or intermittently at different discharge 

rate on basis of this solution we derive an analytical solution to analyse the residual drawdown 

data after pumping test with step-wise or intermittently changing discharge rates. 

Keywords: Recovery equations, residual drawdown, skin factor, Variable discharges 

Page 24 

1- G.P. Kruseman and N.A. de Ridder. (1994) Analysis and Evaluation of PumpingTest Data 

2- Theis, C.V. 1935. The relation between the lowering of the piezometric surface and the rate and 

duration of discharge of well using groundwater storage. Trans. Amer. Geophys. Union, Vol. 16, pp. 5 19- 

524. 

3-Jacob, C.E. 1940. On the flow of water in an elastic artesian aquifer. Trans. Amer. Geophys. Union, 

Vol.21, Part 2, pp. 574-586. 

4-Jacob, C.E. 1944. Notes on determining permeability by pumping tests under water table conditions. 

U.S.Geol. Surv. open. file rept. 

5-Jacob, C.E. 1947. Drawdown test to determine effective radius of artesian well. Trans. Amer. Soc. of 

Civil. Engrs., Vol. 112, Paper 2321, pp. 1047-1064. 

6- A. Atangana and E. Alabaraoye (2012) Groundwater flow described by prolate spheroid coordinates 

and new analytical solution for flow model in a confined aquifer under Theis conditions. Paper accepted 

for publication in AMMS 

7-Ramey, H. J. (1982) Well loss function and the skin effect: A review. In: Narasimhan, T.N., (ed) Recent 

trends in hydrogeology, Geol. Sos. Am, Special paper 189, 265-271 

8- De Marsily G. (1986) Quantitative hydrogeology. Academic Press, London, 440 

9- Matthews, C.S and D. G. Russell. (1967) Pressure build up and flow tests in wells. Soc. Petrol. Engrs. Of 

Am. Inst. Min. Met. Engrs., Monograph 1, 67 

10- Birsoy V. K. And Summer W. K, (1980) Determination of aquifer parameters from step tests and 

intermittent pumping data. Groundwater

Bright and dark soliton solutions for the variable 

coe¢ cient generalizations of the KP equation 

Ahmet Bekir a , Özkan Güner a , Adem Cengiz Çevikel b 

a Eskisehir Osmangazi University, Art-Science Faculty, 

Department of Mathematics and Computer Science, 

Eskisehir-TURKEY 

b Yildiz Technical University, Faculty of Education, 

Department of Mathematics Education, 

Istanbul-TURKEY, 

Email : abekir@ogu.edu.tr; ozkanguner@hotmail.com; acevikel@yildiz.edu.tr 

July 12, 2012 


In this paper, by using a solitary wave ansatz in the form of sech p and tanh p 

functions, we obtain the exact bright (non-topological) and dark (topological) 

soliton solutions for the variable coe¢ cient generalizations of the KP (GVCKP) 

equation, respectively. Note that, it is always useful and desirable to construct 

exact analytical solutions especially soliton-type envelope for the understanding 

of most nonlinear physical phenomena. The physical parameters in the soliton 

solutions are obtained as functions of the dependent coe¢ cients. 

Keywords: Solitons, bright and dark soliton, variable-coe¢ cient general- 

izations of the KP (GVCKP) equation 

References 

PACS (2006) : 02.30 Jr, 02.70 Wz, 05.45 Yv, 94.05 Fg. 

[1] M.J., Ablowitz, H., Segur, Solitons and inverse scattering transform, SIAM, 

Philadelphia, (1981). 

[2] W., Mal‡iet, W., Hereman, The tanh method. I: Exact solutions of nonlinear 

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Applied Mathematics and Computation, 154, 3 (2004) 713-723. 

[4] S.A., El-Wakil, M.A., Abdou, New exact travelling wave solutions using modi…ed 

extended tanh-function method, Chaos, Solitons & Fractals, 31, 4, (2007) 840- 

852. 

Corresponding Author. Tel.: +90 222 2393750; Fax: +90 222 2393578. E-mail address: 

abekir@ogu.edu.tr (A.Bekir) 

1 

Page 25

[5] A.M, Wazwaz, The extended tanh method for new soliton solutions for many 

forms of the …fth-order KdV equations, Applied Mathematics and Computation, 

184, 2 (2007) 1002-1014. 

[6] A.M.,Wazwaz, A sine-cosine method for handling nonlinear wave equations, 

Mathematical and Computer Modelling, 40, (2004) 499-508. 

[7] A., Bekir, New solitons and periodic wave solutions for some nonlinear physical 

models by using the sine–cosine method, Physica Scripta, 77, 4 (2008) 501-504. 

[8] E., Fan, H., Zhang A note on the homogeneous balance method, Phys. Lett. A, 

246, (1998) 403-406. 

[9] M.L., Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. 

A, 213, (1996) 279-287. 

[10] Z.S., Feng,The …rst integral method to study the Burgers-KdV equation, J. Phys. 

A: Math. Gen. 35 (2002) 343-349. 

[11] N., Taghizadeh, M., Mirzazadeh, The …rst integral method to some complex 

nonlinear partial di¤erential equations, Journal of Computational and Applied 

Mathematics, 235, 16 (2011) 4871. 

[12] E., Fan, J., Zhang, Applications of the Jacobi elliptic function method to specialtype 

nonlinear equations, Phys. Lett. A, 305, (2002) 383-392. 

[13] S., Liu, Z., Fu, S., Liu, Q., Zhao, Jacobi elliptic function expansion method and 

periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289, (2001) 

69-74. 

[14] M.L., Wang, X., J., Li Zhang, The G0 

G -expansion method and travelling wave 

solutions of nonlinear evolution equations in mathematical physics, Physics Letters 

A, 372, 4 (2008) 417-423. 

[15] A., Bekir, Application of the G0 

G -expansion method for nonlinear evolution 

equations, Physics Letters A, 372, 19 (2008) 3400-3406. 

[16] M.A., Abdou, The extended F-expansion method and its application for a class 

of nonlinear evolution equations, Chaos, Solitons and Fractals, 31 (2007) 95-104. 

[17] J.L., Zhang, M.L., Wang, Y.M., Wang, Z.D., Fang, The improved F-expansion 

method and its applications, Phys. Lett. A, 350, (2006) 103-109. 

[18] Z., Lü, F., Xie, Explicit bi-soliton-like solutions for a generalized KP equation 

with variable coe¢ cients, Mathematical and Computer Modelling 52 (2010) 1423 

1427 

[19] Y., Liang, G., Wei, X., Li, Transformations and multi-solitonic solutions for a 

generalized variable-coe¢ cient Kadomtsev–Petviashvili equation, Computers and 

Mathematics with Applications 61 (2011) 3268–3277 

[20] Z.N., Zhu, Soliton-like solutions of generalized KdV equation with external force 

term, Acta Phys. Sinica 41 (1992) 1561–1566 

[21] J.J., Mao, J.R., Yang, A new method of new exact solutions and solitary wave-like 

solutions for the generalized variable coe¢ cients Kadomtsev–Petviashvili equation, 

Chin. Phys. 15 (2006) 2804–2808. 

2 

Page 26

[22] H., Triki, A.M.,Wazwaz, Bright and dark soliton solutions for a K(m,n) equation 

with t-dependent coe¢ cients. Phys. Lett. A 373, (2009) 2162–2165. 

[23] H., Triki, A.M.,Wazwaz, A one-soliton solution of the ZK(m, n, k) equation with 

generalized evolution and time-dependent coe¢ cients, Nonlinear Analysis: Real 

World Applications 12 (2011) 2822–2825 

[24] H., Triki, M.S., Ismail, Soliton solutions of a BBM(m, n) equation with generalized 

evolution, Applied Mathematics and Computation, 217, 1 (2010) 48-54. 

[25] H., Triki, A.M.,Wazwaz, Dark solitons for a combined potential KdV and 

Schwarzian KdV equations with t-dependent coe¢ cients and forcing term, Applied 

Mathematics and Computation, 217 (2011) 8846–8851. 

[26] A., Biswas, 1-soliton solution of the K(m, n) equation with generalized evolution, 

Phys. Lett. A 372 (2008) 4601-4602. 

[27] A., Biswas, H., Triki, M., Labidi, Bright and Dark Solitons of the Rosenau- 

Kawahara Equation with Power Law Nonlinearity, Physics of Wave Phenomena,19, 

1 (2011) 24–29. 

3 

Page 27

A Characterization of Compactness in Banach Spaces with Continuous Linear 


Representations of the Rotation Group of a Circle. 

Abdullah Çavu¸s and Mehmet Kunt 

cavus@ktu.edu.tr, mkunt@ktu.edu.tr 

Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey 

Let H be a complex Banach space, T be the unit circle {z ∈ C : |z| = 1}, SO(2) be the group of all 

rotations of T, GL(H) be group of all invertible bounded linear operators on H, α : SO(2) → GL(H) be 

a continuous linear representation, x ∈ H. For all n ∈ Z, n-th Fourier coefficient of x with respect to the 

α is defined by 

Pn(x) = 1 

� 

e 

2π T 

−int α(t)(x)dt 

and the Fourier series of x with respect to the α is defined by 

+∞� 

n=−∞ 

Pn(x). (1) 

The convergence of this series and some properties of Pn(x) are investigated in [5]. In this work, a 

characterization of compactness in Banach space H is given by means of Fourier coefficients Pn(x). One 

of the main results is as follows: 

Theorem :Suppose that dimHn < +∞ for all n ∈ Z. Then a closed subset A ⊂ H is compact if and 

only if for any ε > 0 there exists a natural number N(ε) such that� n 

n+1 σn(x) − x� < ε for all x ∈ A and 

n ≥ N(ε). 

Where, for all n ∈ N ∪ {0}, σn(.) : H → H is a linear bounded operator which is defined by 

σn(x) = 1 

n + 1 

n� 

Sk(x) 

for all x ∈ H, Sk(x) is the k-th partial sum of (1) for all k ∈ N ∪ {0} and 

for all n ∈ Z. 

References 

k=0 

Hn := {x ∈ H : α(t)(x) = e int x, ∀t ∈ T} 

[1] Edwards R. E., Fourier Series : A Modern Introduction, Springer-Verlag, Berlin/Heydelberg/New 

York, 1982. 

[2] Kislyakov S. V., Classical themes of Fourier analysis, Commutative harmonic analysis I, General 

survey, Classical aspects, Encycl. Math. Sci., 15, 113-165 1991. 

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idence, R. I. American Mathematical Society, (AMS), 2001. 

[4] Khadjiev Dj., Çavu¸s A., The imbedding theorem for continuous linear representation of the rotation 

group of a circle in Banach spaces, Dokl. Acad. Nauk of Uzbekistan, N 7, 8-11, 2000. 

Page 28

[5] Khadjiev Dj., Çavu¸s A., Fourier series in Banach spaces, Inverse and Ill-Posed Problems Series, 

Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis, Proceedings of the Inter- 

national Conference, Samarcand, Uzbekistan, Editor-in-Chief : M. M. Lavrent’ev, VSP, Utrecht-Boston, 

71-80, 2003. 

[6] Khadjiev Dj., The widest continuous integral, J. Math. Anal. Appl. 326, 1101-1115, 2007. 

Acknowledgement. This work was supported by the Commission of Scientific Research Projects of Karadeniz Tech- 

nical University, Project number: 2010.111.3.1. 

Page 29

The approximate solutions of linear Goursat Problems via Homotopy Analysis Method 

Aytekin Ery¬lmaz 1 , Musa Ba¸sbük 2 ; HüseyinTuna 3 


1;2 Department of Mathematics, Nevsehir University, Nevsehir Turkey 

3 Department of Mathematics, Mehmet Akif University, Burdur, Turkey 

In this study we investigate the linear Goursat problems that arise in linear partial di¤erential equa- 

tions with mixed derivatives. The standart form of Goursat Problem is given by 

uxt = f(x; t; u; ux; ut); 0 6 x 6 a; 0 6 t 6 b; 

u (x; 0) = g(x); u(0; t) = h(t); 

u(0; 0) = g(0) = h(0): 

The aim of this work is to present an e¢ cient numerical procedure, namely Homotopy Analysis Method, 

for solving homogeneous and inhomogeneous linear Goursat problems. The reliability and e¢ ciency of 

the proposed method are demonstrated by some numerical examples and performed on the computer 

algebraic system Mathematica 7. 

References 

[1] Wazwaz, A., The variational iteration method for a reliable treatment of the linear and the non- 

linear Goursat problem, Applied Mathematics and Computation, 193 (2007), 455–462. 

[2] Wazwaz, A., Partial Di¤erential Equations and Solitary Waves Theory, Higher Education Press, 

Beijing and Springer-Verlag Berlin Heidelberg, 2009. 

[3] Liao, SJ., Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, 

Boca Raton, Chapman and Hall, 2003. 

[4] Y¬ld¬r¬m, A., Odaba¸s¬, M., The homotopy perturbation method for solving the linear and the 

nonlinear Goursat problems, International Journal For Numerical Methods In Biomedical Engineering, 

27 (2011), 1139–1148. 

Page 30

Paths of Minimal Length on Suborbital Graphs with Recurrence Relations 

A.H. Deger 1 , M. Besenk 1 and B.O. Guler 1 


1 Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey 

In this paper, we study suborbital graphs for congruence subgroup Γ0 (N) of the modular group Γ to 

have vertices of the graph Fu,N and hyperbolic paths of minimal length with recurrence relations give 

rise to a special continued fraction. 

References 

[1] Jones G.A., Singerman D. and Wicks K., The Modular Group and Generalized Farey Graphs, 

London Math. Soc. Lecture Note Ser., 316-338, 1991. 

[2] Deger A.H., Besenk M. and Guler B.O., On Suborbital Graphs and Related Continued Fractions, 

Appl. Math. and Comp., 746-750, 2011. 

[3] Sims C.C., Graphs and Finite Permutation Groups, Math. Zeitschr., 76-86, 1967. 

[4] Akbas M., On Suborbital Graphs for the Modular Group, Bull. Lond. Math. Soc., 647-652, 2001. 

[5] Neumann P.M., Finite Permutation Groups, Edge-Coloured Graphs and Matrices, Topics in Group 

Theory and Computation, Academic Press, New York, 1977. 

[6] Tsukuzu T., Finite Groups and Finite Geometries, Cambridge University Press, Cambridge, 1982. 

[7] Biggs N.L. and White A.T., Permutation Groups and Combinatorial Structures, London Math. 

Soc. Lecture Note Ser., Cambridge, 33. CUP, Cambridge, 1982. 

[8] Cuyt A., Petersen V.B., Verdonk B., Waadeland H. and Jones W.B., Handbook of Continued 

Fractions for Special Functions, Springer Science + Business Media B.V., 2008. 

Page 31

Riesz Basis Property of Eigenfunctions of One Boundary-Value Transmission Problem 


A. Hayati OL ¯GAR 1 and O. Sh. MUKHTAROV 2 

1 Department of Mathematics, Gaziosmanpasa University, Tokat, Turkey 

2 Department of Mathematics, Gaziosmanpasa University, Tokat, Turkey 

We consider a Sturm-Liouville equation together with eigendependent boundary conditions and two 

supplementary transmission condition at the one inner point. Note that some special cases of the con- 

sidered problem arise after an application of the method of separation of variables to the heat transfer 

problems, in vibrating string problems when the string is loaded additionally with point masses, in dif- 

fraction problems etc. We introduce a new inner product in the Sobolev Spaces W 1 2 (a, b) and show that 

eigenfunctions of our problem form a Riesz basis of this modified space. 

References 

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Translations of Mathematical Monographs, vol.18, American Mathematical Society, Providence, Rhode 

Island, 1969. 

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1989. 

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I, second edn. Oxford Univ. press, London (1962). 

[6] Walter, J. , Regular eigenvalue problems with eigenvalue parameter in the boundary condition. 

Math. Z., 133:301-312, 1973. 

Page 32

On A SUBCLASS OF UNIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS 

ABDUL RAHMAN S. JUMA 1 and HAZHA ZIRAR 2 

1 Department of Mathematics, Alanbar University, Ramadi, Iraq 2 Department of Mathematics , 


University of Salahaddin, Erbil, Kurdistan, Iraq. 

In this paper, we have introduced the subclass of univalent functions defined in the open unit disc 

and derived some interesting properties like coefficient estimates, distortion theorem, extreme points and 

radii of close- to- convexity , starlikness and convexity. 

References 

[1] Aouf, M. K. and Salagean ,G. S. , Generalization of certain subclass of convex functions and 

corresponding subclass of starlike functions with negative coeffictients, Mathematica, 50(73), (2008), 

119-138. 

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S. Math. Anal. Appl. 38(1972), 746- 765. 

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Appl. Math. Soc. 45(2000), 647- 657. 

[4] Ronning,F. , On starlike functions associated with parabolic region, Ann. Univ Mariae Curie 

Sklodowska Sect. Aus (1991), 117- 122. 

Page 33

Fine spectra of upper triangular triple-band matrices over the sequence space ℓp, 

(0 

Ali KARASA 

Department of Mathematics, Necmettin Erbakan, Konya, Turkey 

Abstract The operator A(r, s, t) on sequence space on ℓp is defined A(r, s, t)x = (rxk−1 + sxk + 

txk+1) ∞ k=0 where x = (xk) ∈ ℓp, with (0 in purpose of this paper is to determine the 

fine spectrum with respect to the Goldberg’s classification of the operator A(r, s, t) defined by a triple 

sequential band matrix over the sequence space ℓp. Additionally, we give the approximate point spectrum, 

defect spectrum and compression spectrum of the matrix operator A(r, s, t) over the space ℓp. 

References 

[1] A.M. Akhmedov, F. Ba¸sar, On the fine spectrum of the Cesàro operator in c0, Math. J. Ibaraki Univ. 

36(2004), 25–32. 

[2] F. Ba¸sar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, 

Ukrainian Math. J. 55(1)(2003), 136–147. 

[3] H. Bilgiç, H. Furkan, On the fine spectrum of the operator B(r, s, t) over the sequence spaces ℓ1 and 

bv, Math. Comput. Modelling 45(2007), 883–891. 

H. Furkan, H. Bilgiç, F. Ba¸sar, On the fine spectrum of the operator B(r, s, t) over the sequence 

spaces ℓp and bvp, (1 

[4] S. Goldberg, Unbounded Linear Operators, Dover Publications, Inc. New York, 1985. 

[5] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc. New York 

· Chichester · Brisbane · Toronto, 1978. 

[6] J.I. Okutoyi, On the spectrum of C1 as an operator on bv, Commun. Fac. Sci. Univ. Ank. Ser. A1. 

41(1992), 197–207. 

[7] J.B. Reade, On the spectrum of the Cesaro operator, Bull. Lond. Math. Soc. 17(1985), 263–267. 

[8] B.E. Rhoades, The fine spectra for weighted mean operators, Pacific J. 

[9] P.D. Srivastava, S. Kumar, Fine spectrum of the generalized difference operator ∆uv on sequence 

space l1, Appl. Math. Comput. in press. 

[10] R.B. Wenger, 

[11] M. Yıldırım, On the fine spectrum of the Rhaly operators on ℓp, East Asian Math. J. 20(2004), 

153–160. 

Page 34

Approximation by Certain Linear Positive Operators of Two Variables 

A.K. Gazanfer 1 and ·I. Büyükyaz¬c¬ 2 

1 Department of Mathematics, Bülent Ecevit University, Zonguldak, Turkey 2 Department of 


Mathematics, Ankara University, Ankara, Turkey 

In this study, we introduce positive linear positive operators which are combined the Chlodowsky 

and Szász type operators and study some approximation properties of these operators in the space of 

continuous functions of two variables on a compact set. The rate of convergence of this operators are 

obtained by means of the modulus of continuity. And we also obtain weighted approximation properties 

for these positive linear operators in a weighted space of functions of two variables and …nd the rate of 

the convergence for this operators by using weighted modulus of continuity. 

References 

[1] A.D. Gadjiev, R.O. Efendiev and E. Ibikli, Generalized Bernstein-Chlodowsky polynomials, Rocky 

Mountain J. Math. 28 (1998). 

[2] A.D. Gadjiev, Linear positive operators in weighted space of functions of several variables, Izvestiya 

Acad. of Sciences of Azerbaijan, N4, 1980. 

[3] A.D. Gadjiev, H. Hac¬saliho¼glu, Convergence of the sequences of linear positive operators., Ankara, 

1995 (in Turkish). 

[4] E.A. Gadjieva and E. Ibikli, On generalization of Bernstein-Chlodowsky polynomials, Hacettepe 

Bull. Natur. Sci. Engrg. 24 (1995), 31 40. 

[5] N.·Ispir, Ç.Atakut, Approximation by modi…ed Szász–Mirakjan operators on weighted spaces, Proc. 

Indian Acad. Sci. (Math. Sci.), 12(4) (2002) 571–578. 

[6] F. Ta¸sdelen, A. Olgun, G. B. Tunca, Approximation of functions of two variables by certain linear 

positive operators, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 117, No. 3, August 2007, pp. 387–399. 

[7] V. I. Volkov , On the convergence of sequences of linear positive operators in the space of continuous 

functions of two variable, Math. Sb. N. S 43(85) (1957) 504 (Russian). 

[8] Z. Walczak, On certain modi…ed Szász–Mirakjan operators for functions of two variable, Demon- 

stratio Math. 33(1) (2000) 92–100. 

Page 35

Impulsive differential equations with variable 

times 

A.Lakmeche (1) and F.Berrabah (2) 

(1) Djillali Liabes University, Sidi Bel Abbes, Algeria, lakmeche@yahoo.fr 

(2) Djillali Liabes University, Sidi Bel Abbes, Algeria, berrabah f@yahoo.fr 


In this paper, Schauder-Tychonoff’s fixed point theorem and the notion of 

upper and lower solutions are used to investigate the existence of solutions for 

first order impulsive equations. 

Keywords:Impulsive equations; upper and lower solutions; fixed point. 

References 

[1] J. Dugundji and A, Granas, Fixed point Theory, Springer-Verlag, New York, 2003 

[2] V. Lakshmikantham, D.D. Bainov and P.S.Simeonov, Theory of impulsive Differential 

Equations, World Scientific, Singapore, 1989 

Page 36

PARABOLIC PROBLEMS WITH PARAMETER OCCURING IN 

ENVIRONMENTAL ENGINEERING 

AIDA SAHMUROVA 

Okan University, Department of Adminstration of Health, Ak…rat, Tuzla 

34959 Istanbul, Turkey, E-mail: aida.sahmurova@okan.edu.tr 

VELI B. SHAKHMUROV 

Okan University, Department of Mechanical Engineering, Ak…rat, Tuzla 34959 

Istanbul, Turkey, E-mail: veli.sahmurov@okan.edu.tr 


In this work, the uniform well possedenes of singular perturbation problems 

for parameter dependent parabolic di¤erential-operator equations are obtained. 

These problems occur in phytoremediation modelling. 

Key Word: Singular perturbation, Initial value problems; Di¤erentialoperator 

equations; Abstract parabolic equation; Interpolation of Banach spaces; 

Semigroups of operators; phytoremidation modelling 

AMS: 34G10, 35J25, 35J70 

1. Introduction 

Remediation techniques have been based on either immobilization, extraction 

by physick-chemical methods, landholding, or burial. These method often 

have some shortcoming: requiring special equipment, expensive, can remove 

biological activity from the soil, and can important a¤ect the soil physical properties. 

The model describing in this projet is developed in three parts. First, the 

dynamic portion will be developed using a a reaction-di¤usion systems. Next, 

the cost function will involve the dynamic state variables and …nally the desired 

EPA target will be de…ned as mathematical property. Assume u1 (t; x) ; 

u2 (t; x) ; u3 (t; x) are amount of heavy metal in the environment in the roots and 

in the shoots at t months on x = (x1; x2; x3) place, respectively. Since the plant 

toxicant interaction dynamic occurs during a harvest season, we need to describe 

the process one harvest cycle. The initial amount of metal in di¤erent harvest 

cycle depends on what is remaining in the soil at the end of the cycle. The 

mathematical description of this process can be obtained as the following initial 

value problem (IVP) for systems of delay parabolic equation with parameter 

s @ui 

@t + 

3X 

bj (s; t) uj (t j; x) = fi (t; x) , 0 < t T; 

j=1 

ui (t; x) = gi (t; x) ; j t 0; x 2 [a; b] ; i; j = 1; 2; 3: 

1 

Page 37

ON DARBOUX HELICES IN MINKOWSKI SPACE R 3 1 

A. ¸Senol 1 , E. Z¬plar 2 and Y. Yayl¬ 2 

1 Department of Mathematics, Çank¬r¬Karatekin University, Çank¬r¬, Turkey 2 Department of 


Mathematics, Ankara University,Ankara, Turkey 

In the present study, we give the conditions for a curve in the Minkowski space to be a Darboux 

helix. We show that is a Darboux helix if there exists a …xed direction d in R 3 1 such that the function 

hW (s); di is constant. We give the relation between slant helice and Darboux helice. As a particular case, 

if we take kwk =constant, the curves are constant precession. Some more particular cases of constant 

precession curves are studied. 

References 

[1] Ahmad T. Ali and Lopez R., Slant Helices in Minkowski space R 3 1, J. Korean Math. Soc., 48 , No. 

1, 159-167, 2011. 

[2] M. do Carmo, Di¤erential Geometry of Curves and Surfaces, Prentice Hall, 1976. 

[3] Izumiya, S and Tkeuchi, N., New special curves and developable surfaces, Turk J. Math., 28, 

153-163, 2004. 

2008. 

[4] Sco…eld, P.D. Curves of constant precession. Am. Math. Montly 102 (1995), 531-537 

[5]W. Kuhnel, Di¤erential Geometry: Curves, Surfaces, Manifolds, Weisbaden: Braunschweing, 1999. 

[6]R. Lopez, Di¤erential geometry of curves and surfaces in Lorentz-Minkowski space,arXiv:0810.3351v1, 

[7]Kula, L and Yayl¬Y. 2005 On slant helix and its spherical indicatrix. Applied Mathematics and 

computation 169, 600-607. 

[8]J. Walrave, Curves and surfaces in Minkowski space, Doctoral Thesis, K.U. Leuven, Fac. Sci., 

Leuven, 1995. 

[9]Z¬plar E, Senol A,Yayl¬Y., On Darboux Helices in Euclidean 3-space, submitted. 

[10]Yayl¬Y, Hac¬saliho¼glu H.H, Closed curves in the minkowski 3-spaceHadronic journal 23, 259-272 

(2000). 

Page 38

On the Numerical Solution of a Diffusion Equation Arising in Two-phase Fluid Flow 

A.S. Erdogan and A.U. Sazaklioglu 



Many applied problems in fluid mechanics and mathematical biology were formulated as the math- 

ematical model of partial differential equations. Fluid flow inside capillaries were also considered with 

mathematical models [1]-[3]. But it is known that due to the lack of some data and/or coeffi cients, many 

real-life problems are modeled as inverse problems [4]-[5]. In this paper, specific modeling of the fluid 

flow for an unknown pressure is modeled as a two-phase flow equation. The unknown pressure acting in 

the model can be identified by using the overdetermined condition. Difference schemes are constructed 

for obtaining approximate solutions of this inverse problem. Stability estimates for the solution of these 

difference schemes are established. 

References 

[1] F. Loth, P.F. Fischer, N. Arslan, C.D. Bertram, S.E. Lee, T.J. Royston, W.E. Shaalan and H.S. 

Bassiouny, Transitional flow at the venous anastomosis of an arteriovenous graft: Potential activation of 

the ERK1/2 mechanotransduction pathway, Journal of Biomechanical Engineering , 125, 49-61, 2003. 

[2] N. Arslan, A.S. Erdogan, A. Ashyralyev, Computational fluid flow solution over endothelial cells 

inside the capillary vascular system, International Journal for Numerical Methods in Engineering, 74(11), 

1679-1689, 2008. 

[3] A.S.Erdogan, A note on the right-hand side identification problem arising in biofluid mechanics, 

Abstract and Applied Analysis, 2012. 

[4] V.T. Borukhov, P.N. Vabishchevich, Numerical solution of the inverse problem of reconstructing 

a distributed right-hand side of a parabolic equation, Computer Physics Communications, 126, 32-36, 

2000. 

[5] A. Ashyralyev, A.S. Erdogan, On the numerical solution of a parabolic inverse problem with the 

Dirichlet condition, International Journal of Mathematics and Computation, 11(J11), 73-81, 2011. 

Page 39

On the Solution of a Three Dimensional Convection Diffusion Problem 

Abdullah Said Erdogan ve Mustafa Alp 

Department of Mathematics, Fatih University, 34500, Buyukcekmece, 

Istanbul, Turkey 

Department of Mathematics, Faculty of Arts and Sciences, Duzce University 

81620, Duzce, Turkey, 


In this paper, the Rothe difference scheme and the Adomian Decomposition 

method are presented for obtaining the approximate solution of 

three dimensional convection-diffusion problem. Stability estimates for 

the difference problem is presented. 

Keywords: Finite difference method, Adomian Decomposition Method, 

Convection-diffusion equation 

1 Introduction 

In many important applications in engineering such as transport of air and 

water pollutants, convection-diffusion problems arises. An example of this kind 

of problem is a forced heat transfer. Several numerical methods are proposed 

for solving three dimensional convection diffusion problem (see [1]-[11] and the 

references therein). In this paper, we focus on the following mixed problem for 

the three dimensional convection-diffusion equation 

⎧ ∂u 

∂t ⎪⎨ 

⎪⎩ 

+ b1 (x, y, z) ∂u 

∂x + b2 (x, y, z) ∂u 

∂y + b3 (x, y, z) ∂u 

� 

∂z 

− a1 ∂2u ∂x2 + a2 ∂2u ∂y2 + a3 ∂2u ∂z2 � 

= f (t, x, y, z) , in Ω × P, 

(1) 

u (x, y, z, t) = 0, on ∂Ω × P , 

u (x, y, z, 0) = g (x, y, z) , in Ω, 

where Ω = (0, 1)×(0, 1)×(0, 1) , P = (0, T ), b1(x, y, z), b2(x, y, z), b3(x, y, z), g (x, y, z) 

are suffi ciently smooth functions and a1, a2,a3 are positive constants. Here, 

b1 (x, y, z) , b2 (x, y, z), b3 (x, y, z) , a1, a2 and a3 are velocity components of the 

fluid in the directions of the axes at the point (x, y, z) at time t. 

References 

[1] M. M. Gupta, and J. Zhang, Applied Mathematics and Computation 113, 

249-274 (2000). 

[2] V. John, and E. Schmeyer, Comput. Methods Appl. Mech. Engrg 198, 475— 

494 (2008). 

1 

Page 40

[3] Y. Ma, and Y. Ge, Applied Mathematics and Computation 215, 3408— 

3417(2010) 

[4] J. Zhang, L. Gea, and J. Kouatchou, Mathematics and Computers in Simulation 

54, 65—80(2000) 

[5] P. Theeraek, S. Phongthanapanich, and P. DechaumphaiMathematics and 

Computers in Simulation 82, 220—233(2011) 

[6] Xue-Hong Wu, Zhi-Juan Chang, Yan-Li Lu, Wen-Quan Tao, and Sheng- 

Ping Shen Engineering Analysis with Boundary Elements 36, 1040—1048 

(2012). 

[7] Hans-Görg Roos, and H. Zarin, Journal of Computational and Applied 

Mathematics 150, 109—128 (2003). 

[8] K.J. in’t Hout, and B.D. Welfert, Applied Numerical Mathematics 57, 19— 

35(2007). 

[9] F.S.V. Bazan, Applied Mathematics and Computation200, 537—546 (2008). 

[10] M. Dehghan, Mathematical Problems in Engineering 2005(1), 61—74 

(2005). 

[11] Y.Tanaka, T.Honma and I. Kaji, Appl. Math. Modelling 10, 170-175 (1986). 

[12] P.E. Sobolevski, Dokl. Akad. Nauk. SSSR 201(5), 1063-1066 (1971). 

[13] Kh. A. Alibekov, and P. E. Sobolevskii, Ukrain. Math. Zh. 31(6), 627—634 

(1979). 

[14] A. Ashyralyev, and P.E. Sobolevskii, Well-Posedness of Parabolic Difference 

Equations, Basel, Boston, Berlin: Birkhäuser Verlag, 1994. 

[15] http://www.fatih.edu.tr/~aserdogan/AA/cdp.m 

[16] S. Momani, Turk J Math 32, 51-60 (2008). 

[17] G. Adomian, Solving Frontier problems of Physics: The decomposition 

method, Kluwer Academic Publishers, 1994. 

2 

Page 41

A Fuzzy Max-Min Approach to Multi Objective, Multi Echelon Closed Loop Supply Chain 

B. Ahlatcioglu Ozkok 1 , E. Budak 1 and S. Ercan 2 

1 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey 2 Department of 


Mathematics, Firat University, Elazig, Turkey 

In today’s competitive markets, optimizing the process of delivering products from suppliers of raw 

materials to the customers for the firms formalizes an important problem in the literature. Increasingly 

contaminated world and limited sources of energy in recent years are regarded, it is inevitable for the 

mathematical models of any supply chain to have an environmentalist perspective. Hence, closed loop 

supply chain method has an increasing importance. In this study, a multi-objective linear model is given 

for the multi-echelon closed loop supply chain and the solution is obtained by utilizing Zimmermann’s 

”min” operator with a fuzzy approach in which the minimum satisfactions of objectives are maximized. 

The model is to determine the locations of facilities and distribution quantity on the network regarding 

three objective functions, which are; minimizing time and cost, maximizing rating. 

References 

[1] Wei, J., Zhao, J., Pricing decisions with retail competition in a fuzzy closed-loop supply chain, 

Expert Systems with Applications, Vol. 38, pp. 11209-11216, 2011. 

[2] Pishvaee. M.S., Razmi, J., Environmental supply chain network design using multi-objective fuzzy 

mathematical programming, In Press, Corrected Proof, Available online 20 October 2011. 

[3] Zimmerman, H.J., Fuzzy Set Theory and its applications, Kluwer Academic Publishers, Boston/ 

Dordrecht/ London, 1992. 

Page 42

A Fuzzy Approach to Multi Objective Multi Echelon Supply Chain 

B. Ahlatcioglu Ozkok 1 , S. Ercan 2 and E. Budak 1 

1 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey 2 Department of 


Mathematics, Firat University, Elazig, Turkey 

Recent times, companies are being forced by hard marketing conditions to make significant and 

strategic decisions on their supply chains. In this context, companies are trying to optimize supply 

chains towards customer demands and trying to prevent costs that caused by number of inactive facilities. 

In this study, by using AHP we make decision about potential establishment of a number of potential 

warehouses and distributions centers at regions to be selected from a set of possible candidates with 

certain possibilities of customer demands in the supply chain network of a company that is importing 

and exporting cleaning materials. The proposed model attempts to simultaneously minimize total cost 

and maximizing rating candidate locations using mixed integer linear programming. To obtain solution 

fuzzy decision making method is used and numerical example is illustrated. 

References 

[1] Tsiakis, P., Shah, N., Pantelides, C.C., Design of multi-echelon supply chain network under demand 

uncertainty, Ind. Eng. Chem. Res., 40, 3585-3604, 2001. 

[2] Amid, A., Ghodsypour, S.H., O’Brien, C., Fuzzy multi objective linear model for supplier selection 

in a supply chain, Int. J. Production Economics, 104, 394-407, 2006. 

[3] Zimmerman, H.J., Fuzzy Set Theory and its applications, Kluwer Academic Publishers, Boston/ 

Dordrecht/London, 1992. 

Page 43 

[4] Kokangul, A., Susuz, Z., Integrated analytical hierarch process and mathematical programming to 

supplier selection problem with quantity discount, Applied Mathematical Modeling, 33, 1417-1429, 2009.

Modeling Voting Behavior in the Eurovision Song Contest 

B. Dogru 1 

1 Department of Economics, Gumushane University, Gumushane, Turkey 


Modeling voting behavior or determinants of voting in a popular music competition such as 

Queen Elizabeth Piano Contest and Eurovision Song Contest have been growing tremendously after 

2000s. (See [1], [4], [5], [9]). The aim of this study is also to model , voting behavior of juries 

and public opinion (via televoting system) of country in evaluating the singer of country 

( where is the total number of participants in the Eurovision Song Contest (ESC). We 

modeled voting behavior taking into consideration the individual characteristic of performer and 

voter, as well as quality of song. Characteristic properties of performer and 

characteristics of voter together affect votes given to a performer, as well as 

exchange of votes between two countries. Voting equation can be improved with these factors as 

below: 

Where are parameters to be estimated. The last two parameters of right-hand 

side of the equation (1) are affinity and objective quality of song. These two parameters together 

indicate some individual characteristics of singer and voter such as gender (male, female and duet), 

the “language” in which song is performed (English, English +national language, French, National 

language), the order of “appearance” in the contest, whether the song is performed “alone” or in a 

“group”, a dummy for “host” country ( if singer represents the host country, the variable takes 1 and 

0 for other), and a dummy variable to capture “cultural block” ties’ effect on voting (Western, 

Scandinavia, Former Yugoslavia, Former Socialist and Independents). Geographic effect 

and quality of a song are computed as below 

Page 44 

Estimation result of the linear voting equation 4 (including neighborhood and quality 

variable) shows that not only quality of the song is an important part of voting but also affinity 

variables are very crucial determinants of voting equation. Estimation result also indicates that order 

of appearance in the contest, the language of the song and the gender of the performing artist turn 

out to be quite important parameters in explaining voting behavior. 

References 

[1] Dekker, A. (2007). The Eurovision Song Contest as a ‘Friendship’ Network. 

Connectıons, 27(3), 53-58. 

[4] Ginsburgh, V., & Noury, A. (2004). Cultural Voting The Eurovision Song 

Contest. Discussion Papers: http://www. core. ucl. 

[5] Haan, M., Dijkstra, S. G., & Dijkstra, P. T. (2005). Expert Judgment Versus 

Public Opinion – Evidence from the Eurovision Song Contest. Journal of Cultural 

Economics(29), 59-78. 

[9] Yair, G., & Maman, D. (1996). The Persisitent Structure of Hegomony in the 

Eurovision Song Contest. Acta Sociologica, 39, 309-325.

Derivation and numerical study of relativistic Burgers 

equations posed on Schwarzschild spacetime 

Baver Okutmustur 

Middle East Technical University (METU) 

baver@metu.edu.tr 

We consider nonlinear hyperbolic balance laws posed on a curved spacetime 

endowed with a volume form and identify a unique (up to normalization) hyperbolic 

balance law that enjoys the Lorentz invariance property also shared 

by the Euler equations of relativistic compressible fluids. The proposed model 

can be viewed as a relativistic version of Burgers equation and provides us with 

a simplified model on which numerical methods for hyperbolic equations can 

be developed and analyzed. This model is also compared with a second model 

derived directly from the relativistic Euler equations. We then introduce a finite 

volume scheme for the approximation of discontinuous solutions to the Burgerstype 

model when the background is chosen to be (a subset of) the Schwarzschild 

spacetime. Our scheme is formulated geometrically and is consistent with the 

natural divergence form of the balance law and applies to weak solutions containing 

shock waves. Most importantly, our scheme is well-balanced in the sense 

that it preserves static equilibrium solutions. Numerical experiments demonstrate 

the convergence of the proposed finite volume scheme and its relevance 

for computing late-time asymptotics of (possibly) discontinuous solutions on a 

curved background. 

This presentation is based on the joint paper [2]. 

References 

[1] P. Amorim, P.G. LeFloch, and B. Okutmustur, Finite volume schemes on 

Lorentzian manifolds, Comm. Math. Sc., 6. (2008), pp. 1059–1086. 

[2] P.G. LeFloch, H. Makhlof, and B. Okutmustur, Relativistic Burgers equations 

on a curved spacetime. Derivation and finite volume approximation, 

SIAM J. NUMER. ANAL.,Vol. 50, No. 4, (2012), pp. 21362158. 

[3] P.G. LeFloch and B. Okutmustur, Hyperbolic conservation laws on spacetimes. 

A finite volume scheme based on differential forms, Far East J. 

Math. Sci. 31. (2008), pp. 49–83. 

[4] G. Russo and A. Khe, High–order well–balanced schemes for systems of 

balance laws, in “ Hyperbolic problems: theory, numerics and applications”, 

Proc. Sympos. Appl. Math., Vol. 67, Part 2, Amer. Math. Soc. 

(2009), pp. 919–928. 

Joint work with: Philippe LeFloch and Hasan Makhlof (Université Pierre et Marie 

Curie) 

Page 45

Newton-Pade Approximations for Univariate and Multivariate Functions 

C. Akal 1 , A. Lukashov 2 

1;2 Department of Mathematics, Faculty of Arts and Science, Fatih University, Istanbul, Turkey 


The Newton-Padé approximants are a particular case of the multipoint Padé approximants, corre- 

sponding to the situation when the sets of interpolation points are nested. 

One may consult papers [1-11] for the theory of those approximations for univariate functions. Re- 

cently, the authors [13] found a new form for the Newton-Padé approximations and used it in their 

convergence study. In [12] a multivariate generalization of the Newton-Padé approximations was intro- 

duced. 

The goal of this note is two-fold. Firstly, we will give short extract from our forthcoming paper [13]. 

Next, we present generalizations of main lemmas for the case of multivariate functions. For the sake of 

simplicity we restrict ourselves to the case of two variables because the generalization to more than two 

variables is straightforward. 

References 

[1] H. E. Salzer, An osculatory extension of Cauchy’s rational interpolation formula, Zamm-Z. Angew. 

Math. Mech. 64(1) (1984) 45-50. 

[2] J. Meinguet, On the solubility of the Cauchy interpolation problem, Approximation Theory, ed. 

Talbot, A., Academic Press, London 1970, 535-600. 

[3] M. H. Gutknecht, The multipoint Padé table and general recurrences for rational interpolation, Acta 

Appl. Math. 33 (1993) 165-194. 

[4] S. Tang, L. Zou, C. Li, Block based Newton-like blending osculatory rational interpolation, Anal. 

Theory Appl. 26(3) (2010) 201–214. 

[5] Q. Zhao, J. Tan, Block-based Thiele-like blending rational interpolation, J. Comput. Appl. Math. 

195 (2006) 312–325. 

[6] A. M. Fu, A. Lascoux, A Newton type rational interpolation formula, Adv. Appl. Math. 41 (2008) 

452-458. 

[7] M.A. Gallucci, W.B. Jones, Rational approximations corresponding to Newton series (Newton- Padé 

approximants), J. Approx. Theory 17 (1976) 366-392. 

[8] G. Claessens, The rational Hermite interpolation problem and some related recurrence formulas, 

Comput. Math. Appl. 2 (1976) 117-123. 

[9] A. Draux, Formal orthogonal polynomials and Newton-Padé approximants, Numer. Alg. 29 (2002) 

67-74. 

[10] D. D. Warner, Hermite interpolation with rational functions, Ph.D. Thesis, Univ. of California 

(1974). 

Page 46

[11] G.A. Baker, P. Graves-Morris, Pade approximants, vol.2 1981. 

[12] A. Cuyt, B. Verdonk, General Order Newton-Padé Approximants for Multivariate Functions, Nu- 

merische Mathematik, 43 (1984) 293-307. 

[13] A. Lukashov, C. Akal, Determinant form and a test of convergence for Newton-Padé approximations, 

Journal of Computational Analysis and Applications, (to be appear) January 2013. 

Page 47

Finite Difference Method for The Reverse Parabolic Problem 

with Neumann Condition 

Charyyar Ashyralyyev∗,†, Ayfer Dural∗∗ and Yasar Sozen‡ 

∗Department of Computer Technology of the Turkmen Agricultural University, 74400, 

Gerogly Street, Ashgabat,Turkmenistan, E-mail: charyar@gmail.com 

†Department of Mathematical Engineering, Gumushane University, 29100, 

Gumushane,Turkey 

∗∗Gaziosmanpasa Lisesi, 34245, Istanbul, Turkey, E-mail:ayfer_drl@hotmail.com 

‡Department of Mathematics, Fatih University, 34500, Istanbul, Turkey, 

Email:ysozen@fatih.edu.tr 

Abstract. A finite difference method for the approximate solution of the reverse multidimensional parabolic differential 

equation with a multipoint boundary condition and Neumann condition is applied. Stability, almost coercive stability, and 

coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The 

theoretical statements are supported by the numerical example. 

The present paper considers the multipoint nonlocal boundary value problem for then multidimensional parabolic equation 

with Neumann condition 

n 

� 

�ut 

( t, x) �� ar ( x) ux � x ( , ), ( 1,..., 

) , 0 1, 

r r � f x t x � x xn �� t� r�1 

� 

p � 

�u(1, 

x) � � �iu( �i, x) �� ( x), x ��, 0 � �1��2�... �� P�1, 

�1� � 

i�1 

��u( 

x, t) 

� � 0, x � S, 0 � t �1 

� �n 

under the condition 

p 

� 

k �1 

� �1. 

Here, ar(x), (x∈Ω), � ( x) 

(x∈ � ), f(t,x) (t∈(0,1),x∈Ω) are given smooth functions and ar(x)≥a>0, Ω=(0,1)×⋯×(0,1) is the 

unit open cube in the n-dimensional Euclidean space with boundary S, � =Ω∪S, and n is the normal vector to Ω. 

The first and second order of accuracy in t and the second order of accuracy in space variables for the approximate solution 

of problem (1) are presented. The stability, almost coercive stability, and coercive stability estimates for the solution of these 

difference schemes are obtained. The modified Gauss elimination method for solving these difference schemes in the case of 

one-dimensional parabolic partial differential equations is used. 

REFERENCES 

1. O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Transl 

Math Monogr, Providence, 1968. 

2. O.A. Ladyzhenskaya, and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, London, New 

York,1968. 

3. M.L. Vishik, A.D. Myshkis, and O.A. Oleinik, Partial Differential Equations, in Mathematics in USSR in the Last 40 

Years, 1917-1957, Fizmatgiz, Moscow, 1959, pp. 563 (Russian). 

4. A. Ashyralyev, Nonlocal boundary value problems for partial differential equations: well-posedness, in AIP Conference 

Proceedings Global Analysis and Applied Mathematics, edited by K. Tas, D. Baleanu, O. Krupková, and D. Krupka 

International Workshop on Global Analysis, 2004, pp. 325. 

5. A. Ashyralyev, J. Evol Equ 6, 1–28 (2006). 

6. A. Ashyralyev, A. Hanalyev, and P.E. Sobolevskii, Abstr. Appl. Anal. 6, 53–61(2001). 

7. A. Ashyralyev, A. Dural, and Y. Sözen, Vestnik of Odessa National University: Math and Mech 13, 1–12 (2009). 

8. A. Ashyralyev, A. Dural, and Y. Sözen, Iran J. Optim. 1, 1–25 (2009). 

9. A. Ashyralyev, A. Dural, and Y. Sözen, Abst. Appl. Anal. accepted. 

10. A. Ashyralyev, and P.E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhäuser, Basel, 

Switzerland, 1994. 

11. Ph. Clement, and S. Guerre-Delabrire, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9, 245–266 (1999). 

12. A.V. Gulin, N.I. Ionkin, and V.A. Morozova, Differ. Equ. 37, 970–978 (2001) (Russian). 

13. X.Z. Liu, X. Cui, and J.G. Sun, J. Comput. Appl. Math. 186, 432–449 (2006). 

14. J. Martin-Vaquero, Chaos Solitons Fractals 42, 2364–2372 (2009). . 

15. J. Martin-Vaquero, and J. Vigo-Aguiar, Appl. Numer. Math. 59, 2507–2514 (2009). 

16. M. Sapagovas, J. Comput. Appl. Math. 88, 89–98 (2003). 

17. M. Sapagovas, J. Comput. Appl. Math. 92, 77–90 (2005). 

k 

Page 48

Three Models based Fusion Approach in the Fuzzy Logic Context for the 

Segmentation of MR Images : A Study and an Evaluation 


C. Lamiche 1 and A. Moussaoui 2 

1 Department of Computer Science, M'sila University, M'sila, Algeria 

2 Department of Computer Science, Setif University, Setif, Algeria 

In this work we present a study and an evaluation of three models based fusion approach in the 

fuzzy logic context for the segmentation of MR images. The process of fusion consists of three 

parts : (1) information extraction, (2) information combination, and (3) decision step. 

Information provided by T1-weighted,T2-weighted and PD-weighted images is extracted and 

modeled separately in each one using FPCM (Fuzzy Possibilistic C-Means) algorithm, fuzzy 

maps obtained are combined with an operator of fusion which can managing the uncertainty and 

ambiguity in the images and the final segmented image is constructed in decision step. Some 

results are presented and discussed. 

References 

Page 49 

[1] Gonzalez R. C. and Woods R. E., Digital Image Processing. Massachusetts: Addison- 

Wesley, 1992. 

[2] Hata Y., Kobashi S. and Hirano S., Automated segmentation of human brain MR images 

aided by fuzzy information granulation and fuzzy inference”, IEEE Trans. SMC, Part C, 30, pp. 

381-395, 2000. 

[3] Van Leemput K., Maes F., Vandermeulen D. and Suetens P., Automated model-based 

tissue classification of MR images of the brain, IEEE Trans. Medical Imaging, 18, pp. 897-908, 

1999. 

[4] Wang Y., Adali T., Xuan J. and Szabo Z., Magnetic resonance image analysis by 

information theoretic criteria and stochastic models, IEEE Trans, Information Technology in 

Biomedicine, 5, pp. 150-158, 2001. 

[5] Bloch I. and Maitre H., Data Fusion in 2D and 3D Image Processing: An overview,” in 

Proceedings of X Brazilian Symposium on Computer Graphics and Image Processing, Brazil, pp. 

127-134, 1997. 

[6] Dou W., Ruan S., Chen Y., Bloyet D. and Constans J. M., A framwork of fuzzy information 

fusion for the segmentation of brain tumor tissues on MR images,” Image and vision 

Computing, 25, pp. 164-171, 2007

Approximate Solutions of The Cauchy Problem for The Heat Equations 


Deniz A˘gırseven 1 

1 Department of Mathematics, Trakya University, 22030, Edirne, TURKEY 

Homotopy Analysis Method (HAM) [1-2] is applied to the problem of the one-dimensional heat equa- 

tions with a non-linear heat source subject to the temperature and the heat flux given at a single boundary 

to obtain the analytical solutions. Solutions obtained take an important place for one-dimensional heat 

flow as applied to a few regular geometries such as slabs, cylinders and spheres. Some of the test problems 

are presented to show the efficiency of HAM. 

References 

[1 ]Liao S.J., The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. 

Thesis, Shanghai Jiao Tong University, 1992. 

[2] Liao S.J., Beyond perturbation: introuction to the Homotopy Analysis Method, Boca Raton:Chapman 

Hall/CRC Press, 2003. 

[3] Abbasbandy S., Homotopy analysis method for heat radiation equations, Int.Commun. Heat Mass 

Transfer, , 34, 380-387, 2007. 

[4] Lesnic D., Decomposition methods for non-linear, non-characteristic Cauchy heat problems Com- 

munications in Nonlinear Science and Numerical Simulation, 10, 581-596, 2005. 

[5] Adomian G., Rach R. , Noise terms in decomposition series solution, Comput. Math. Appl., 24, 

61-64, 1992. 

[6]Wazwaz A.M., A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. 

Comput., 118, 287-310, 2001. 

[7] Iqbal S., Javed A., Application of optimal asymptotic method for the analytic solution of singular 

Lane Emden type equation, Appl. Math. Comput, 217, 7753-7761, 2011. 

[8] Ashyralyev A., Erdogan A.S:, Arslan N., On the numerical solution of the diffusion equation with 

variable space operator, Appl. Math. Comput., 189 , 682-689, 2007. 

[9] Agirseven D., Ozis T., An analytical study for Fisher type equations by using homotopy pertur- 

bation method, Computers and Mathematics with Application, 60 , 602-609, 2010. 

[10] Sami Bataineh A., Noorani M.S.M., Hashim I., Solutions of time dependent Emden- Fowler type 

equations by homotopy analysis method, Physics Letters A, 371, 72-82, 2007. 

[11] Ozis T., Agirseven D., He’s homotopy perturbation method for solving heat-like and wavelike 

equations with variable coefficients , Physics Letters A, 372 , 5944-5950, 2008. 

[12] Shidfar A., Karamali G.R., Damirchi J, An inverse heat conduction problem with a nonlinear 

source term, Nonlinear Analysis, 65, 615-621, 2006. 

[13] Shidfar A., Molabahrami A., A weighted algorithm based on the homotopy analysis method: 

Application to inverse heat conduction problems, Communications in Nonlinear Science and Numerical 

Simulation, 15, 2908-2915, 2010. 

Page 50

The normal inverse Gaussian distribution: exposition and applications to modeling asset, 


index and foreign exchange closing prices 

D. Teneng 1 and K. Pärna 2 

1 Institute of Mathematical Statistics, University of Tartu, Tartu, Estonia 

2 Institute of Mathematical Statistics, University of Tartu, Tartu, Estonia 

We expose the unique properties of the normal inverse Gaussian distribution (NIG) useful for mod- 

eling asset, index and foreign exchange closing prices. We further demonstrate that traditional beliefs 

in asset, index, and foreign exchange closing prices not being independently identically distributed ran- 

dom variables are fundamentally flawed. Best models are selected using a novel model selection strategy 

proposed by Käärik and Umbleja (2011). Our results show that closing prices of Baltika and Ekpress 

Grupp (companies trading on Tallinn stock exchange), FTSE100, GSPC and STI (major world indexes), 

CHF/JPY, USD/EUR, EUR/GBP, SAR/CHF, QAR/CHF and EGP/CHF (Foreign Exchange rates) can 

be modeled by NIG distribution. This means their underlying stochastic properties can fully be captured 

by NIG; very useful for predicting price movements, pricing models, underwriting and trading derivatives 

etc 

Acknowledgement Research supported by Estonian Science foundation grant number 8802 and Esto- 

nian Doctoral School in Mathematics and Statistics. 

References 

[1] Barndorff-Nielsen O. E., Processes of normal inverse Gaussian type, Systems & Finance and 

Stochastics, 2, (1998), 42-68. 

[2] Figueroa-López J. E., Jump diffusion models driven by Lévy Processes, Springer Handbooks of 

Computational Statistics, (2012), 61-88. 

[3] Käärik M., Umbleja M., On claim size fitting and rough estimation of risk premiums based on 

Estonian traffic example, International Journal of Mathematical Models and Methods in Applied Sciences, 

Issue 1, vol. 5, 17-24, (2011). 

[4] Lo A. W. and Mackinlay A. C., Stock market prices do not follow random walks: Evidence from 

a simple specification test, Review of Financial studies, Vol. 1, 41-66, (1998). 

[4] Necula C., Modelling heavy-tailed stock index returns using the generalized hyperbolic distribution, 

Romanian Journal of Economic Forecasting,Vol. 6(2), 610-615, (2009) 

[5] Schoutens W., Lévy Processes in Finance, John Wiley & Sons Inc., New York, (2003). 

[6] Teneng D., NIG-Levy process in asset price modelling: case of Estonian companies, Proc. 30 th 

International Conference on Mathematical Methods in Economics - MME 2012,Karvina, 1-3 Sept. 2012, 

to appear. 

[7]Rydberg T. H., The Normal Inverse Gaussian Levy Process : Simulation and approximation, Comm. 

Stat.: Stoch. Models, Vol. 13 (4), 887-910, (1997). 

Page 51

Radial Basis Functions Method for determining of unknown coe¢ cient in parabolic 


equation 

E. Can 

Department of Physics, Kocaeli University, Kocaeli 41380, Turkey 

Electro Optic Systems Engineering, Kocaeli University, Kocaeli 41380, Turkey 

In this paper, we consider an inverse problem of …nding unknown source parameterp(t) and u(x; t) 

satisfy equation 

with the initial-boundary conditions 

ut = uxx + p(t)u + f (t; x) ; 0 6 x 6 1; 0 < t 6 T; (1) 

subject to the overspeci…cation over the spatial domain 

u(x; 0) = '(x); 0 6 x 6 1 (2) 

(0; t) = 1(t); 0 < t 6 T (3) 

u(1; t) = 2(t); 0 < t 6 T (4) 

u(x ; t) = E(t); 0 < x 6 1; 0 < t 6 T (5) 

where f(x; t); '(x); 1(t); 2(t) and E(t) 6= 0 are known functions, x is a …xed prescribed interior point in 

(0,1). If p (t) is known then direct initial boundary value problem (1) (4) has a unique smooth solution 

u (x; t) [1]: If u represent a temperature distribution, then (1) (4) can be interpreted as a control 

problem with source parameter. Based on the idea of the radial basis functions (RBF) approximation , 

a fast and highly accurate meshless method is developed for solving an inverse problem with a control 

parameter [2]. Some numerical examples using the proposed algorithm are presented. 

References 

[1] Isakov V., Inverse Problems for Partial Di¤erential Equations, Applied Mathematical Sciences, 

Springer-Verlag, vol. 127, 1997. 

[2] Limin Ma and Zongmin Wu, Radial Basis functions method for parabolic inverse problem, Int. J. 

of Computer Math., 88(2), 383-395, 2011. 

Page 52

Compact and Fredholm Operators on Matrix Domains of Triangles in the Space of Null 

Sequences 

E. Malkowsky 


Department of Mathematics, University of Giessen, Giessen, Germany 

Abstract The matrix domain XA of an infinite matrix A = (ank) ∞ n,k=0 

subset X of the set ω of all complex sequences is the set of all x = (xk) ∞ k=0 

of complex numbers in a 

∈ ω for which the series 

Anx = � ∞ 

k=0 ankxk converge for all n and Ax = (Anx) ∞ n=0 ∈ X. Also, if X and Y are subsets of ω 

then (X, Y ) denotes the set of all infinite matrices that map X into Y , that is, A ∈ (X, Y ) if and only 

if X ⊂ YA. Let c0 denote the set sequences x ∈ ω that converge to zero, and T = (tnk) ∞ n,k=0 and 

˜T = (˜tnk) ∞ k,k=0 be triangles, that is, tnk = ˜tnk = 0 for k > n and tnn = ˜tnn �= 0 (n = 0, 1, . . . ). We 

characterise the class ((c0)T , (c0) ˜ T ). Furthermore we obtain an explicit formula for the Hausdorff measure 

of noncompactness of operators LA given by a matrix A ∈ (c0)T , (c0) ˜ T ), that is, for which LA(x) = Ax 

for all x ∈ (c0)T . From this result, we obtain a characterisation the class of compact operators given by 

matrices in ((c0)T , (c0) ˜ T ). Finally we give a sufficient condition for an operator given by a matrix to be 

a Fredholm operator on (c0)T . 

References 

[1] Djolović I. and Malkowsky E., A note on compact operators on matrix domains, Journal of Math- 

ematical Analysis and Applications, 340(1), 291–303, 2008 

[2] Djolović I. and E. Malkowsky, Matrix transformations and compact operators on some new m th 

order difference sequences, Applied Mathematics and Computations, 198(2), 700–714, 2008 

[3] de Malafosse B. and Rakočević V., Application of measure of noncompactness in operators on the 

spaces sα, s 0 α, s c α, ℓ p α, Journal of Mathematical Analysis and Applications, 323(1), 131–145, 2006 

[4] Malkowsky E. and Rakočević V., An Introduction into the Theory of Sequence Spaces and Measures 

of Noncompactness, Zbornik radova, Matematički institut SANU, Belgrade, 9(17), 143–234, 2000 

[5] Malkowsky E. and Rakočević V., On matrix domains of triangles, Applied Mathematics and 

Computations, 189(2), 2007, 1146–1163 

[6] M.Schechter, Principles of Functional Analysis, Academic Press, New York and London, 1973 

[7] A. Wilansky, Summability through Functional Analysis, North–Holland Mathematics Studies No. 

85, North–Holland, Amsterdam, New York, Oxford, 1984 

Page 53

Compact Operators on Spaces of Sequences of Weighted Means 

E. Malkowsky 1;2 , F. Özger 1 


2 Mathematisches Institut, Universität Giessen, Arndstrasse 2, D–35392, Giessen Germany 


We reduce the spaces a r 0, a r c, a r 0( ) and a r c( ) and simplify their dual spaces and the characterisations 

of matrix transformations on them in [3]. We also obtain an estimate and a formula for the Hausdor¤ 

measure of noncompactness of some matrix operators on the spaces a r 0 and a r c, and the corresponding 

characterisations of compact matrix operators. 

References 

[1] C. Ayd¬n, and F. Ba¸sar, Hokkaido Mathematical Journal 33, 383–398 (2004) 

[2] C. Ayd¬n, and F. Ba¸sar, Applied Mathematics and Computation 157, 677–693 (2004) 

[3] E. Malkowsky, and F. Özger, Filomat 26 (3), 511–518 (2012) 

[4] I. Djolović, J. Math. Anal. Appl. 318, 658–666 (2006) 

[5] I. Djolović, and E. Malkowsky, J. Math. Anal. Appl. 340, 291–303 (2008) 

[6] B. de Malafosse, and V. Rakoµcević, J. Math. Anal. Appl. 323 (1), 131–145 (2006) 

[7] E. Malkowsky, Rendi. Circ. Mat. Palermo II, Suppl. 68, 641–655 (2002) 

[8] E. Malkowsky, and V. Rakoµcević, Applied Mathematics and Computation 189, 1148–1163 (2007) 

[9] A. Wilansky, Summability through Functional Analysis, North–Holland Mathematics Studies No. 85, 

North–Holland, Amsterdam, New York, Oxford, 1984 

Page 54


Extended Eigenvalues of Direct Integral of Operators 

E. Otkun Çevik 1 and Z.I. Ismailov 2 

1 Institute of Natural Sciences, Karadeniz Technical University, Trabzon, Turkey 


In this work, a connection between extended eigenvalues of direct integral of operators in the direct 

integral of Hilbert spaces and their coordinate operators has been investigated. 

References 

[1] Biswas A., Lambert A. and Petrovic S., Extended eigenvalues and the Volterra operator, Glasgow 

Math. J., 44, 521-534, 2002. 

2004. 

[2] Lambert A., Hyperinvariant subspaces and extended eigenvalues, New York J. Math., 10, 83-88, 

[3] Karaev M. T., Invariant subspaces, cyclic vectors, commutant and extended eigenvectors of some 

convolution operators, Methods Func. Anal. Topology, 11, 48-59, 2005. 

[4] Karaev M. T., On extended eigenvalues and extended eigenvectors of some operator classes, Proc. 

Amer. Math. Soc., 134(8), 2883-2392, 2006. 

[5] Biswas A. and Petrovic S., On extended eigenvalues of operators, Int. equat. oper. theory, 55, 

233-248, 2006. 

[6] Petrovic S., On the extended eigenvalues of some Volterra operators, Int. equat. oper. theory, 57, 

593-598, 2007. 

2007. 

Page 55 

[7] Shkarin S., Compact operators without extended eigenvalues, J. Math. Anal. Appl., 332, 455-462,

Exponential decay and blow up of a solution for a system of nonlinear higher-order wave 


equations 

E. Pi¸skin and N. Polat 

Department of Mathematics, Dicle University, Diyarbakir, Turkey 

This work studies a initial-boundary value problem of the weak damped nonlinear higher-order wave 

equations. Under suitable conditions on the initial datum, we prove that the solution decays exponentially 

and blows up with negative initial energy. 

References 

[1] Adams R. A. and Fournier J. J. F., Sobolev Spaces, Academic Press, 2003. 

[2] Georgiev V. and TodorovaG., Existence of a solution of the wave equation with nonlinear damping 

and source terms, J. Di¤erential Equations, 109 (2), 295–308, 1994. 

[3] Zhou Y., Global existence and nonexistence for a nonliear wave equation with damping and source 

terms, Math. Nacht, 278, 1341-1358, 2005. 

[4] Messaoudi S. A. and Said-Houari B., Global nonexistence of positive initial-energy solutions of a 

system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl. 

365, 277–287, 2010. 

Page 56

A New General Inequality for double integrals 

Erhan Set 1 , Mehmet Zeki Sarıkaya 1 and Ahmet Ocak Akdemir 2 

1 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey 

2 A˘grı ˙ Ibrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, 04100, 


A˘grı, Turkey 

The inequality of Ostrowski gives us an estimate for the deviation of the values of a smooth function 

from its mean value. More precisely, if f : [a, b] → R is a differentiable function with bounded derivative, 

then � 

�� 

f(x) − 1 

∫b 

b − a 

a 

� 

� 

� 

f(t)dt� 

� 

� ≤ 

[ 

1 

4 

a+b (x − 2 + )2 

(b − a) 2 

] 

(b − a) ∥f ′ ∥∞ for every x ∈ [a, b]. Moreover the constant 1/4 is the best possible. Inequality (1) has wide applications 

in numerical analysis and in the theory of some special means; estimating error bounds for some special 

means, some mid-point, trapezoid and Simpson rules and quadrature rules, etc. Hence inequality (1) 

has attracted considerable attention and interest from mathematicans and researchers. Due to this, over 

the years, the interested reader is also refered to ([1]-[9]) for integral inequalities in several independent 

variables. In addition, the current approach of obtaining the bounds, for a particular quadrature rule, 

have depended on the use of peano kernel. The general approach in the past has involved the assumption 

of bounded derivatives of degree greater than one. 

In this paper, we obtain a new general inequality involving functions of two independent variables by 

defining the peano kernel K(x, y; s, t) as the following: 

⎧ ( 

b − a 

t − (a + λ 

6 

⎪⎨ 

K(x, y; t, s) = 

⎪⎩ 

) 

) ( ( )) 

d − c 

s − c + λ for a ≤ t ≤ x, c ≤ s ≤ y, 

( 

6 

b − a 

t − (a + λ 

6 ) 

) ( ( )) 

d − c 

s − d − λ for a ≤ t ≤ x, y ≤ s ≤ d, 

( 

6 

b − a 

t − (b − λ 

6 ) 

) ( ( )) 

d − c 

s − c + λ for x ≤ t ≤ b, c ≤ s ≤ y, 

( 

6 

b − a 

t − (b − λ 

6 ) 

) ( ( )) 

d − c 

s − d − λ for x ≤ t ≤ b, y ≤ s ≤ d. 

6 

This inequality is a new generalization of the inequalities of Simpson and Ostrowski type obtained by 

Zhongxue in [9]. 

References 

[1] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applica- 

tions for cubature formulae, Soochow J. Math., 27(1), (2001), 109-114. 

[2] M.E. Özdemir, H. Kavurmacı and E. Set, Ostrowski’s type inequalities for (α, m)−convex functions, 

Kyungpook Math. J., 50 (2010), 371-378. 

[3] B. G. Pachpatte, On a new Ostrowski type inequality in two independent variables, Tamkang J. 

Math., 32(1), (2001), 45-49. 

[4] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. 

LXXIX, 1(2010), pp. 129-134. 

[5] M.Z. Sarıkaya, E. Set and M.E. 

Özdemir, On new inequalities of Simpson’s type for s-convex 

functions, Computers & Mathematics with Applications, 60 (2010), 2191-2199. 

Page 57 

(1)

[6] E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second 

sense via fractional integrals, Computers and Mathematics with Applications, 63(7), (2012), 1147-1154. 

[7] E. Set and M.Z. Sarıkaya, On The Generalization Of Ostrowski And Grüss Type Discrete Inequal- 

ities, Computers & Mathematics with Applications, 62, (2011), 455–461. 

[8] N. Ujević, Sharp inequalities of Simpson type and Ostrowski type, Computers & Mathematics 

with Applications, 48 (2004) 145-151. 

[9] L. Zhongxue, On sharp inequalities of Simpson type and Ostrowski type in two independent 

variables, Computers & Mathematics with Applications, 56 (2008) 2043-2047. 

Page 58

Exact Solutions of the Schrödinger Equation with Position Dependent Mass 

for the solvable Potentials 


F. Aricak 1 and M.Sezgin 2 

1 Department of Physics, Trakya University, Edirne, Turkey 

2 Department of Mathematics, Trakya University, Edirne, Turkey 

In this work the infinitesimal operators of the regular representations of the group SL(2,R) are 

considered. According to these infinitesimal operators the Casimir operator is expressed. The 

Hamiltonian H is related to Casimir operator C of the group. The energy eigenvalues and the 

corresponding eigenfunctions are given for the solvable potentials 

References 

Page 59 

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Sturm Liouville Problem with Discontinuity Conditions at Several Points 

F.H¬ra 1 and N. Alt¬n¬¸s¬k 2 

1;2 Department of Mathematics, Ondokuz May¬s University, Samsun,Turkey 

In this paper we deal with the computation of the eigenvalues of Sturm Liouville problem with 

several discontinuity conditions (transmission conditions) inside a …nite interval and parameter dependent 

boundary conditons.By using an operator theoretic interpretation we extend some classic results for 

regular Sturm Liouville problems.A symmetric linear operator A is de…ned in an appropriate Hilbert 

space such that the eigenvalues of such a problem coincide with those of A. Also,we obtained asymptotic 

formulaes for the eigenvalues and corresponding eigenfunctions. 

Consider the following Sturm Liouville problem with discontinuity conditions at several points inside 

a …nite interval, 

Tk (u) := 

(u) := u 00 + q (x) u = u; x 2 (x0; x1) (1) 

B2 (u) := ( 0 1u (x1) 

0 

1 

B1 (u) := 1u (x0) + 2u 0 (x0) = 0 (2) 

@ u ( k + 0) 

u 0 ( k + 0) 

0 

2u 0 (x1)) + 1u (x1) 2u 0 (x1) = 0 (3) 

A Dk 

0 

@ u ( k 

u 0 ( k 

0) 

0) 

1 

A = 0; k = 1; m (4) 

where x0 = 0 < 1 < ::: < m < m+1 = x1; q 2 L2 (x0; x1) ; is a complex spectral parameter. 

2 

We shall assume that 1 + 2 2 

2 6= 0; 1 + 2 8 

0 

< 1 , if 1 + 

2 6= 0; > 0;where = 

: 

0 

2 = 0 

and 

0 

0 

1 2 2 1 , otherwise 

0 1 

Dk = 

@ 

1k 2k 

3k 4k 

References 

A ; ik 2 R; i = 1; 4; jDkj > 0 for k = 1; m: Let D0 be the 2 2 identity matrix. 

[1] Alt¬n¬¸s¬k N., Kadakal M., Mukhtarov O. Sh., Eigenvalues and eigenfunctions of discontinuous 

Sturm Liouville problems with eigenparameter dependent boundary conditions, Acta Math. Hung., 102 

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[3] Chanane B., Sturm Liouville problems with impulse e¤ects, Appl. Math.Comput. 190, 610-626, 

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[6] Kadakal M, Mukhtarov O. Sh, Sturm–Liouville problems with discontinuities at two points, Com- 

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[7] Kobayashi M., Eigenfunction expansions: A discontinuous version, SIAM J. Appl. Math. 50 (3), 

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Page 60

[9] Titeux I., Yakubov Y., Completeness of root functions for thermal conduction in a strip with 

piecewise continuous coe¢ cients, Math. Models Methods Appl. Sc., 7:7, 1035–1050, 1997. 

[10] Walter J., Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, 

Math. Z., 133, 301-312, 1973. 

[11] Wang A., Sun J., Hao X., Yao S., Completeness of eigenfunctions of Sturm Liouville problems 

with transmission conditions, Methods and Applications of Analysis,16 (3) 299-312, 2009. 

Page 61

Characterization of Three Dimensional Cellular Automata over Zm 

Ferhat S¸ah 1 , I. S¸iap 2 and H. Akın 3 

1 Department of Mathematical Engineering, Yildiz Technical University,Istanbul, Turkey 


2 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey 

3 Department of Mathematics, Education Faculty, Zirve University, Gaziantep, Turkey 

Three dimensional cellular automata wasn’t much studied by researches. Tsalides et al. characterized 

three dimensional cellular automata in [1] and then Hemmingsson investigated quasi periodic behavior 

of three dimensional cellular automata in [2]. In this work we study the algebraic behavior of three 

dimensional linear cellular automata over Zm. we provide necessary and sufficient conditions for a three 

dimensional linear cellular automata over the ring Zm to be reversible or irreversible. As a consequence of 

our result we characterize three dimensional linear cellular automata under the null boundary conditions. 

Acknowledgements: The work is supported by T ÜB˙ ITAK (Project Number: 110T713). 

References 

[1] P. Tsalides, P. J. Hicks, and T.A. York, Three-Dimensional Cellular Automata and VLSI Appli- 

cations, IEE Proceedings, 136 (6), 490-495 (1989). 

[2] J. A. Hemmingsson, Totalistic Three-Dimensional Cellular Automaton with Quasiperiodic Be- 

haviour, Physica A, 183 (3), 255-261, (1992). 

Page 62

Positivity of Elliptic Difference Operators and its Applications 

G.E. Semenova, semgalya@mail.ru 

Department of Differential Equations, Institute of Mathematics and Informatics of the North-Eastern 

Federal University, Russia 

As is well-known that the investigation of well-posedness of various types of parabolic and elliptic 

differential and difference equations is based on the positivity of elliptic differential and difference 

operators in various Banach spaces and on the structure of the fractional spaces generated by these 

positive operators. An excellent survey of works in the theory of fractional spaces generated by positive 

multidimensional difference operators in the space and its applications to partial differential equations 

was given in [1]-[2]. In a number of works (see, e.g., [3]-[11], and the references therein) difference 

schemes were treated as operator equations in a Banach space and the investigation was based on the 

positivity property of the operator coeffi cient. 

In the present paper, we consider the difference operator 

(−1) n ∂ 2n 

hp + Ah, 

where Ah is the self-adjoint positive definite operator in L2h. Applying the method of paper [3] the 

positivity of this difference operator in the Holder spaces is established. In applications, the wellposedness 

of the Cauchy problem for parabolic differential and difference equations is investigated. 

References 

[1] A. Ashyralyev, and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, 

Birkhäuser Verlag, Basel, Boston, Berlin, 1994. 

[2] A. Ashyralyev, and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, 

Birkhäuser Verlag, Basel, Boston, Berlin, 2004. 

[3] P. E. Sobolevskii, The Coercive Solvability of Difference Equations, Dokl. Acad. Nauk. SSSR 

201(5), 1063—1066 (1971) (Russian). 

[4] Kh. A. Alibekov, Investigations in C and Lp of Difference Schemes of High Order Accuracy for 

Apporoximate Solutions of Multidimensional Parabolic Boundary Value Problems, Ph.D. thesis, 

Voronezh State University, Voronezh, 394006 (1978) (Russian). 

[5] Kh. A. Alibekov, and P. E. Sobolevskii, Stability and Convergence of Difference Schemes of a High 

Order for Parabolic Differential Equations, Ukrain. Math. Zh. 31(6), 627—634 (1979) (Russian). 

[6] A. Ashyralyev, and P. E. Sobolevskii, The Linear Operator Interpolation Theory and the Stability 

of the Difference Schemes, Dokl. Acad. Nauk SSSR 275(6), 1289—1291 (1984) (Russian). 

[7] A. Ashyralyev, Method of Positive Operators of Investigations of the High Order of Accuracy 

Difference Schemes for Parabolic and Elliptic Equations, Doctor of Sciences Thesis, Inst. of Math. 

of Acad. Sci. Kiev, Kiev, 01601 (1992) (Russian). 

[8] B. A. Neginskii, and P. E. Sobolevskii, Difference Analogue of Theorem on Inclosure an Interpolation 

Inequalities, in Proceedings of Faculty of Mathematics, Voronezh State University, Voronezh, 

1970, pp. 72—81. 

[9] Yu. A. Simirnitskii, and P. E. Sobolevskii, Positivity of Multidimensional Difference Operators in 

the C−norm, Usp. Mat. Nauk. 36(4), 202—203 (1981) (Russian). 

[10] S. I. Danelich, Fractional Powers of Positive Difference Operators, Ph.D. thesis, Voronezh State 

University, Voronezh, 394006 (1989) (Russian). 

[11] A. Ashyralyev, and B. Kendirli, Positivity in Ch of One Dimensional Difference Operators with 

Nonlocal Boundary Conditions, in Some Problems of Applied Mathematics, edited by A. Ashyralyev, 

and H. A. Yurtsever, Fatih University, Istanbul, Turkey, 2000, pp. 45—60. 

1 

Page 63

On (α, β)-derivations in BCI-algebras 

G. Muhiuddin 

Department of Mathematics, University of Tabuk, P. O. Box 741, Tabuk 71491, Saudi Arabia 


The notion of (regular) (α, β)-derivations of a BCI-algebra X is introduced, some 

useful examples are discussed, and related properties are investigated. Condition for 

a (α, β)-derivation to be regular is provided. The concepts of a d(α,β)-invariant (α, β)- 

derivation and α-ideal are introduced, and their relations are discussed. Finally, some 

results on regular (α, β)-derivations are obtained. 

References 

[1] H.A.S. Abujabal and N.O. Al-Shehri, On Left Derivations of BCI-algebras, Soochow 

J. Math. 33(3) (2007), 435—444. 

[2] M. Aslam and A.B. Thaheem : A note on p-semisimple BCI-algebras, Math. Japon. 

36 (1991), 39-45. 

[3] N. Aydin and A. Kaya: Some generalization in prime rings with (σ, τ)-derivation, 

Doga Turk. J. Math. 16 (1992), 169-176. 

[4] G. Mudiuddin and A. M. Al-roqi, On t-derivations of BCI-algebras, Abstract and 

Applied Analysis, (In Press) (2012). 

[5] M. A. Ozturk, Y. Ceven and Y. B. Jun : Generalized Derivations of BCI-algebras, 

Honam Math. J. 31 (4) (2009), 601-609. 

[6] J. Zhan and Y. L. Liu, On f-derivations of BCI-algebras, Int. J. Math. Math. Sci. 

2005(11) (2005), 1675—1684. 

Page 64

Transient and Cycle Structure of Elementary Rule 150 with Reflective Boundary 


H. Akın 1 , I. S¸iap 2 and M.E. Koro˘glu 2 

1 Department of Mathematics, Faculty of Education, Zirve University, Istanbul, Turkey 

2 Department of Mathematics, Yildiz Technical University, Istanbul, Turkey 

Cellular automata are simple mathematical representation of complex dynamical systems. Therefore 

there are several applications of cellular automata in many areas such as coding, cryptography, VLSI 

design etc. [1,2]. In this study, a recurrence relation for computation minimal polynomial of transition 

matrix of linear elementary rule 150 with reflective boundary condition [3] was obtained. Then, the 

maximum transient and cycle lengths of this rule were calculated by algorithm in [4]. 

Acknowledgements: The work is supported by T ÜB˙ ITAK (Project Number: 110T713). 

References 

[1] P.P. Chaudhuri, D.R. Choudhury, S. Nandi and S. Chattopadhyay, Additive Cellular Automata 

Theory and Applications, Vol.1, (IEEE Computer Society Press, 1997 Los Alamitos). 

[2] J. L. Schiff, Cellular Automata: A Discrete View of the World (Wiley Sons, Inc., 2008 Hoboken, 

New Jersey). 

[3] H. Akın, F. S¸ah, I. S¸iap, On 1D reversible cellular automata with reflective boundary over the 

prime field of order p, International Journal of Modern Physics C, 23 (1), pp. 1-13, (2012) 

[4] J. Stevens, R. Rosensweig, A. Cerkanowicz, Transient and cyclic behavior of cellular automata 

with null boundary conditions, J. Statist. Phys. 73, 159.174 (1993). 

Page 65

Numerical Solution of a laminar viscous flow boundary layer equation using Haar Wavelet 

Quasilinearization Method 

Harpreet Kaur 1 , R.C. Mittal 2 and Vinod Mishra 1 

1 Department of Mathematics, Sant Longowal Institute of Engineering and 

Technology,Longowal-148106(Punjab), India 

2 Department of Mathematics, Indian Institute of Technology, Roorkee-247667(Uttrakhand), India 


In this paper,we propose a wavelet method to solve the well known Blasius equation. The method 

is based on the Haar wavelet operational matrix defined over the interval [0, 1]. In this method,we have 

used the coordinate transformation for converting the problem on a fixed computational domain. The 

generalized Blasius equation arises in the various boundary layer problems of hydrodynamics and in fluid 

mechanics of laminar viscous flows. Comparison is made with existing solutions in literature. Haar 

Wavelet Quasilinearization Method is of high accuracy even in the case of a small number of grid points 

and without any iteration. 

References 

[1] C. Cattani, Haar wavelet spline, J. Inter. Math. 35-47,4(2001). 

[2] S. Abbasbandy, A Numerical Solution of Blasius Equation by Adomian’s Decomposition Method 

and Comparison with Homotopy Perturbation Method, Chaos, Solitons and Fractals, 257-260, 31(2007). 

[3] C.H. Hsiao, State analysis of linear time delayed system via Haar wavelets, Math. Comput. Simu. 

457-470, 44(1997). 

[4] G Hariharan, K. Kannan and K. R. Sharma, Haar wavelet method for solving Fisher’s equation, 

Appl, Math. Compu. 284-292, 211(2009). 

[5] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math.909- 

996,41(1998). 

[6] S. J. Liao, A An Explicit, Totally Analytical Approximate Solution for Blasius Viscous Flow 

Problem, International Journal of Non-Linear Mechanics, 34(1999). 

[7] A.I. Ranasinghe and F. B. Majid, Solution of Blasius Equation by Decomposition, Applied Math- 

ematical Sciences, 605-611, 3(2009). 

Page 66


Characterizations of Slant Helices According to Quaternionic Frame 

H. Kocayi¼git, M. Önder and B. B. Pekacar 

Department of Mathematics, Celal Bayar University, Manisa, Turkey 

In this study, we give the characterizations of slant helices according to quaternionic frame in 3- and 

4-dimensional Euclidean spaces. Furthermore, we obtain some necessary and su¢ cient conditions for a 

space curve to be a slant helix according to quaternionic frame. 

References 

[1] Ali, A., López, R., “On Slant Helices in Minkowski Space E 3 1”, arXiv:0810.1464v1 [math.DG], 8 

Oct 2008. 

[2] Ali, A., Turgut, M., “Some Characterizations of Slant Helices in the Euclidean Space E n ”, 

arXiv:0904.1187v1 [math.DG], 7 Apr 2009. 

[3] Gök, ·I., Okuyucu, O. Z., Kahraman, F., Hac¬saliho¼glu, H. H., "On the Quaternionic B2 Slant 

Helices in the Euclidean Space E 4 ", Adv. Appl. Cli¤ord Algebras 21, 707-719, 2011. 

[4] Bharath¬, K., Nagaraj, M., “Quaternion Valued Function of a Real Variable Serret-Frenet For- 

mulea”, Indian J. Pure appl. Math., 18(6): 507-511, June 1987. 

Page 67

Some Characterizations of Constant Breadth Tımelıke Curves in Minkowskı 4-space 

Hüseyin Kocayiğit, Mehmet Önder,Zennure Çiçek 

Celal Bayar University,Manisa,TURKEY 


In this study, the differential equation characterizations of timelike curves of constant breadth 

4 

are given in Minkowskı 4-space E . Furthermore, a criterion for a curve to be the timelike 

4 

curve of constant breadth in E is introduced. As an example, the obtained results are applied 

to the case that the curvatures k1, k2, k 3 and are discussed. 

References 

[1] Ball, N. H., On Ovals, American Mathematical Monthly, 27 (1930), 348-353. 

[2] Barbier, E., Note sur le probleme de l'aiguille et le jeu du point couvert. J. Math. Pures 

Appl., II. Ser. 5, 273–286 (1860). 

[3] Blaschke, W., Konvexe bereichee gegebener konstanter breite und kleinsten inhalt, Math. 

Annalen, B. 76 (1915), 504-513. 

[4] Blaschke, W., Einige Bemerkungen über Kurven und Flächen konstanter Breite. Ber. 

Verh. sächs. Akad. Leipzig, 67, 290-297 (1915). 

[5] Breuer, S., and Gottlieb, D., The Reduction of Linear Ordinary Differential Equations to 

Equations with Constant Coefficients, J. Math. Anal. Appl., 32 (1970), no. 1, 62-76. 

[6] Chung, H.C., A Differential-Geometric Criterion for a Space Curve to be Closed, Proc. 

Amer. Math. Soc. 83 (1981), no. 2, 357-361. 

[7] Euler, L., De Curvis Triangularibus, Acta Acad. Prtropol., (1778), (1780), 3-30. 

[8] Fujivara, M., On space curves of constant breadth, Tohoku Math., J., 5 (1914), 179-784,. 

[9] Kazaz, M., Önder, M., Kocayiğit H., Spacelike curves of constant breadth in Minkowski 

4-space, Int. Journal of Math. Analysis, 2 (2008), no. 22, 1061 – 1068. 

[10] Kocayiğit, H., Önder, M., Spacelike curves of constant breadth in Minkowski 

3-space, Annali di Matematica, DOI 10.1007/s10231-011-0247-5. 

[11] Köse, Ö., On space curves of constant breadth, DOĞA Tr. J. Math., 10 (1986), no. 1, 11- 

14. 

[12] Mağden A., Köse, Ö., On the curves of constant breadth in 

Page 68 

4 

E 

4 

E space, Turkish J. Math., 

21 (1997), no. 3, 277-284. 

[13] Mellish, A. P., Notes on Differential Geometry, Annals of Math. J., 5 (1914), 179-184. 

[14] O’Neil, B. Semi Riemannian Geometry with Applications to Relativity, Academic Press, 

New York, (1983). 

[15] Önder, M., Kocayiğit, H., Candan, E., Differential Equations Characterizing Timelike 

and Spacelike Curves of Constant Breadth in Minkowski 3-spaceJ. Korean Math. So c. 48 

(2011), No. 4, pp. 849-866. 

[16] Reuleaux, F., The Kinematics of Machinery, Trans. By A.B.W. Kennedy, Dover, Pub. 

Nex York, (1963). 

[17] Ross, S.L., Differential Equations, John Wiley and Sons, Inc., New York, (1974), 440- 

468. 

[18] Sezer, M., Differential equations characterizing space curves of constant breadth and a 

criterion for these curves, Turkish, J. of Math. 13 (1989), no. 2, 70-78. 

[19] Smakal, S., Curves of constant breadth, Czechoslovak Mathematical Journal, 23 (1973), 

no. 1, 86–94.

Page 69 

[20] Struik, D. J., Differential Geometry in the Large, Bulletin Amer. Mathem. Soc., 37 

(1931), 49-62. 

[21] Tanaka, H., Kinematics Design of Com Follower Systems, Doctoral Thesis, Columbia 

Univ., (1976). 

[22] Walrave, J., Curves and surfaces in Minkowski space. PhD. thesis, K. U. Leuven, Fac. of 

Science, Leuven (1995). 

[23] Yılmaz, S., Turgut, M., On the Time-like Curves of Constant Breadth, Math. Combin. 

Vol.3 (2008), 34–39.

Using Inverse Laplace Transform for the solution of a Flood Routing Problem 

H. Saboorkazeran 1 and M.F. Maghrebi 1,2 

1 Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 

2 Member of Iranian Structural Engineering Organization Province of Khorasan Razavi 


The inverse Laplace transform is of significant importance in mathematical sciences when an 

analytical solution exists in Laplace domain. So, a new solution of the linearized St. Venant 

equations (LSVE) has been obtained for flood routing in open channels. The LSVE has been 

previously used by many researchers [1] and in Laplace domain are in the matrix form 

�� 

�� 

� A � B� 

� 0 

�t 

�x 

where � is transfer matrix includes deviations of discharge q ( x, 

t) 

and depth y ( x, 

t) 

around the 

reference values ( 0 , 0 ) Y Q . In this new formulation, the Manning formula is linearized as boundary 

condition besides the St. Venant equations to get a Laplace transformable, simplified set of 

equations in Laplace domain as follows 

where 

k v 

�Q 

�Y 

ˆ( L, 

s) 

� k yˆ 

( L, 

s) 

(2) 

q v 

� . A method for Laplace inversion, which provides a great convergence, very 

accurate response for flood routing problem is used here. As previously this method has been 

used for difuusion waves model [2, 3], the results show the improved De Hoog algorithm [4] 

provide a solution with zero error for discharge, and very small percent of error for depth. 

Applying the well-known Preissmann implicit scheme on the LSVE for equal condition shows 

that the De Hoog algorithm is in complete agreement with the numerical solution of the LSVE. 

References 

Page 70 

[1] Litrico X. and Fromion V., Frequency modeling of open-channel flow, Journal of 

Hydraulic Engineering, 130, 806-815, 2004. 

[2] Kazezyılmaz-Alhan C.M., An improved solution for diffusion waves to overland flow, 

Applied Mathematical Modelling (in press), 2011. 

[3] Ahsan M., Numerical solution of the advection diffusion equation using Laplace transform 

finite analytical method, 3rd International Conference on Managing Rivers in the 21st Century: 

Sustainable Solutions for Global Crisis of Flooding, 204-215, 2011. 

[4] De Hoog F.R., Knight J.H. and Stokes A.N., An improved method for numerical inversion 

of Laplace transforms, SIAM, Journal of scientific and statistical computing, 3, 357-366, 1982. 

(1)

Applied Mathematics Analysis of the Multibody Systems 

H. Sahin 1 , A. Kerim Kar 2 and E. Tacgin 2 

1 Istanbul Ulasim A.S., Istanbul, Turkey 

2 Department of Mechanical Engineering, Faculty of Engineering, Marmara University, Istanbul, 

Turkey 


In this work, A methodology is developed for the analysis of the multibody systems that is 

applied on the vehicle as a case study. The previous study is emphasized on the derivation of the 

multibody dynamics equations of motion for bogie [see 2]. In this work, we have developed a 

guide-way for the analysis of the dynamics behavior of the multibody systems for mainly 

validation, verification of the realistic mathematical model and partly for the design of the 

alternative optimum vehicle parameters . 

Derivation of the DAEs 

Lagrange method is used with trajectory coordinate system as seen by equation 1. to derive 

generalized equation of motion for the differential algebraic equations [see 4]. These generalized 

equations programmed in the Matlab’s Symbolic Mathematics Toolbox. The size of the DAE’s 

are 44 for the bogie and about 156 for the whole railway vehicle. 

A methodology is developed for applied mathematics analysis of the multibody systems. 

This methodology can be used to compare with the symbolically derived DAEs of the motions 

with the previous studies for validation or the optimization of the vehicle dynamical parameters 

[see 1 and 3]. Case studies of the railway vehicle multibody mathematical model is tested for 

this methodology with a success. Although the most critical and influential symbolically varied 

parameter of the velocity is picked for the case study one can pick the rest of the other 

parameters such as mass, inertia or dimensions of the vehicle to design vehicle or mechatronic 

system for purposes such as stability (critical velocity for railway case) and comfort criteria. 

Keywords: Computational differential-algebraic equations (CDAEs), Multibody dynamics 

(MBD), Eigenvalue analysis, Lagrange dynamics, Railway vehicles. 

(1) 

Page 71

ACKNOWLEDGEMENTS 

Authors would like to acknowledge the financial support of the TUBITAK with the project 

numbered 110M561. Authors are grateful to the BTE Department of the TUBITAK MAM 

Research Institute for the permition to have the real and simulated data from the TRENSIM 

project completed for the Turkish State Railways (TCDD) for the E43000 Locomotive 

Simulator. 

References 

1. C. Smitke and P. Goossens, “Symbolic Computation Techniques for Multibody Model 

Development and Code Generation” in ECCOMAS Thematic Conference on Multibody 

Dynamics, Brussels, Belgium, 4-7 July 2011. 

2. H. Sahin , A. Kar, and E. Tacgin, "Analysis of the Differential-Algebraic Equations 

(DAEs) for Multibody Dynamics", Proceedings of the International Conference on Applied 

Analysis and Algebra, Istanbul, Turkiye, June 20-24, 2012. 

3. T. Kurz, P. Eberhard., & S. C. Henninger, From Neweul to Neweul-M2: symbolic 

equations of motion for multibody system analysis and synthesis. Multibody System Dynamics , 

2010, 25-41. 

4. Greenwood, D. T. Advanced Dynamics. Cambridge University (2003). 

Page 72

Multibody Railway Vehicle Dynamics Using Symbolic Mathematics 

H. Sahin 1 , A. Kerim Kar 2 and E. Tacgin 2 

1 Istanbul Ulasim A.S., Istanbul, Turkey 

2 Department of Mechanical Engineering, Faculty of Engineering, Marmara University, Istanbul, 

Turkey 


In this work, the Equations of Motion (EOMs) of the Multibody Dynamics is derived for a 

railway vehicle. The previous work of the authors is related to derive the Multibody Dynamics 

model of the bogie with 44 DAEs (see [1]). Lagrange dynamics is used as common approach in 

applied mathematics and mechanics for computational differential-algebraic equations (CDAEs). 

Differential equations of motions are formulized as in the generalized symbolic mathematics and 

applied in the Matlab’s MuPad Symbolic Math Toolbox. 

The size of the railway vehicle’s DAEs is about 156. Finally, the results are compared using 

eigenvalues with previous studies in the same area with a success. The symbolic mathematics is 

currently used for derivation of the multibody dynamics EOMs (see [2] and [4]). Langrange 

dynamics for the trajectory coordinate is applied to derive generalized EOMs for the multibody 

dynamics. Following Equation 1 is one of the generalized equation used to derive the state space 

representation of the EOMs for the railway vehicle (see [3]). 

Keywords: Computational differential-algebraic equations (CDAEs), Multibody dynamics (MBD), 

Eigenvalue analysis, Lagrange dynamics, Railway vehicles. 

ACKNOWLEDGEMENTS 

Authors would like to acknowledge the financial support of the TUBITAK with the project 

numbered 110M561. Authors are grateful to the BTE Department of the TUBITAK MAM 

Research Institute for the permition to have the real and simulated data from the TRENSIM 

project completed for the Turkish State Railways (TCDD) for the E43000 Locomotive 

Simulator. 

(1) 

Page 73

References 

Page 74 

[1] H. Sahin , A. Kar, and E. Tacgin, "Analysis of the Differential-Algebraic Equations (DAEs) 

for Multibody Dynamics", Proceedings of the International Conference on Applied Analysis and 

Algebra, Istanbul, Turkiye, June 20-24, 2012. 

[2] T. Kurz, P. Eberhard., & S. C. Henninger, From Neweul to Neweul-M2: symbolic equations 

of motion for multibody system analysis and synthesis. Multibody System Dynamics , 2010, 25- 

41. 

[3] Rieveley, R. J. The Effect of Direct Yaw Moment on Human Controlled Vehicle Systems. 

Dissertation . Windsor, Ontario, Canada: University of Windsor, 2010, May 19. 

[4] C. Smitke and P. Goossens, “Symbolic Computation Techniques for Multibody Model 

Development and Code Generation” in ECCOMAS Thematic Conference on Multibody 

Dynamics, Brussels, Belgium, 4-7 July 2011.

Existence of Global Solutions for a Multidimensional Boussinesq-Type Equation with 


Supercritical Initial Energy 

H. Taskesen 1 and N. Polat 1 

1 Department of Mathematics, Dicle University, Diyarbak¬r, Turkey 

In this work, global weak solutions of the multidimensional Boussinesq-type equation with power 

type nonlinearity juj p and supercritical initial energy is given by potential well method. Classical 

energy methods can not guarantee the global existence for this type of nonlinearity. As is known the 

functional de…ned for potential well method includes only the initial displacement, and by use of sign 

invariance of this functional one can only prove the global existence for critical and subcritical initial 

energy. In the case of supercritical initial energy such a functional fails to prove the global existence. A 

new functional is de…ned, which contains not only initial displacement, but also initial velocity. 

References 

[1] Y-Z. Wang, Y-X. Wang, Existence and nonexistence of global solutions for a class of nonlinear 

wave equations of higher order, 72 (2010) 4500-4507. 

[2] S. Wang, G. Xu, The Cauchy problem for the Rosenau equation, Nonlinear Anal. 71 (2009) 

456-466. 

[3] H. Taskesen, N. Polat, A. Erta¸s, On global solutions for the Cauchy problem of a Boussinesq-type 

equation, Abst. Appl. Anal. in press. 

[4] H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the 

form P u = Autt + F (u), TAMS 192 (1974) 1–21. 

Page 75

Dissipative Extensions of Fourth Order Di¤erential Operators in the Lim- 3 case 

Hüseyin Tuna 


Department of Mathematics, Mehmet Akif Ersoy University, Burdur, Turkey 

Extensions of symmetric operators arise in many areas of mathematical physics, like solvable models 

of quantum mechanics and quantization problems. Let us consider the scalar fourth order di¤erential 

operators generated by di¤erential expression 

where q (x) is a real continuous function in [0; 1): 

l (y) = y (4) + q (x) y; 0 x < +1 

In this paper, a space of boundary value is constructed for scalar fourth order di¤erential operators in 

the Lim 3 case. We describe all maximal dissipative, acretive, self adjoint and other extensions in terms 

of boundary conditions. 

References 

[1] Allahverdiev B. P., On extensions of symmetric Schrödinger operators with a matrix potential, 

Izvest. Ross. Akad. Nauk. Ser . Math. 59; (1995) ; 19 54; English transl. Izv. Math. 59; (1995) ; 45 62. 

[2] Bruk V.M , On a class of boundary –value problemswith a spectral parameter in the boundary 

conditions, Mat. Sb.,100; (1976) ; 210 216: 

442: 

[3] Calkin J. W., Abstract boundary conditions, Trans. Amer. Math. Soc.,Vol 45;No. 3; (1939) ; 369 

[4] Fulton C.T., Parametrization of Titchmarsh"s m ( ) functions in the limit circle case, Trans. 

Amer. Math. Soc. 229; (1977) ; 51 63: 

[5] Fulton C.T., The Bessel-squared equation in the lim-2, lim-3, and lim-4 cases, Quart. J. Math. 

Oxford (2); 40; (1989) ; 423 456. 

[6] Gorbachuk M. L., On spectral functions of a second order di¤erential operator with operator 

coe¢ cients, Ukrain. Mat. Zh. 18; (1966) ; no.2; 3 21; English transl: Amer. Math. Soc. Transl. Ser. 

II 72; (1968) ; 177 202: 

[7] Gorbachuk M. L., Gorbachuk V. I., Kochubei A.N., The theory of extensions of symmetric operators 

and boundary-value problems for di¤erential equations’, Ukrain. Mat. Zh. 41; (1989) ; 1299 1312; 

English transl. in Ukrainian Math. J. 41(1989); 1117 1129: 

[8] Gorbachuk M. L., Gorbachuk V. I., Boundary Value Problems for Operator Di¤erential Equations, 

Naukova Dumka, Kiev, 1984; English transl. 1991;Birkhauser Verlag. 

[9] Guse¼¬nov I. M. and Pashaev R. T., Description of selfadjoint extensions of a class of di¤erential 

operators of order 2n with defect indices (n + k; ; n + k); 0 < k < n; Izv.Akad.Nauk Azerb. Ser. Fiz. 

Tekh. Mat. Nauk, No.2; (1983) ; 15 19 (in Russian). 

Page 76

On the stability of the steady-state solutions of cell equations in a tumor growth model 

I. Atac and S. Pamuk 

Department of Mathematics, University of Kocaeli, Umuttepe Campus, 41380, Kocaeli - TURKEY 


In this study, we provide the stability analysis of the steady state solutions of endothelial, pericyte 

and macrophage cell equations in a mathematical model in tumor angiogenesis. We do this by studying 

phase plane analysis of the system of ordinary differential equations obtained from the cell equations. We 

also discuss the biological importance of the analysis in tumor angiogenesis. 

Keywords: Angiogenesis, Phase plane analysis, Tumor cells 

References 

[1] W.E. Boyce, R.C. DiPrima, Elemantary Differential Equations and Boundary Value Problems, John 

Wiley and Sons, Inc., USA, (1992). 

[2] L. Edelstein-Keshet, Mathematical Models in Biology, Random House, NY, (1988). 

[3] H.A. Levine, B.D. Sleeman, M. Nilsen-Hamilton, A mathematical model for the roles of pericytes 

and macrophages in the initiation of angiogenesis. I.The role of protease inhibitors in preventing 

angiogenesis, Math. Biosc. 168(2000) 77-115. 

[4] S. Pamuk, Qualitative analysis of a mathematical model for capillary formation in tumor angiogen- 

esis, Math. Models and Methods Apll. Sci. 13(1)(2003) 19-33. 

[5] S. Pamuk, A. Guven, Stability Analysis of the Steady-State Solution of a Mathematical Model in 

Tumor Angiogenesis, Global Analysis and Applied Mathematics, 729(2004) 369-373. 

[6] N. Paweletz, M. Knierim, Tumor-Related Angiogenesis, Critical Reviews in Oncology/Hematology, 

9(1989) 197-242. 

Page 77

The Cuttings Transport Modelling with Couette Flow 

Ishak Cumhur 

Department of Mathematics, Recep Tayyip Erdogan University, Rize, Turkey 


In the petroleum production, when an oil well is drilled, rock cuttings are transported up to 

the surface. As current mathematical models of the flow and transport neglect the effect of 

drillstring rotation, it is necessary to have a model that includes rotation effects. Predicting 

effective cuttings transport mechanism requires all of the parameters to be considered 

simultaneously. To beter understand the cuttings transport mechanism, a mechanistic model is 

used for cuttings in Couette and Poiseuille flow, as well as the helical flow being the 

superposition of Couette and Poiseuille flows. 

In this paper, we present the approximate solution of cuttings transport model(Couette 

Flow Model) being only direction of rotation by combining Modified Differential Transform 

Method and Adomian Decompositon. 

Couette flow velocity profile will be used in 

model equation 

instead of , where is fluid velocity at location, is dynamic viscosity, is particle 

size and (Kurzweg, 1995). Nondimensional parameters are 

and 

. 

References 

1. Alexandrou, A., Principles of Fluid Mechanics, Prentice Hall, New Jersey, 2001. 

2. Bolchover, P., Allwright, D., Coşkun, E., Jones, G. ve Ockendon, J., Cuttings Transport with 

Drillstring rotation. http://www.maths-in-industry.org/miis/135/1/ cuttings.pdf 12 Mayıs 2008 

3. Cumhur, İ., Cuttings Transport Model and Its Analysis in Annular Region between two Cylinders, 

PhD dissertation, Karadeniz Technical University, Trabzon, 2011. 

4. Çengel, Y., A. ve Cimbala, J., M., Akışanlar Mekaniği Temellleri ve Uygulamaları, Güven Bilimsel, 

İzmir, 2007. 

5. Kurzweg, U., H, Stokes’ Drag. http://www.mae.ufl.edu/~uhk/HOMEPAGE.html 29 Haziran 2009 

6. Papanastasiou, T., C., Georgiou, G. ve Alexandrou, A., N., Viscous Fluid Flow, CRC Press, Florida, 

2000. 

Page 78 

7. Venkatarangan, S., N. ve Rajalakshmi, K., A Modification of Adomian’s Solution for Nonlinear 

Oscillatory Systems, Computers Math. Applic., 6, 29 (1995) 67-73. 

8. Zhou, J., K., Differential Transformation and Its Application for Electrical Circuits, Huazhong 

University Press, Wuhan, China, 1986. 

9. Zhu, Y., Chang, Q. ve Wu, S., A New Algorithm for calculating Adomian Polynomials, Applied 

Mathematics and Computation, 169 (2005) 402-416.

On the Numerical Solution of Diffusion Problem with 

Singular Source Terms 

Irfan Turk ∗ and Maksat Ashyraliyev † 

∗ Department of Mathematics, Fatih University, Istanbul, Turkey, irfanturk@gmail.com 

† Department of Mathematics, Bahcesehir University, Istanbul, Turkey, maksat.ashyralyyev@bahcesehir.edu.tr 

Abstract. 

Partial differential equations with singular (point) source terms arise in many different scientific and engineering applications. 

Singular means that within the spatial domain the source is defined by a Dirac delta function. Our interest is this type 

of problems is motivated by mathematical modelling of forecasting and development of new gas reservoirs [1, 2, 3, 4]. 

Solutions of the problems having singular source terms generally have lack of smoothness, which is an obstacle for standard 

numerical methods. Therefore, solving these type of problems numerically requires a great deal of attention [5, 6, 7]. 

In this paper we discuss the numerical solution of initial-boundary value problem with singular source terms 

⎧ 

⎪⎨ 

⎪⎩ 

ut = D uxx + k1δ(x − a1)+k2δ(x − a2), 0 < x < 1, t > 0, 0 < a1 < a2 < 1, 

u(t,0) = uL, u(t,1) = uR, t ≥ 0, 

u(0,x) = ϕ(x), 0 ≤ x ≤ 1, 

where δ(x) is a Dirac delta function. We follow the standard finite volume approach based on the integral form of (1). We 

consider this approach more natural than the finite difference one directly based on the differential form, since for the integral 

form the treatment of the Dirac delta function expression is mathematically clear. For ease of presentation, we assume that 

there are only two source terms. The presented material is extendable to the case with more than two source terms. Finally, 

this study can be readily extended to the case with time-dependent source terms. 

Keywords: Diffusion Equation; Singular Source Terms; Finite Volume Method 

PACS: 02.30.Jr, 02.60.Cb, 02.60.Lj, 87.10.Ed 

REFERENCES 

Page 79 

1. S. N. Zakirov and V. I. Vasilyev, Forecasting and development of gas reservoirs, Nedra, Moscow, 1984 (in Russian). 

2. K. S. Basniev, A. M. Vlasov, I. N. Kochina and V. M. Maksimov, Underground Hydrodynamics, Nedra, Moscow, 1986 (in 

Russian). 

3. P. G. Bedrikovetskii, E. V. Manevich and R. Esedulaev, Fluid Dynamics 28 (2), 214–222 (1993). 

4. B. Annamukhamedov, N. V. Avramenko, K. S. Basniev, P. G. Bedrikovetskii, E. N. Dedinets and M. S. Muradov, Journal of 

Engineering Physics 58 (4), 475–483 (1990). 

5. J. Santos and P. de Oliveira, Journal of Computational and Applied Mathematics 111, 239–251 (1999). 

6. A. K. Tornberg and B. Engquist, Journal of Computational Physics 200, 462–488 (2004). 

7. M. Ashyraliyev, J. G. Blom and J. G. Verwer, Journal of Computational and Applied Mathematics 216, 20–38 (2008). 

8. W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer 

Series in Computational Mathematics, Vol. 33, Springer, Berlin, 2003. 

(1)

One Boundary-Value Problem Perturbed by Abstract Linear Operator 


K. Aydemir 1 and O. Sh. Mukhtarov 2 

1 Department of Mathematics, Gaziosmanpaa University, Tokat, Turkey 

2 Department of Mathematics, Gaziosmanpaa University, Tokat, Turkey 

The investigation of regular boundary value problems for which the eigenvalue parameter appears 

in both the ordinary differential equation and the boundary conditions originates from the Birkhoff’s 

work [3]. In recent years, more and more researchers are interested in the discontinuous Sturm-Liouville 

problems. Various physics applications of this kind of problems are found in many literatures (see [1], 

[2], [6]). The purpose of this paper is to study a Sturm-Liouville problem with discontinuities in the case 

when an eigenparameter appears not only in the differential equation but also in the boundary conditions. 

Morever, the ”differential equation” contained also an abstract linear operator (unbounded in general) 

in the Hilbert space L2(−1, 0) ⊕ L2(0, 1). We apply a different approach for investigation some spectral 

properties of this problem. 

References 

[1] Akdoan Z., Demirci M. and Mukhtrov O. Sh., Normalized Eigenfunction of Discontinnuous Sturm- 

Liouville Type Problem with Transmission Conditions,Applied Mathematical Sciences, Vol.1, no. 52, 

2573-2591, 2007. 

[2] Fulton C. T., Two-point boundary value problems with eigenvalue parameter contained in the 

boundary conditions, Proc. roy. soc. edin. , 77A, 293-308, 1977. 

[3]Birkhoff G. D. , On the asymptotic character of the solution of the certain linear differential 

equations containing parameter, Trans. Amer. Math. Soc., 9, p.219-231, 1908. 

[4] Mukhtarov O. Sh. and Kadakal M., Some spectral properties of one Sturm-Liouville type problem 

with discontinuous weight, Sib. Math. J., Vol. 46, 681-694, 2005. 

[5] Titchmarsh E. C., Eigenfunction expensions associated with second order diferential equations I, 

(2nd edn) London: Oxford Univ. Press., 1962. 

[6]Tikhonov, A. N. and Samarskii, A. A.,Partial Equation of Mathematical Physics , Vol 1, San 

Francisko, Translated from the Russian, Moscow, pp.380, 1962. 

Page 80

Using expanding method of (G ′ /G) to find the travelling wave solutions of nonlinear 

partial differential equations and solving mkdv equation by this method 

K. Nojoomi 1 , M. Mahmoudi 2 and A. Rahmani 1 

1 Department of Mathematics, SheikhBahaee University, Esfahan, Iran 2 Department of Mathematical 


Finance, SheikhBahaee University,Esfahan, Iran 

Expanding method of (G ′ /G) can be implemented to find survey solutions of travelling wave of 

some nonlinear partial differential equations. The answers depend on hyperbolic functions, trigonometric 

functions and rational functions (see [1]). 

This method, converts nonlinear partial differential equations into a plane differential equation. It is 

possible to use this method to solve integrable equations and non-integrable equations. In this paper, by 

describing the method we analysis the application of it to solve mkdv equation (see [2-3]). 

Phenomenon in physics and other fields are often described by nonlinear partial differential equations. 

During 40 years ago, finding survey solutions of nonlinear partial differential equations by implementing 

different methods have been the target of many researchers and the powerful methods of inverse diffusion 

method, homogeneous equilibrium,.expanding method of (G ′ /G) are proposed that are based on assump- 

tions that the solutions of travelling wave of nonlinear partial differential equations can be expressed 

in (G ′ /G) by polynomial and G = G(ξ)is correct in second order linear ordinary differential equations 

(LODE). The degree of polynomial can be obtained by balance between the highest-order derivative of 

the dependent variable in linear part of the differential equation with highest-order of dependent variable 

in nonlinear part that is appeared in ODE. 

Definitions and Basic preliminaries: 

1.Balance number 

balance number of m can be obtained by balance between the highest-order derivative of the dependent 

variable in linear part of the differential equation with highest-order of dependent variable in nonlinear 

part that is appeared in ODE. 

2.Explaining of (G ′ /G) Expanding Method 

We consider nonlinear differential with independent variable of x and t 

P (u, ut, ux, utt, uxt, uxx, ...) = 0 (1) 

Which u = u(x, t) is an unknown function, P is a polynomial in u = u(x, t) and has been its various 

partial derivative that include higher order derivative and nonlinear parts. 

3.Solving mkdv Method by (G ′ /G) Expanding Method 

In this section, we consider mkdv as the following 

We intend to find the solution of above traveling wave equation 

Page 81 

ut − u 2 ux + δuxxx = 0 δ > 0 (2) 

u(x, t) = u(ξ) ξ = x − vt (3)

The speed of V will be determined later. 

In this paper, (G ′ /G) Expanding Method proposed by Wang, is used to solve mkdv method. It 

is clear that solving nonlinear partial differential equations needs suitable change of variable and after 

solving this kind of equation, we reach a solution. As we observed, by using (G ′ /G) Expanding Method, 

it is possible to solve these equations and have more solutions without considering specific change of 

variables. This method has various applications; as it is a direct and survey method to find travelling 

wave solutions of nonlinear partial differential equations and the outcome results can affect the future 

researches significantly. 

References 

[1] M.L. Wang, X.Z. Li, J.L. Zhang., The expansion method (G ′ /G) traveling wave solutions of 

nonlinear evolution equations in mathematical Physics, phys.Lett., A 372, 417-423, A 372 2008. 

[2] F.Calogero,W.Eckhaus., Nonlinear evolution equations, rescalings,model PDES and their integra- 

bility., I.Inv Probl.3, 229-262, 1987. 

[3] F. Calogero, The evolution partial differential equation ut = uxxx + 3(uxxu 2 + 3u 2 xu) + 3uxu 4 ., 

Math.Phys.28, 538-555, 1987. 

[4] A.R.Mohamed,S.E.Thlaat., Numerical treatment for the modified Burgers equation, Math Comput 

Simulat.90-98, 70 ,2005. 

[5] H.Wilhelmsson., Explosive instabilities of rection diffusion equation,Phsy., Rev. A 36, (1987) 

965-966, 202.2008. 

Page 82

Weighted Bernstein Inequality for Trigonometric Polynomials on a Part of The Period 


Mehmet Ali Aktürk 1 and Alexey Lukashov 1 


In this study we give a weighted Bernstein inequality for trigonometric polynomials on a part of the 

period. 

References 

[1] Bernstein S. N., Sur l’ordre de la meilleure approximation des functions continues par des polynômes 

de degré donné, Memoires de l’Académie Royale de Belgique, 4, 1–103, 1912. 

[2] Bernstein S. N., Collected Works: Vol. I, Constr. Theory of Functions, 1905-1930, English 

Translation, Atomic Energy Commision, Springfield, VA, 1958. 

[3] Riesz M., Eine trigonometrische Interpolations-formel und einige Ungleichungen für Polynome, 

Jahresbericht des Deutschen Mathematiker-Vereinigung, 23, 354–368, 1914. 

[4] Videnskii V.S., Extremal estimates for the derivative of a trigonometric polynomial on an interval 

shorter than its period, Soviet Math. Dokl., 1, 5-8, 1960. 

1925. 

[5] Borwein P., and Erdélyi T., Polynomials and Polynomial Inequalities, Springer: New York,1995. 

[6] Priwaloff I., Sur la convergence des séries trigonométriques conjuguées, Mat. Sb., (32)2, 357-363, 

[7] Lukashov A.L., Inequalities for derivatives of rational functions on several intervals, Izv. Math., 

68, 543–565, 2004. 

[8] Dubinin V. N. and Kalmykov S. I., A majoration principle for meromorphic functions, Mat. Sb., 

198(12), 37–46, 2007. 

[9] Kalmykov S. I., Majoration principles and some inequalities for polynomials and rational functions 

with prescribed poles, J. Math. Sci., 157(4), 2009. 

[10] Peherstorfer F., and Steinbauer R., Strong asymptotics of orthonormal polynomials with the aid 

of Greens function, SIAM J. Math. Anal., 32(2), 385-402, 2000. 

Page 83

MIXED PROBLEM FOR A DIFFERENTIAL EQUATION WITH INVOLUTION 


UNDER BOUNDARY CONDITIONS OF GENERAL FORM 

Sadybekov M.A. 1 , A.M.Sarsenbi 2 

1 Institute of Mathematics, Informatics and Mechanics, Kazakhstan 

2 South-Kazakhstan State University by M.Auezov, Almaty, Kazakhstan 

To solve the mixed problem for a partial differential equation with involution and a symmetric potential 

there was found an explicit analytical representation by the Fourier method. The problem was considered 

under general boundary conditions with constant coeffi cients by a space variable. At the same we used 

the methods for avoiding the termwise differentiation of a functional series and applying the minimal 

conditions on initial data of the problem. 

References 

[1] Andreev A.A., On the correctness of boundary value problems for some equations in partial deriv- 

atives with the Carleman shift, A.A.Andreev, Differential equations and their applications: Proceedings 

of the 2nd Int. workshop.- Samara, 2,5-18 pp, 1998. 

[2] Khromov A.P. ,Mixed problem for a differential equation with involution potential of special 

type, A.P.Khromov, Proceedings of Sarat. University. New series, - V. 10, - Mathematics. Mechanics. 

Informatics, iss, 4. -17 - 22 pp,2010. 

[3] Chernyatin A.V., Justification of the Fourier method in mixed problem for equations in partial 

derivatives, A.V.Chernyatin, 112 pp,1991. 

Page 84


An Application on Suborbital Graphs 

M. Besenk 1 , A.H. Deger 1 , and B.O. Guler 1 


In this paper, we investigate suborbital graphs for the action of the normalizer of Γ0 (N) in PSL(2, R), 

where N will be of the form 28p2 , p> 3 and p is a prime. In addition we give the conditions to be a 

� � 

forest for normalizer in the suborbital graph F . 

References 

∞, u 

2 8 p 2 

[1] Akbas M. and Singerman D., The Signature of the Normalizer of Γ0 (N), London Math. Soc. 

Lecture Note Ser., 77-78, 1992. 

[2] Akbas M., On Suborbital Graphs for the Modular Group, Bull. Lond. Math. Soc., 647-652, 2001. 

[3] Biggs N.L. and White A.T., Permutation Groups and Combinatorial Structures, London Math. 

Soc. Lecture Note Ser., Cambridge, 33. CUP, Cambridge, 1982. 

2006. 

[4] Keskin R., Suborbital Graphs for the Normalizer of Γ0 (m), European J. Combinatorics, 193-206, 

[5] Keskin R. and Demirtürk B., On Suborbital Graphs for the Normalizer of Γ0 (N), Electronic J. 

Combinatorics, 1-18, 2009. 

[6] Sims C.C., Graphs and Finite Permutation Groups, Math. Zeitschr., 76-86, 1967. 

[7] Conway J.H. and Norton S.P., Monstrous Moonshine, Bull. London Math. Soc., 308-339, 1979. 

[8] Jones G.A., Singerman D. and Wicks K., The Modular Group and Generalized Farey Graphs, 

London Math. Soc. Lecture Note Ser., 316-338, 1991. 

[9] Schoeneberg B., Elliptic Modular Functions, Springer, Berlin, 1974. 

Page 85 

[10] Tsukuzu T., Finite Groups and Finite Geometries, Cambridge University Press, Cambridge, 1982.


Cellular Automata Based Byte Error Correcting Codes over Finite Fields 

Mehmet E. Koroglu 1 , Irfan Siap 1 and Hasan Akin 2 

1 Department of Mathematics, Yildiz Technical University, Istanbul-Turkey 

2 Department of Mathematics, Education Faculty, Zirve University, Gaziantep, Turkey 

Reed-Solomon codes are very convenient for burst error correcting codes, but as the number of errors 

increase, the circuit structure for Reed-Solomon codes become very complex. The modular and regular 

structure of cellular automata can be constructed with VLSI economically. Therefore, in recent years, 

cellular automata have became an important tool for error correcting codes. For the first time cellular 

automata based byte error correcting codes analogous to extended Reed-Solomon codes over binary fields 

was studied by Chowdhury et al. in [1] and Bhaumik et al. improved that coding-decoding scheme in 

[2]. In this study cellular automata based double-byte error correcting codes are generalized from binary 

fields to primitive finite fields Zp. 

References 

[1] D. R. Chowdhury, I. Sen Gupta and P.P. Chaudhuri, CA-Based Byte Error- Correcting Code, 

IEEE Transaction on Computers 44 (3), 371-382, 1995. 

[2] J. Bhaumik, D. R. Chowdhury, and I. Chakrabarti, An Improved Double Byte Error Correcting 

Code Using Cellular Automata, In Proc. 8th Int. Conf. Cellular Automat for Res. Ind. (ACRI), LNCS 

5191, 463—470, 2008. 

Page 86

On The First Fundamental Theorem for Special Dual Orthogonal Group SO(2, D) And its 

Application to Dual Bezier Curves 

M.Incesu 1 , O. Gursoy 2 

1 Department of Mathematics Education, Mus Alparslan University, Mus, Turkey 2 Department of 


Mathematics,Maltepe University, Istanbul, Turkey 

Let D be set of dual numbers. In this work we study the first fundamental theorem for special dual 

orthogonal transformations group SO(n, D) in case of n = 2 . Then our getting results compared the 

special orthogonal transformations group SO(4, R) in R 4 because D 2 is isomorph to R 4 So we showed 

that the minimal conditions of the dual vectors are more less than minimal conditions of real vectors. 

References 

[1] H. Weyl, The Classical Groups Their Invariants and Representations,2 nd ed.,with suppl., Prince- 

ton, Princeton University Press, 1946. 

[2] H.H.Hacisalihoglu, Hareket GHeometrisi ve Kuaterniyonlar Teorisi, Gazi niversitesi Fen Edebiyat 

Fakltesi Yaynlar, Ankara 1983. 

[3] Dj. Khadjiev, An Application of the Invariant theory to the Differential Geometry of Curves, Fan, 

Tahkent, 1988. (in Russian) 

[4] Dj. Khadjiev, Some Questions in Theory of Vector Invariants, Math. USSR-Sbornic, 1,3 (1967), 

383-396. 

[5] M.Incesu, The Complete System of Point Invariants in the Similarity Geometry, Ph.D. Thesis, 

Karadeniz Technical University, Graduate School of Natural and Applied Sciences, 2008. 

[6] M. Incesu and O. Gursoy, The similarity Invariants of Bezier Curves and Surfaces, XX. th National 

Mathematics Symposium, Ataturk University, 03-06 September 2007, Erzurum. 

[7] I. Oren, Invariants of Points for the Orthogonal Group O(3, 1),Ph.D. Thesis, Karadeniz Technical 

University, Graduate School of Natural and Applied Sciences, 2007. 

[8] Y. Sagiroglu,Affine Diferential Invariants of Parametric Curves, Ph.D. Thesis, Karadeniz Technical 

University, Graduate School of Natural and Applied Sciences, 2002. 

[9] A. Schrijver, Tensor Subalgebras and First Fundamental Theorems in Invariant Theory, Journal 

of Algebra, 319 (2008),1305-1319. 

Page 87

On Euler’s differential method for continued fractions 

M. Jafari Shah Belaghi 1 and A. Bashirov 1 

1 Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Turkey (TRNC) 


A continued fraction is an expression of the form 

a0 + K ∞ [ ] 

bk 

k=1 = a0 + 

ak a1 + 

b1 

b2 

a2+ b 3 

a 3 +··· 

where a0, a1, a2, . . . and b1, b2, b3, . . . are two sequences of real or complex numbers. It is remarkable that 

rational and irrational numbers can be clearly distinguished by continued fractions. In the 19th century, 

theory of continued fractions was one of the most popular areas of investigation in mathematics. The 

great mathematicians such as Karl Jacobi, Oscar Perron, Charles Hermit, Karl Friderich Gauss, Augustin 

Cauchy, Thomas Stieltjes etc. have contributed to the theory [1]. 

[ ] 

k+t , which depends on the parameter −1 < t < ∞. 

We study the continued fraction f(t) = K ∞ k=1 

k 

This continued fraction was studied by Euler. Using the Euler’s differential method, which was not used 

by mathematicians for a long time, we derive a new formula 

f(t) = 

∫ 1 

dp 

(1 − x)p−t 

0 

∫ 1 

0 (1 − x)p−t dp dxp (xtex ) dx 

where p = 0, 1, 2, . . . . For the integer values t = p = 1, 2, . . . , 

where 

dxp (xt+1ex ) dx 

, p − 1 < t ≤ p + 1, 

f(p) = (p + 1) 

∑p−1 ap,k 

k=0 p−k+1 

∑p−1 k=0 ap,k 

⎛ 

ap,k = ⎝ p 

⎞ 

⎠ 

1 

· 

k (p − k − 1)! . 

Previously, it was proved by Euler that f(0) = (e − 1) −1 . Using numerical methods it is found that the 

function σ(t) = √ t − f(t) is slowly increasing and limt→∞ σ(t) = 0.25. 

References 

[1] Khrushchev, S.V., Orthogonal Polynomials and Continued Fractions from Euler’s Point of View, 

Encyclopedia of Mathematics and Its Applications, Vol. 122., Cambridge University Press 2008. 

, 

, 

Page 88


Almost Convergence and Generalized Weighted 

M. Kirişçi 

1 Hasan Ali Yücel Faculty of Education, Istanbul University, Istanbul, Turkey 

In this paper, we investigate some new sequence spaces which naturally emerge from the 

concepts of almost convergence and generalized weighted mean. The object of this paper is to 

introduce to the new sequence spaces obtained as the matrix domain of generalized weighted 

mean in the spaces of almost null and almost convergent sequences. Furthermore, the beta and 

gamma dual spaces of the new spaces are determined and some classes of matrix transformations 

are characterized. 

References 

Page 89 

[1] B. Altay, F. Başar, Some paranormed sequence spaces of non-absolute type derived by 

weighted mean, J.Math. Anal. Appl. 319(2)(2006), 494-508. 

[2] F. Başar, M. Kirişçi, Almost convergence and generalized difference matrix, Comput. Math. 

Appl. 61(3)(2011), 602-611. 

[3] M. Candan, Almost convergence and double sequential band matrix, under communication. 

[4] A.M. Jarrah, E. Malkowsky, BK spaces, bases and linear operators, Rendiconti Circ. Mat. 

Palermo II 52(1990), 177-191. 

[5] K. Kayaduman, M. Şengönül, The spaces of Cesàro almost convergent sequences and core 

theorems, Acta Math. Sci. in press. 

[6] H. K zmaz, On certain sequence spaces, Canad. Math. Bull. 24(2)(1981), 169-176. 

[7] M. Kirişçi, F. Başar, Some new sequence spaces derived by the domain of generalized 

difference matrix, Comput. Math. Appl. 60(5)(2010), 1299-1309. 

[8] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80(1948), 

167-190. 

[9] E. Malkowsky, E. Savaş, Matrix transformations between sequence spaces of generalized 

weighted means, Appl. Math. Comput. 147(2004), 333-345. 

[10] A. Sönmez, Some new sequence spaces derived by the domain of the triple band matrix, 

Comput. Mat. Appl. 62(2)(2011), 641-650. 

[11] A. Sönmez, Almost convergence and triple band matrix, Math. Comput. Model. in press.

Wavelet-based prediction of crude oil prices 

M. Mahmoudi 1 , K. Nojoomi 2 and A. Rahmani 2 

1 Department of Mathematical Finance, SheikhBahaee University, Esfahan, Iran 2 Department of 


Mathematics, SheikhBahaee University,Esfahan, Iran 

There are two kinds of transactions in the crude oil markets; one is based on immediate delivery while 

the other one on future delivery. The spot market is dependent on the first kind of transactions and the 

future market is associated to the second one. Market condition ( e.g. market risk, irrational trading, 

etc. ) along with other factors ( e.g. credit risk, insurance risk, seasonal factors and etc. ) is often the 

main cause of uncertainty in the crude oil markets. Therefore, the future markets ( leading markets ) are 

built up to provide a cover structure for these uncertainties. Also crude oil future contracts, determine 

definitive prices in future deadlines to buy or sell according to specific criteria of delivery and payment. 

On the other hand, future prices reflects the markets expectations about future conditions. Consequently, 

large differences between futures and spot prices is often used to describe the overall market conditions. 

Wavelets are used as a legitimate alternative alternative for irregular situations such as data or signals 

with scaled features, or containing discontinuities and sharp edges and so on (see [1-2]). 

In this study, we are going to use the wavelets as a suitable tool to investigate its performance in the 

crude oil futures markets (see [3]). We intend to provide forecasts over different forecasting horizons by 

introducing a prediction procedure and predicting future prices based on the wavelets by utilizing a series 

of data from the crude oil market and at last putting the results in comparison with the crude oil future 

markets data. Definitions and Basic preliminaries 

1.Multi-scale analysis 

multi-scale analysis with a sequence of involute sub-space Vj of functional space of procedure V with 

null common point and at dense in L2(R) . This analysis is a discretion at different levels of scalability, 

which requires two-scale relationship such as f(x) ∈ Vj ⇐⇒ f(2x) ∈ Vj−1 (see [4-5]). 

2.discrete wavelet transform(DWT) 

discrete wavelet transformation enables us to discrete a time based sequence to subsequences with 

different scales in order to extract important hidden information and unstable features (see [4-5]). 

We present a procedure to predict crude oil prices for time series of 1, 2, 3 and 4 month and then 

compare the predicted values with actual expected prices of future market in mentioned time series and 

as for 1 month time series the result are shown in below figure: 

Page 90

Forecast results in contrast with observed values 

Forecasting horizon Wavelet-based forecast Futures 

1 month ahead 0.992 0.952 

2 months ahead 0.998 0.903 



And as you can see in below figure, wavelet based prediction procedure is more efficient for sample with 

a value bigger than 100 . 

Applicable procedure will be created by some main key properties of wavelets and is established based 

on discrete wavelet transformation (DWT) on Average monthly time series of crude oil. Wavelet based 

prediction procedure which is used in this study, can be applied to examination of the dynamic properties 

of various financial and economical phenomenon, like economic time series. also predicted crude oil prices 

based on wavelets can be used to determine oil prices in future contracts. 

References 

[1] Cao L, Hong Y, Zhao H, Deng S., Predicting economic time series using a nonlinear deterministic 

technique, Comput., Econom 9(2):14978. 1996 

[2] Ramsey J.B., Wavelets in economics and finance: past and future, Stud. Nonlinear Dynam., 

Economet 6(3):127. 2002. 

[3] Shahriar Yousefi, Ilona Weinreich, Dominik Reinarz., Wavelet-based prediction of crude oil prices., 

Chaos, Solitons and Fractals 265-275, 25(2005). 

[4] Albert Boggess, Francis J. Narcowich., A First Course in Wavelets with fourier analysis, Prentice 

Hall, 2001. 

[5] Donald B. Percival, Andrew T. Walden., Wavelet Methods for Time Series Analysis, Cambridge 

University Press, 2006. 

Page 91

Numerical solution of a time-fractional Navier–Stokes Equation with modified 

Riemann-Liouville derivative 

Mehmet Merdan 1 , Ahmet Gökdoğan 1 

1 Gümüşhane University, Department of Mathematical Engineering, 

29100-Gümüşhane, Turkey 


In this paper, fractional variational iteration method (FVIM) is implemented to give an approximate 

analytical solution of a time-fractional Navier–Stokes Equation. Fractional derivatives are described in 

the Riemann-Liouville derivative. A new application of fractional variational iteration method (FVIM) 

was extended to derive analytical solutions in the form of a series for these equations. By using an 

initial value, the explicit solution of the equation has been presented in the closed form and then its 

numerical solution has been showed graphically.The behavior of the solutions and the effects of 

different values of fractional order � are indicated graphically. The results obtained by the FVIM 

reveal that the method is performs extremely well in terms of efficiency and simplicity method for 

nonlinear differential equations with modified Riemann-Liouville derivative. 

Keywords: Fractional variational iteration method, A time-fractional Navier–Stokes 

Equation, Riemann-Liouville derivative, Fractional calculus 

References 

Page 92 

[1] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. 

[2] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 

[3] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional 

Differential Equations, Elsevier, Amsterdam, 2006. 

[4] Podlubny I (1999). Fractional Differential Equations, Academic Press, San Diego. 

[5] Caputo M (1967). Linear models of dissipation whose Q is almost frequency independent, 

Part II, J. Roy. Astr. Soc., 13: 529. 

[6] A. A. Kilbas, H. H. Srivastava, J. J. Trujillo, Theoryand Applications of Fractional 

Differential Equations, Elsevier, TheNetherlands, 2006. 

[8] Miller KS, Ross B (1993). An Introduction to the Fractional Calculus and Fractional 

Differential Equations, Wiley, New York. 

[9] Samko SG, Kilbas AA, Marichev OI (1993). Fractional Integrals and Derivatives: Theory 

and Applications, Gordon and Breach, Yverdon. 

[10] G.M.Zaslavsky, Hamiltonian Chaosand Fractional Dynamics, Oxford University Press, 

2005. 

[11] M.Merdan, A.Yıldırım, A.Gökdoğan, Numerical solution of time-fraction Modified 

Equal Width Wave Equation, Engineering Computations, 2011 (in press) 

[12] Merdan M., Solutions of time-fractional reaction-diffusion equation with modified 

Riemann-Liouville derivative, International Journal of Physical Sciences . 7(15), pp. 

2317 - 2326 (2012). 

[13] Merdan M., Mohyud-Din S.T., A New Method for Time-fractionel Coupled-KDV 

Equations with Modified Riemann-Liouville Derivative, Studies in Nonlinear Sciences, 2 

(2), pp. 77-86 (2011). 

[14] Merdan M., Gökdoğan A., Yıldırım., Mohyud-Din S.T., Numerical simulation of 

fractional Fornberg-Whitham equation by differential transformation method, Abstract 

and Applied Analysis, Article ID 965367 (2012).

Page 93 

[15] M. El-Shahed, A. Salem, On the generalized Navier–Stokes equations, Appl. Math. 

Comput. 156 (1) (2004) 287–293. 

[16] S. Momani , Z. Odibat, Analytical solution of a time-fractional Navier–Stokes equation 

by Adomian decomposition method, Applied Mathematics and Computation 177 (2006) 

488–494. 

[17] Z. Z. Ganji, D. D. Ganji, Ammar D. Ganji, M. Rostamian, Analytical Solution of Time- 

Fractional Navier–Stokes Equation in Polar Coordinate by Homotopy Perturbation 

Method, Numer Methods Partial Differential Eq 26: 117–124, 2010 

[18] J.H. He, Variational iteration method- a kind of non-linear analytical technique: Some 

examples, Int. J. Nonlinear Mech. 34 (1999) 699-708. 

[19] J.H. He, X.H. Wu, Variational iteration method: New development and applications, 

Comput. Math. Appl. 54 (7-8) (2007) 881-894. 

[20] J.H. He, Some applications of nonlinear fractional differential equations and their 

approximations, Bull. Sci. Technol. 15 (2) (1999) 86-90. 

[21] G. Jumarie, Stochastic differential equations with fractional Brownian motion input. Int. 

J. Syst. Sci. 6, (1993), 1113–1132. 

[22] G. Jumarie, 2006. New stochastic fractional models for Malthusian growth, the 

Poissonian birth process and optimal management of populations. Math. Comput. Model. 

44, (2006) 231–254. 

[23] G. Jumarie, Laplace’s transform of fractional order via the Mittag–Leffler function and 

modified Riemann–Liouville derivative, Applied Mathematics Letters 22 (2009) 1659- 

1664. 

[24] G. Jumarie, 2009. Table of some basic fractional calculus formulae derived from a 

modified Riemann–Liouvillie derivative for non differentiable functions. Applied 

Mathematics Letters 22 (2009) 378-385. 

[25] G. Jumarie, On the solution of the stochastic differential equation of exponential growth 

driven by fractional Brownian motion, Applied Mathematics Letters 18 (2005) 817–826. 

[26] M.-J. Jang, C.-L. Chen, and Y.-C. Liu, Two-dimensional differential transform for partial 

differential equations, Applied Mathematics and Computation, vol. 121, no. 2-3, pp. 261– 

270, 2001. 

[27] Faraz N., Khan Y., Jafari H., Yildirim A., Madani M., Fractional variational iteration 

method via modified Riemann–Liouville derivative, J. King. Saud. Univ.(Science) 23, 

pp.413-417 (2011).

The Modified Simple Equation Method for Solving Some Nonlinear Evolution 

Equations 


M.Mızrak and A.Ertaş 

Department of Mathematics, Dicle University, Diyarbakır, Turkey 

In this paper we applied modified simple equation method (MSEM) for solving some 

nonlinear evolution equations which are very important in applied sciences. 

We consider a nonlinear evolution equation: 

( ) 

F u, u , u , u ,... = 0 

(1) 

t x xx 

where F is a polynomial in u and its partial derivatives. 

Step 1. Using the wave transformation 

u = u( ξ), ξ = x− t 

(2) 

From (1) and (2) we have the following ODE: 

' '' ''' ( ) 

P u, u , u , u ,... = 0 

(3) 

where P is a polynomial in u and its total derivatives and ' 

Step 2. We suppose that Eq. (3) has the formal solution: 

( ) 

( ) 

N ⎛ψ′ ξ ⎞ 

u( ξ ) = ∑ Ak ⎜ ⎟ 

k= 

0 ψ ξ ⎟ 

⎝ ⎠ 

k 

d 

= . 

dξ 

where Ak are arbitrary constants to be determined such that AN ≠ 0 while ψ ( ξ ) is an unknown 

function to be determined later. 

Page 94 

Step 3. We determine the positive integer N in (4) by balancing the highest order derivatives 

and the nonlinear terms in Eq. (3). 

Step 4. We substitute (4) into (3), we calculate all the necessary derivatives u′ , u′′ 

,.. . and then we 

ψ ′ ( ξ ) 

account the function ψ ( ξ ) . As a result of this substitution, we get a polynomial of and 

ψ ξ 

its derivatives. In this polynomial, we equate with zero all the coefficients of it. This operation 

( ) 

(4)

yields a system of equations which can be solved to find and ψ ξ . Consequently, we can 

get the exact solution of Eq. (1). 

References 

A ( ) 

[1] Murray J.D, Mathematical Biology I, Springer-Verlag New York, USA , 2002 

[2] Hereman W. and Nuseir A, Symbolic methods to construct exact solutions of nonlinear partial 

differential equations. 

[3] Jawad A.J.M., Petkovic M. D., Biswas A., 2010 Modified simple equation method for nonlinear 

evolution equations Applied Mathematics and Computation 217 869-877, 2010. 

[4] Zayed E. M. E., A note on the modified simple equation method applied to Sharma–Tasso–Olver 

equation , Applied Mathematics and Computation 218 3962–3964, 2011 

k 

Page 95

Application of Cross Efficiency in Stock Exchange 

Mozhgan Mansouri Kaleibar a and Sahand Daneshvar b 

a Young Researchers Club, Tabriz Branch, Islamic Azad University, Tabriz,Iran 

Email: Mozhganmansouri953@gmail.com 

b Tabriz Branch, Islamic Azad University, Tabriz, Iran 

Email: Sahanddaneshvar@yahoo.com 


This paper firstly revisits the cross efficiency evaluation method which is an extension 

tool of data (envelopment analysis. In this paper, we consider the DMUs as the 

players (institutions) in a cooperative game, where the characteristic function values 

of institutions are defined to compute the Shapley value of each DMU (institution), 

and the common weights associate with the imputation of the Shapley values are used 

to determine the ultimate cross efficiency scores for institution of Stock Exchange of 

Tehran. This paper introduces the models for computing benefit for each institution. 

Using shapely value we obtain the effect of each institution, and through determining 

common weight for each company we find out the ultimate weight which shows 

how much the existence or not existence of that institution affects the interesting 

competence. 

Keywords: Data Envelopment Analysis (DEA), Cross efficiency, Cooperative game, Shapley 

value, Common weights, stock exchange. 

References 

[1] G. Owen, On the Core of Linear Production Games, Mathematical Programming, 

1975, No. 9, 358- 370. 

[2] J. Wu, L. Liang and F. Ynag, Determination of The Weights for The Ultimate Cross 

Efficiency Using Shapley Value in Cooperative Game, Expert Systems with Applications, 

2009, No. 36, 872-876. 

[3] K. Nakabayashi and K. Tone, Egoist’s Dilemma: a DEA Game, The International 

Journal of Management Science, 2006, No. 36, 135-148. 

[4] S. Daneshvar and M. Mansouri Kaleibar, The Minimal Allocated Cost and Maximal 

Allocated Benefit, Presented at the Int Conf. Engineering System Management and 

Application Sharjah, UAE, 2010. 

[5] W. Cooper, L.M. Seiford and K. Tone, Data Envelopment Analysis, Boston: Klawer 

Academic publishers, 2000. 

1 

Page 96


Application of the Trial Equation Method for some Nonlinear 

Evolution Equations 

M. Odabasi 1, 2 and E. Misirli 2 

1 Tire Kutsan Vocational School, Ege University, Izmir, Turkey 

2 Department of Mathematics, Ege University, Izmir, Turkey 

Nonlinear partial differential equations have important applications in physics, engineering and 

applied mathematics. Mathematical modelling of physics and engineering problems usually 

results in nonlinear partial differential equations. To find the travelling wave solutions of 

nonlinear evolution equations several methods [1-6] have been proposed. This study presents an 

application of the trial equation method for nonlinear partial differential equations. The trial 

equation method is used to obtain exact travelling wave solutions of some nonlinear evolution 

equations arising in mathematical physics. 

References 

Page 97 

[1] Ablowitz M.J. and Clarkson P.A., Solitons, Nonlinear Evolutions and Inverse Scattering, 

Cambridge University Press, Cambridge, 1991. 

[2] Oliver P.J., Applications of Lie Group to Differential Equations. Springer, New York, 1993. 

[3] Malfliet W., Solitary wave solutions of nonlinear wave equations, 

American Journal of Physics, Vol. 60 (7), 650–654, 1992. 

[4] Liu C.S., A New Trial Equation Method and Its Applications, Communications in 

Theoretical Physics, 45, 395–397, 2006. 

[5] Du X.H., An irrational trial equation method and its applications, Pramana Journal of 

Physics, Vol. 75 (3), 415–422, 2010. 

[6] Gurefe Y., Sönmezoğlu A. and Mısırlı E., Application of the trial equation method for 

solving some nonlinear evolution equations arising in mathematical physics, Pramana 

Journal of Physics, Vol. 77 (6), 1023–1029, 2011.

Paranormality of Some Class of Differential Operators for First Order 

M. Sertba¸s 1 , L. Cona 2 


2 Department of Mathematical Engineering, Gumushane University, Gumushane, Turkey 


In this work, the paranormality properties of some class direct sum of differential operators for first 

order in the Hilbert space of vector-functions in the finite interval are investigated. Finally, a spectrum 

of these operators is researched. 

Keywords: Selfadjoint and Paranormal Operator; Direct Sum of Operators and Hilbert Spaces; Spec- 

trum. 

2000 AMS Classification: 47A20, 47A10 

References 

[1] Fruta T., On the Class of Paranormal Operators, Proc. Japan Acad. 43, 594-598, 1967. 

[2] Jablonski Z.J. and Stochel J., Unbounded 2-Hyperexpansive Operators, Proceedings of the Edinburgh 

Mathematical Society, 44, 613-629, 2001 

[3] Ismailov Z.I., Otkun Çevik E., Unluyol E., Compact Inverses of Multipoint Normal Differential Op- 

erators for First Order, Electronic Journal of Differential Equations, 89, 1-11, 2011. 

Page 98

Oscillation Theorems for Second-Order Damped Dynamic Equation on Time Scales 

M. Tamer S¸enel 

Department of Mathematics, Faculty of Science, Erciyes University, 38039, Kayseri , TURKEY 


Much recent attention has been given to dynamic equations on time scales, or measure chains, and 

we refer the reader to the landmark paper of S. Hilger [1] for a comprehensive treatment of the subject. 

A book on the subject of time scales by Bohner and Peterson [2] also summarizes and organizes much of 

the time scale calculus. 

In this paper we shall study the oscillations of the following nonlinear second-order dynamic equations 

with damping 

(r(t)Ψ(x ∆ (t)) ∆ + p(t)Ψ(x ∆ (t)) + q(t)f(x σ (t)) = 0, t ∈ T, (1) 

where Ψ(t), f(t), p(t), q(t) and r(t) are rd-continuous functions. By using a generalized Riccati transfor- 

mation and integral averaging technique, we establish some new sufficient conditions which ensure that 

every solution of this equation oscillates. Throughout this paper, we will assume the following hypotheses: 

(H1) p(t), q(t) ∈ Crd(R, R + ), 

(H2) Ψ : T → R is such that Ψ 2 (u) ≤ κuΨ(u) for κ > 0, u �= 0, 

(H3) f : R → R is such that f(u) 

u ≥ λ > 0, and uf(u) > 0, u �= 0, 

(H4) r(t) ∈ C1 rd ([t0, ∞), R + ), � ∞ 1 ( 

References 

t0 

r(t) 

e −p(t) (t, t0))∆t = ∞. 

r(t) 

[1] S. Hilger, Analysis on measure chains A unified approach to continuous and discrete calculus, 

Results Math., 18 , 18-56, 1990. 

[2] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, 

Birkhäuser, Boston, 2001. 

[3] Samir H. Saker, Ravi P. Agarwal, Donal O’Regan, Oscillation of second-order damped dynamic 

equations on time scales, J.Math.Anal. and App., 330, 1317-1337, 2007. 

[4] Taher S. Hassan, Lynn Erbe, Allan Peterson, Oscillation Theorems of Second Order Superlinear 

Dynamic Equations with Damping on Time Scales, Com. Math. Appl., 59, 550-558, 2010. 

[5] M. T. S¸enel, Oscillation theorems for dynamic equation on time scales, Bull. Math. Anal. Appl., 

3, no.4, 101-105, 2011. 

[6] M. T. S¸enel, Kamenev-Type Oscillation Criteria for the Second-Order Nonlinear Dynamic Equa- 

tions with Damping on Time Scales, Abstract and Applied Analysis, Vol. 2012, Article ID 253107, 18 

pages, doi:10.1155/2012/253107. 

This work was supported by Research Fund of the Erciyes University. Project Number:FBA-11-3391 

Page 99

On the fine spectrum of the $\Lambda$ operator defined by a lambda matrix 

over the sequence space $c_{0}$ and $c$ 


M. Yeşilkayagil 1 and F. Başar 2 

1 Department of Mathematics, Uşak University, Uşak, Turkey 


The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg's 

classification of the operator $\Lambda$ defined by a lambda matrix over the sequence spaces 

$c_{0}$ and $c$. As a new development, we give the approximate point spectrum, defect 

spectrum and compression spectrum of the matrix operator $\Lambda$ on the sequence spaces 

$c_{0}$ and $c$. 

References 

Page 100 

1.A.M. Akhmedov, F. Başar, On the fine spectrum of the Ces\`{a}ro operator in $c_0$}, Math. J. 

Ibaraki Univ.(36)(2004), 25-32. 

2.B. Altay, F. Başar, On the fine spectrum of the generalized difference operator $B(r,s)$ over 

the sequence spaces $c_0$ and $c$}, Int. J. Math. Math. Sci. 2005:(18) (2005), 3005-3013. 

3.B. Altay, F. Başar, On the fine spectrum of the difference operator $\Delta$ on $c_0$ and $c$, 

Inform. Sci. (168)(2004), 217-224. 

4.B. Altay, M. Karakuş, On the spectrum and fine spectrum of the Zweier matrix as an operator 

on some sequence spaces, Thai J. Math. (3)(2005), 153-162. 

5.J. Appell, E. Pascale, A. Vignoli, Nonlinear Spectral Theory, de Gruyter Series in Nonlinear 

Analysis and Applications 10, Walter de Gruyter, Berlin, New York, 2004. 

6.S. Goldberg, Unbounded Linear Operators, Dover Publications, Inc. New York, 1985. 

7.V. Karakaya, M. Altun, Fine spectra of upper triangular double-band matrices, J. Comput. 

Appl. Math. (234)(2010), 1387-1394. 

8.E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley \& Sons Inc. 

New York, Chichester, Brisbane, Toronto, 1978. 

9. J.B. Reade, On the spectrum of the Cesaro operator, Bull. Lond. Math. Soc. (17)(1985), 263- 

267. 

10. R.B. Wenger, The fine spectra of H\"older summability operators, Indian J. Pure Appl. Math. 

(6)(1975), 695-712.


On Hadamard Type Integral Inequalities For Nonconvex Functions 

Mehmet Zeki Sarikaya 1 , Hakan Bozkurt 1 and Necmettin Alp 1 

1 Department of Mathematics, Duzce University, Duzce, Turkey 

Convexity plays a central and fundamental role in mathematical …nance, economics, engineering, 

management sciences and optimizastion theory. In recent years, several extensions and generalizations 

have been considered for classical convexity. A signi…cant generalization of convex functions is that of 

'-convex functions introduced by Noor in [3]. In [3] and [6], the authors have studied the basic properties 

of the '-convex functions. It is well-know that the '-convex functions and '-sets may not be convex 

functions and convex sets. This class of nonconvex functions include the classical convex functions and 

its various classes as special cases. For some recent results related to this nonconvex functions, see the 

papers [3]-[6]. In this article, using functions whose derivatives absolute values are '-convex and quasi-'- 

convex, we obtained new inequalities releted to the right and left side of Hermite-Hadamard inequality. 

In particular if ' = 0 is taken as, our results obtained reduce to the Hermite-Hadamard type inequality 

for classical convex functions. 

References 

[1] S.S. Dragomir and R.P. Agarwal, Two inequalities for di¤erentiable mappings and applications to 

special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95. 

[2] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and 

Applications, RGMIA Monographs, Victoria University, 2000. 

171 

[3] M. Aslam Noor, Some new classes of nonconvex functions, Nonl.Funct.Anal.Appl.,11(2006),165- 

[4] M. Aslam Noor, On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. 

Pure Appl. Math., 8(2007), No. 3, 1-6, Article 75. 

[5] M. Aslam Noor, Hermite-Hadamard integral inequalities for log-' convex functions, Nonl. Anal. 

Forum, (2009). 

31-42. 

[6]M. Aslam Noor, On a class of general variotional inequalities, J. Adv. Math. Studies, 1(2008), 

[7] K. Inayat Noor and M. Aslam Noor, Relaxed strongly nonconvex functions, Appl. Math. E-Notes, 

6(2006), 259-267. 

[8] U.S. K¬rmac¬, Inequalities for di¤erentiable mappings and applications to special means of real 

numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146. 

[9] U.S. K¬rmac¬and M.E. Özdemir, On some inequalities for di¤erentiable mappings and applications 

to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153, (2004), 361-368. 

[10] U.S. K¬rmac¬, Improvement and further generalization of inequalities for di¤erentiable mappings 

and applications, Computers and Math. with Appl., 55 (2008), 485-493. 

[11] D.A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, 

Annals of University of Craiova Math. Comp. Sci. Ser., 34 (2007) 82-87. 

[12]C.E.M. Pearce and J. Peµcarić, Inequalities for di¤erentiable mappings with application to special 

means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51–55. 

Page 101

[13] J. Peµcarić, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical appli- 

cations, Academic Press, New York, 1991. 

[14] M. Z. Sarikaya, A. Saglam and H. Y¬ld¬r¬m, New inequalities of Hermite-Hadamard type for 

functions whose second derivatives absolute values are convex and quasi-convex, International Journal of 

Open Problems in Computer Science and Mathematics ( IJOPCM), 5(3), 2012. 

[15] M. Z. Sarikaya, A. Saglam and H. Y¬ld¬r¬m, On some Hadamard-type inequalities for h-convex 

functions, Journal of Mathematical Inequalities, Volume 2, Number 3 (2008), 335-341. 

[16] M. Z. Sarikaya, M. Avci and H. Kavurmaci, On some inequalities of Hermite-Hadamard type for 

convex functions, ICMS Iternational Conference on Mathematical Science. AIP Conference Proceedings 

1309, 852 (2010). 

[17] M. Z. Sarikaya and N. Aktan, On the generalization some integral inequalities and their applica- 

tions Mathematical and Computer Modelling, Volume 54, Issues 9-10, November 2011, Pages 2175-2182. 

[18] M. Z. Sarikaya, E. Set and M. E. Ozdemir, On some new inequalities of Hadamard type involving 

h-convex functions, Acta Mathematica Universitatis Comenianae, Vol. LXXIX, 2(2010), pp. 265-272. 

[19] A. Saglam, M. Z. Sarikaya and H. Yildirim, Some new inequalities of Hermite-Hadamard’s type, 

Kyungpook Mathematical Journal, 50(2010), 399-410. 

Page 102

A geometrical approach of an optimal control problem governed by EDO 

Abstract: 

NEDJOUA DRIAI 

Department of Mathematics, Ferhat abbas University, Setif ,Algeria 

The theory of optimal control is a very important branch of optimization, the resolution of the 

problems controls optimal asks for the intervention of several mathematical tools, in particular the 

partial derivative equations. In this work one gives a geometrical approach of a problem of optimal 

control, it where one calls on the basic notions of the calculation of the variations such as the equation 

of Euler-Lagrange which is a requirement of optimality, the principle of maximum of Pontriagaine 

(PMP), which gives an analytical aspect to the problem controls optimal and makes it possible to study 

unquestionable property of the functions which defines the criterion to be minimized, the regularity of 

the solutions (minimum or maximum). An other very important aspect is well geometrical aspect which 

is used to find the geodetic ones, their natures, their numbers which requires a geometrical luggage 

such as the fields, of vector, the vector spaces, the curve acceptable… Then can about it defines a 

problem controls optimal controlled by EDO geometrically by giving some conditions. 

References: 

[1] NR Burq and P. Gerard; Optimal control of the partial derivative equations. 

[2] Ovidiu Calin Der-Chen Chang; Geometric Mechanics one Riemannian Manifolds; Birkhäuser 

Boston 2005 [3] L.C.Young; readings one the calculus of variations and optimal control theory; 

Chelsea publishing company; N.Y, 1980 

Page 103

Existence of Local Solution for a Double Dispersive Bad Boussinesq-Type Equation 

N. Dündar 1 , N. Polat 1 


1 Department of Mathematics, Dicle University, Diyarbak¬r, Turkey 

In this work, we consider a purely spatial higher order bad Boussinesq-type equation. We obtain the 

existence and uniqueness of the local solutions. The local existence of the solution is given by aid of 

contraction mapping principle. 

References 

[1] L.A. Ostrovskii, A.M Sutin, Nonlinear Waves in Rods, J. Appl. Math. Mech. 41 (1977) 543–549 

(English translation of P.M.M.). 

[2] C.I. Christov, G. A Maugin, On Boussinesq’s Paradigm in Nonlinear Wave Propagation, C. 

R.Mecanique 335 (2007) 521–535. 

[3] N. Polat, A. Erta¸s, Existence and Blow-up of Solution of Cauchy Problem for the Generalized- 

Damped Multidimensional Boussinesq Equation, J. Math. Anal. Appl. 349 (2009) 10-20. 

[4]Y. Liu, Existence and Blow up of a Nonlinear Pochhammer–Chree Equation, Indiana Univ. Math.J. 

45 (1996) 797–816. 

[5] T. Kato, G. Ponce, Commutator Estimates and the Euler and Navier–Stokes Equations, Comm. 

Pure Appl. Math. 41 (1988) 891–907. 

Page 104

A Perturbation Solution Procedure for a Boundary Layer Problem 

N. Elmas 1 , A. Ashyralyev 2 And H. Boyaci 1 

1 Department of Mechanical Engineering, Celal Bayar University 45140 Muradiye, Manisa, Turkey 

2 Department of Mathematics, Fatih University, 34500 Buyukçekmece, Istanbul, Turkey 


A perturbation algorithm using a new transformation is introduced for boundary-value problem with small 

parameter multiplying the derivative terms. To account for the linear and the nonlinear dependence of the 

function, we exhibit the function f for the system. We introduce the transformation f ( x, 

; ) x , where f 

depends on x, and . Results of Multiple Scales method, method of matched asymptotic expansions and our 

method are contrasted. 

We consider the following boundary-value problem 

y 

y( 

0) 

y 

0 

y 

2 

0 

y( 

1) 

Where is a small dimesionless positive number. It is assumed that the equation and boundary conditions have 

been made dimensionless. 

In Direct Perturbation Method, secular terms appear of higher orders of the expansion invalidating the solution. 

In order the avoid this problem a new transformation has been proposed in our study. 

The new transformation is defined as, 

T e 

f ( x, 

; 

Using the chain rule, we transform the derivate accordingly 

dy 

dx 

2 

d y 

2 

dx 

d 

2 

dT 

2 

e 

dy 

dT 

y 

e 

d 

dx 

dTe 

dx 

df 

dx 

dy 

dx 

x 

f 

dy 

dT 

2 

e 

d 

dx 

df 

dx 

dy 

dT 

e 

x 

dy 

dT 

e 

f 

df 

dx 

2 

d f 

2 

dx 

x 

x 

y'( 

f x 

f 

x 

df 

2 

dx 

f ) 

d 

dT 

e 

y'' 

( f 

So, we have obtained a more effective parameter expression T e without losing the original parameter 

x, and . Thus, speeding up and slowing down control of the time parameter will be available as in Method of 

Multiple Scales. 

In Equation (3) first order derivatives according to new variable T e appear in the second order derivative 

expressions according to original time variable x. So, we are able to have information about parameters of 

nonlinear differential equation and to interpret the results. 

) 

dy 

dT 

x 

e 

x 

x 

1 

2 

df 

dx 

f ) 

2 

x 

f 

y'( 

f 

xx 

dTe 

dx 

x 

2 f 

dy 

dT 

x 

) 

e 

d 

dx 

T e 

df 

dx 

x 

f 

Page 105 

(1) 

(2) 

(3)

By this new transformation we have the advantages of both method of matched asymptotic expansions and 

Method of Multi Scales [1-6]. 

Using perturbation algorithm with new transformation, a more effective time expression without losing the 

original time parameter t have been obtained. Information about parameters of nonlinear differential equation 

and interpretation of the results has been achieved. Applying this transformation on boundary value problem 

the results obtained are compared with the results of the studies conducted to time. 

References 

[1] H. Nayfeh, Introduction to Perturbation Techniques, John Wiley and Sons, New York 1981. 

[2] Nayfeh, A.H., Nayfeh S.A. and Mook, D.T., On Methods for continuous systems with quadratic and 

cubic nonlinearities, Nonlinear Dynamics, 3, 145-162, 1992 

[3] Nayfeh, A.H., 1998. Reduced order models of weakly nonlinear spatially continuous systems, Nonlinear 

Dynamics 16,105–125, 1998. 

[4] Ashyralyev A. and Sobolevskii P. E. New Difference Schemesfor Partial Differential Equations. 

Birkhauser Verlag: Basel. Boston. Berlin, 443 p,2004. 

[5] Erwin Kreyszig Advanced Engineering Mathematics, John Wiley & Sons, New York, 1993. 

[6] Perturbation Methods, Instability, Catastrophe and Chaos, C F Chan Man Fong and D D Kee, World 

Scientific Publishing, 1999 

Page 106

Solution of Differential Equations by Perturbation Technique Using any Time Transformation 


N. Elmas 1 and H. Boyaci 1 

1 Department of Mechanical Engineering, Celal Bayar University 45140 Muradiye, Manisa, Turkey 

A perturbation algorithm using any time transformation is introduced. To account for the nonlinear 

dependence of the function, we exhibit the function f of the system in the differential equation. To 

this end, we introduce the transformation f ( w, 

t) 

t , where f is a function that depends on t or w. 

T e 

The problems are solved with new time transformation: Linear damped vibration equation, classical 

Duffing equation and damped cubic nonlinear equation. Results of Multiple Scales, Lindstedt 

Poincare method, new method and numerical solutions are contrasted [1-6]. 

Solution of Differential equations by perturbation technique using any time transformation. In Direct 

Perturbation Method, mostly secular terms appear of higher orders of the expansion invalidating the 

solution. In order the avoid this problem a new time transformation has been proposed in our study. 

The new time transformation is defined as, 

Using the chain rule, we transform the derivate accordingly 

du 

dt 

2 

d u 

2 

dt 

T e 

f ( w, 

t) 

t 

(1) 

u'( 

f� 

( w, 

t) 

t 

u'' 

( f� 

( w, 

t) 

t 

f ( w, 

t)) 

f ( w, 

t)) 

2 

u ( �f 

�( 

w, 

t) 

t 

2 f� 

( w, 

t)) 

So we have obtained a more effective time expression T e without losing the original time parameter 

t using the function f. Thus, speeding up and slowing down control of the time parameter will be 

available as in Method of Multiple Scales. 

Page 107 

(2)

In Equation (2) first order time-derivatives according to new time variable T e appear in second order 

time derivative expressions according to original time variable. So, we are able to have information 

about some parameters of nonlinear differential equation, and to interpret the results. 

By this new time transformation we have the advantages of both Lindstedtpoincare method and 

Method Multi Scales . 

Using a new perturbation algorithm with new time transformation, we showed that, first we have 

obtained a more effective time expression without losing the original time parameter t using the 

function f. We are able to have information about some parameters of nonlinear differential equation, 

and to interpret the results. When we apply this transformation on the known Duffing equation with 

the results of the studies conducted to date have compared the results obtained. 

We found in this new time with the transformation of the solutions are compared with approximate 

solutions do not differ in Results of Multiple Scales, Lindstedt Poincare method and we found that 

the approximate solutions. 

References 

[1] H. Nayfeh, Introduction to Perturbation Techniques, John Wiley and Sons, New York 1981. 

[2] Nayfeh, A.H., Nayfeh S.A. and Mook, D.T., On Methods for continuous systems with quadratic 

and cubic nonlinearities, Nonlinear Dynamics, 3, 145-162, 1992 

[3] Nayfeh, A.H., 1998. Reduced order models of weakly nonlinear spatially continuous systems, 

Nonlinear Dynamics 16,105–125, 1998. 

[4] M. Pakdemirli and M. M. F. Karahan, A New Perturbation Solution for Systems with Strong 

Quadratic and Cubic Nonlinearities, Mathematical Methods in the Applied Sciences 33, 704-712, 

2010. 

[5] Perturbation Methods, Instability, Catastrophe and Chaos, C F Chan Man Fong and D D Kee, 

World Scientific Publishing, 1999 

[6] M. Pakdemirli, M. M. F. Karahan and H. Boyacı, A new perturbation algorithm with 

better convergence properties: Multiple Scales Lindstedt Poincare Method, Mathematical and 

Computational Applications 14 ,31-44, 2009. 

Page 108

Aproximation Properties of a Generalization of Linear Positive Operators in C[0,A] 


N.Gonul 

Department of Mathematics, Bulent Ecevit University, Zonguldak, Turkey 

In this paper we study the order of convergence of a generalization of positive operators by means 

of the functions from Lipschitz class. We use the test functions 

x 

1+x for = 0; 1; 2; a Korovkin type 

theorem given by [1]. Furthermore we estimate the rate of convergence of these operators.Some …gures 

correspond to obtaining results are given. Finally, the algorithm used in the program has been added. 

References 

[1] Cakar O. and Gadjiev A. , On uniform approximation by Bleimann, Butzer and Hahn on all 

positive semiaxis, Tras. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. 19, 21–26, 1999. 

[2] Coskun, T. Weighted approximation of continuous functions by sequences of linear positive oper- 

ators. Proc. Indian Acad. Sci. (Math. Sci.) Vol. 110, No. 4, 357-362, 2000. 

[3] Dogru O., On Bleimann, Butzer and Hahn type generalization of Balázs operators, Dedicated to 

Professor D. D. Stancu on his 75th birthday, Studia Univ. "Babe¸s-Bolyai", Mathematica 47 , 37-45, 

2002. 

[4] Korovkin P.P., Linear Operators and Approximation Theory, Hindustan Publ.Co., Delhi, 1960. 

[5] Ibragimov I.I., Gadziev A. D. On a sequence of linear positive operators,Soviet Math. Dokl., v.11, 

No:4, pp. 1092-1095,1970. 

Page 109

Three-term Asymptotic Expansion for the Moments of 

the Ergodic Distribution of a Renewal-reward Process 

with Gamma Distributed Interference of Chance 

N. Okur Bekar 1 , R. Aliyev 2 and T. Khaniyev 3 

1 Karadeniz Technical University, Faculty of Sciences, Department of Mathematics, 

61080, Trabzon, Turkey 

2 Baku State University, Faculty of Applied Mathematics and Cybernetics, Department of 

Probability Theory and Mathematical Statistics Z. Khalilov 23, Az 1148, Baku, Azerbaijan 

3 TOBB University of Economics and Technology, Faculty of Engineering, Department of 

Industrial Engineering, 06560, Sogutozu, Ankara, Turkey 


In this study, a renewal-reward process with a discrete interference of chance �X(t) � is 

investigated. We assume that (X �(t)) t� 0 

is a renewal-reward process with a gamma distributed 

interference of chance with parameters ( �� , ) . Under the assumption that the process is ergodic, 

the paper provides the three-term asymptotic expansions for the moments 

References 

n 

EX� , n � , as λ� 0. 

[1] Aliyev R.T., Khaniyev T.A., Okur Bekar N., Weak convergence theorem for the ergodic 

distribution of the renewal- reward process with a gamma distributed interference of chance. 

Theory of Stochastic Processes, 15 (31) 2, 42-53, 2009. 

[2] Csenki A., Asymptotic for renewal-reward processes with retrospective reward structure, 

Operation Research and Letters, 26, 201-209, 2000. 

[3] Feller W., An Introduction to Probability Theory and Its Applications II, J. Wiley, New 

York, 1971. 

[4] Gihman I.I., Skorohod A.V., Theory of Stochastic Processes II, Springer, Berlin, 1975. 

[5] Ross S.M., Stochastic Processes, 2nd Ed. New York: John Wiley & Sons, 1996. 

Page 110

Blow up of a solution for a system of nonlinear higher-order wave equations with strong 


damping 

N. Polat and E. Pi¸skin 

Department of Mathematics, Dicle University, Diyarbakir, Turkey 

This work studies a initial-boundary value problem of the strong damped nonlinear higher-order wave 

equations. Under suitable conditions on the initial datum, we prove that the blow up of the solution. 

References 

[1] Agre K. and Rammaha M.A., Systems of nonlinear wave equations with damping and source terms, 

Di¤. Integral Eqns., 19(11), 1235–1270, 2006. 

[2] Yu S., On the strongly damped wave equation with nonlinear damping and source terms, E. J. 

Qualitative Theory of Di¤. Equ., 39, 1-18, 2009. 

2001. 

[3] Messaoudi S. A., Blow up in a nonlinearly damped wave equation, Math. Nachr., 231, 105–111, 

[4] Pi¸skin E. and Polat N., Global existence and exponential decay of solutions for a class of sys- 

tem of nonlinear higher-order wave equations with strong damping, J. Adv. Res. Appl. Math., Doi: 

10.5373/jaram (in press). 

Page 111

Reduction of spectral problem of Cauchy-Riemann operator with homogeneous boundary 


conditions to an integral equation 

N.S.Imanbayev 

H.A.Jassavi IKTU, Turkestan, Kazakhstan 

In this paper the problem on the eigenvalues of the Cauchy-Riemann operator with homogeneous 

boundary conditions is reduced to an integral equation In the functional space C(|z| ≤ 1) we consider 

the operators generated by differential operation of the Cauchy-Riemann 

∂ω (z) 

Kω (z) = , 

∂z 

where z = x + iy, z = x − iy, ∂ 

� � 

1 ∂ ∂ 

∂z = 2 ∂x + i ∂y on the set 

References 

D (K) ⊂ 

� 

� 

∂ω (z) 

ω (z) ∈ C(|z| ≤ 1), ∈ C(|z| ≤ 1) . 

∂z 

[1] Otelbayev M., and Shinibekov A.N., About the correct problems of Bitsadze-Samarskiy type, 

Reports of Academy of Sciences,USSR.-V.265, 4.-pp.815-819,1982. 

[2] Mikhailets V.A., Spectral problems with general boundary conditions,Abstract of doct.of ph.-math, 

Kiev, 29 p,1989. 

[3]Imanbaev N.S., and Kanguzhin B. E , and Kirgizbaev Zh. About Fredholm property of one spectral 

problem related to Cauchy-Riemann operator,Inter-institute collection of scientific proceedings, ”Ques- 

tions of stability, durability and controllability of the dynamic systems”,Moscow: RGOTUPS, P. 54- 

59.2002 

[4] Muskhelishvili N.I., Singular integral equations,Nauka Moscow, 511 p.1968. 

[5] Bitsadze A.V., Fundamentals of theory of analytic functions of complex variable, Nauka, Moscow, 

239, p.1969. 

Page 112

A Note on Some Elementary Geometric Inequalities 

O. Gercek 1 , D. Caliskan 2 , A. Sobucova 1 and F. Cekic 1 

1 Department of Mathematics, Fatih University, Istanbul, Turkey 2 Department of Mathematics 


Teaching, Qafqaz University, Baku, Azerbaijan 

In present paper, solutions of some elementary geometric inequalities are obtained. Firstly, we get a 

more useful inequality by specifying the largest lower and smallest upper bounds, to be able to end the 

inaccuracy of the following inequality. Let a, b, c are lengths of sides of a triangle, and if distances of a 

taking point in the inner region of a triangle to the vertices are x, y, z, then following inequality satis…es 

1 

(a + b + c) < x + y + z < a + b + c: 

2 

Nevertheless it is true and useful, but it has not accurate boundaries. Because neither 1 

2 (a + b + c) 

is the largest lower bound, nor a + b + c is the smallest upper bound of this sum. But in university 

preparation course books and textbooks, which describe this inequality, they resolved by accepted these 

greatest lower bound and least upper bound, so that incorrect results were obtained. In this work, we 

solved this inequality with more useful bounds. Namely, 

1. If distances of a taking point in the inner region of a triangle to the vertices are x, y, z which have 

lengths of sides a, b, c, and the area A, and also measures of all internal angles are smaller than 

120 degrees, then following inequality satis…es 

r 1 

2 a2 + b 2 + c 2 + 4 p 3A x + y + z < max fa + b; a + c; b + cg : 

2. If distances of a taking point in the inner region of a triangle to the vertices are x, y, z which have 

lengths of sides a, b, c, and the area A, and one of the measure of internal angle is greater than or 

equal to 120 degrees, then following inequality satis…es 

min fa + b; a + c; b + cg < x + y + z < max fa + b; a + c; b + cg : 

We have stated and proved some theorems and lemmas that have done before (see in [1-5]). One of 

the our theorems is famous one, namely the Toricelli-Fermat point that was solved in many ways. 

In [6] and [7], we observed that Mustafa Ya¼gc¬have studied such as this work nicely. But, our proofs are 

completely di¤erent and some parts are more simple and clearer. In [7], he gave a problem. He claimed 

the problem he has given can not be solved using only geometry, but calculus is also used to solve the 

problem. We decided to prove this problem given in [7]. After showing the existence and uniqueness of 

the triangle 4XYZ de…ned in the problem, we tried to solve the problem by using only geometrical way 

and we succeeded. The most interesting part of our article is second part, proving this following problem 

given below: 

Problem. For a given point P, on changing plane of the X, Y and Z let be jP Xj = a; jP Y j = b; and 

jP Zj = c: Then the circumference of the triangle 4XYZ has the largest value, when P is at the inner 

center of the triangle. Secondly, what can be the minimum value of the sum of a, b and c? 

Page 113

References 

[1] H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited, MAA, (1967). 

[2] D. Pedoe, Geometry: A Comrehensive Course, Dover, (1970). 

[3] R. Honsberger, Mathematical Gems I, MAA, (1973). 

[4] D. Pedoe, Circles: A Mathematical View, MAA, (1995). 

[5] A. Ostermann and G. Wanner, Geometry by Its History, Springer, (2012). 

[6] M. Ya¼gc¬, Fermat-Toricelli Noktas¬, Matematik Dünyas¬ 2004(1), 58-61 (2004). 

[7] M. Ya¼gc¬, Fermat-Toricelli’ye K¬sa Bir Ziyaret, Matematik Dünyas¬ 2004(4), 79 (2004). 

Page 114


Solving Crossmatching Puzzles Using Multi-Layer Genetic Algorithms 

O. Kesemen 1 and E. Özkul 1 

1 Department of Statistics and Computer Science, Faculty of Science 

Karadeniz Technical University, 61080 Trabzon, Turkey 

okesemen@gmail.com, eda.ozkul.gs@gmail.com 

Nowadays, puzzles are used commonly as a fountain head of our monotonous lives or to spend free 

time. Crossmatching puzzle (CMP) is quite similar to the crossword puzzle (CWP). There are detection 

key table and control key table in crossmatching puzzles instead of questions in crosswrod puzzle which 

are written from left to right and from top to bottom. Letters in each row of the main solution table 

are arranged in an order in detection key table. In the same way, letters in the main solution table are 

arranged in order column by column are put in the control key table. Therefore, the main table is tried 

to solve with crossing of the letters in the detection key table and control key table. 

For solution of cross-matching puzzle can be used depth search algorithm. However, in spite of depth 

search algorithm gives the exact result, size of puzzle as augments computing time of solution increases 

exponentially and it makes the solution of puzzle impossible. In this case, stochastic search is better to 

use instead of deterministic search algorithm. 

In this study, improved genetic algorithm as multi-layer is used as stochastic search method [2]. In 

this algorithm, each letters represent a gene and each rows represent a chromosome. An individual is 

generated by chromosomes as number of rows come together. Fitting function of created individual is 

determined according to fitness of control key table. 

Keywords: Crossmatching Puzzle, Multi-Layer Genetic Algorithm 

References 

[1] Kesemen, O., ve Karakaya, G., Yeni Bir Sayı Yerletirme Oyunu: Sıklık Bulmaca (SıkBul), 9. 

Matematik Sempozyumu, 20-22 Ekim, Trabzon, 2010. 

[2] Mantere, T. and Koljonen, J., Solving and Rating Sudoku Puzzles with Genetic Algorithms, New 

Developments in Artificial Intelligence and the Semantic Web, Proceedings of the 12th Finnish Artificial 

Intelligence Conference Step 2006. 

Page 115


Generate Adaptive Quasi-Random Numbers 

O. Kesemen 1 and N. Jabbari 1 



okesemen@gmail.com, nasim.jabbari@gmail.com 

Random number generation, especially with the development of computer technology has an impor- 

tant place in the world of it.Uniform distribution random number generation can be done in almost all 

programming languages. Also in other distributions number generation can be produced with the help 

of the generated number from uniform distribution [1]. Games, education and simulation such as appli- 

cations programs which frequently used random number generation, is produced in the form of discrete 

data. Sometimes random numbers generation produced with the same values observe consecutive. In 

most applications (such as education) to purify this effect we can reused the resulting number again. But 

it reduces the amount of numbers, the random number will be in facilitates prediction. 

Indeed, as it reduced the probability of the taken number at the same time for making the prediction 

of the next number difficult,variable produced random numbers with their probability [2]. 

The frequency of region of each generated random number (fi) are stored in an increase. The next 

random number which be generated,if it selected in each region,that number is considered to be the 

number. In this method,because of the equal intervals of the numbers they have equal probability.The 

aim here is to reduce the probability of selecting the same number. In this case,the interval between the 

numbers must be narrowed. The process of increasing of the propbability of narrowering the interval 

transferred to other numbers. In this case, interval’s limits will be changed. 

In statistical simulations,deviations from randomness is be reduced by taking too much random num- 

bers. In this work, reduction of the amount of deviation,a set of data as a result of simulation provides 

an approach theoretical curve more rapidly. Thus simulation results can be reached with less data. This 

provides a reduction of the computing time. 

Keywords: Adaptive Quasi Random Number, Statistical Simulation 

References 

[1] Kobayashi, H., Mark, B.L. and Turin, W., Probability, Random Processes,and Statistical Analysis, 

Cambridge University Press, 2012 

[2] Morokoff, W. J. and Caflisch, R. E.,Quasi-Random Sequences And Their Discrepancies, Vol. 6 

(15) 1251-1279, SIAM Journal of Science Computing, 1994. 

Page 116

Polygonal Approximation of Digital Curve Using Artificial Bee Colony Optimization 


Algorithms 

O. Kesemen 1 and S. Vafaei 1 



okesemen@gmail.com, soheyla.vafaei@yahoo.com 

In image processing and pattern recognition, it is an important concept defining two-dimensional 

objects in the image [1]. Firstly, the dominant points of the edges of the object (corner points) are 

determined while objects are defined. Objects with the help of the dominant points compared to a 

polygon, then the number of edges or vertices are determined. The purpose of the dominant point, the 

desired object is to represent, using fewer points. Thus, in practice, it is realized large memory, and 

trading volume. The problem is how to select these points. According to number of dominant point of 

any object, all combinations of boundary pixels is tested. Thus the exact solution is used in polygonal 

approach that gives the least error. If the object is small, and the required number of points is less, 

the exact solution will give the best results. However, if the required number of combinations is more, 

deterministic solution is impossible. Therefore, for solving the problem it is needed a stochastic search 

algorithm are needed. In this case, artificial bee colony (ABC) algorithm selected. ABC algorithm 

has been developed by modeling the bees look for food in bulk [2]. In this study, the advantages and 

shortcomings of the ABC method were examined by comparing ABC method with Genetic algorithm 

method 

rithm 

Keywords: Dominant Point, Digital Curve, Polygonal Approximation, Artificial Bee Colony Algo- 

References 

[1] Yin, P.Y., A new method for polygonal approximation using genetic algorithms, Pattern Recogni- 

tion Letters, (19) 1017-1026, 1998. 

[2] Karaboga, D. and Basturk, B., On the performance of artificial bee colony (ABC) algorithm, 

Applied Soft Computing (8) 687-697, 2008. 

Page 117

Generating Random Points from Arbitrary Distribution In Polygonal Areas 


O. Kesemen 1 and Ü. Ünsal 1 



okesemen@gmail.com, ulkunsal@gmail.com 

Random numbers generation in polygonal area is used in many applications area simulation. These 

application areas can be distribution of living life in a pond, level of pollution in a city, density of tree 

species in a forest, traffic flow in a region, density of flight in air space, diversity of wildlife in a region, 

crime rate in a city etc. [1]. 

Traditionally, acceptance-rejection method is used that in a polygonal area which it has known prob- 

ability density function (f(x,y)) to generate random numbers [2]. For this, at first rectangular area 

boundaries which are surrounding polygon is found. By the help of these boundaries X and Y random 

values are created which they selected from uniform distribution. If created point is out of polygonal 

area, point is rejected. If selected point within polygonal area for adapting selected point to probabil- 

ity density function, a random value which is between zero and highest probability density value within 

polygonal area selected from uniform distribution (z direction). If this value is bigger than value of f(X,Y) 

is rejected, if not it is accepted. Thus, a random point is selected in polygonal area. This procedure can 

be repeated any numbers of random numbers are generated. Used method not only generate unnecessary 

random number but also it causes increased computational time for investigating whether the point in 

polygonal area. 

The basis of proposed method is based on that calculated by dividing triangle pieces of the all area with 

corner points of polygonal area by combining together. By selecting a certain random point in polygonal 

area, triangulation size can be reduced and calculation sensitization can be increased. A plane which is 

in the probability density function value of corner points of each triangle, regarded as probability density 

function of the triangle. Under the probability density functions that they have all triangles volume 

be equal 1 is agreed as the probability basic axiom. Hence volume of triangular prism of each triangle 

formed is gave probability that the selection of the triangle. The probability density function defines in 

a unit triangle which is subtending the probability density function for each triangle. A random number 

generated within this defined triangle is moved by the principle of affine invariance into selected triangle 

area. In this manner any number of random numbers can be generated. 

The method proposed in this study not only prevents unnecessary random number generation but also 

reduces computation time indeed. Specially, when want to generate a large number of random numbers 

it can be used as an effective method. 

Keywords: Random Number, Poligonal Area, Triangulate 

References 

[1] Kesemen, O. and Dogru, F.Z., Cumulative Distribution Functions of Two Variables in Polygonal 

Areas, 7. International Statistical Congress,28 Apr-01 May 2011 Belek-ANTALYA. 

[2] Martinez, W.L. and Martinez, A.R., Computational Statistics Handbook with MATLAB, Chapman 

& Hall Crc, 2002. 

Page 118

Panoramic Image Mosaicing Using Multi-Object Artificial Bee Colony Optimization 


Algorithm 

O. Kesemen 1 and Y. Yegino˘glu 1 



okesemen@gmail.com, yesimyeginoglu@gmail.com 

Sometimes distance, necessary to take wide- angle photography , may not be available. In this case, 

the need may occur combining in accordance photographs taken piece by piece. Nowadays, a lot of 

camera manufacturer tried to solve the problem by using wide-angle lens (fish eye) [1-2]. But in order 

to change perspective it is almost impossible to get a good image. On the other hand, on the basis of 

images taken by a number of different angle (especially video images) may be required to obtain a wide- 

angle image. In this case, in accordance with a multi-image combined panoramic images are obtained. 

However, adaptation research for the realization of suitable attachment can take a very long time. 

In this study, for solving the problem, artificial bee colony algorithm [3] is changed based on adopted 

multi-object search. According to this method, the right side of each image is determined as the food 

region and the left side represents a bee hive. The bees in each hive move to food regions of other images, 

divided into groups that have equal number of bees. Each bee has its own search on food regions. After 

one of the bees which from the first hive reached the highest value of the objective function, tries to pull 

the other bees from other regions. As a result of a particular iteration bees of every hive are kept together 

in a certain region. Thus, it can be determined that which image is positioned in which order and which 

location. 

Keywords: Multi-Object Optimization, Artificial Bee Colony Algorithm,Panoramic Image Mosaicing 

References 

[1] Peleg, S. and Herman, J., Panoramic Mosaics by Manifold Projection, Computer Vision and 

Pattern Recognition, 338-343, 1997. 

[2] Kourogi, M., Kurata, T., Hoshino, J. and Muraoka, Y., Real-time Image Mosaicing from a Video 

Sequence, Image Processing, (4) 133-137, 1999. 

[3] Karaboga, D. and Basturk, B., On the Performance of Artificial Bee Colony (ABC) Algorithm, 

Applied Soft Computing (8) 687-697, 2008. 

Page 119


Some Properties of a Sturm-Liouville-Type Problem and The Green Function 

Okan KUZU, Yasemin KUZU, Mahir KADAKAL 

Department of Mathematics, Ahi Evran University, Kirsehir, Turkey 

In this study we have created Hilbert Space of The Sturm-Liouville Boundary Value 

Problem in [0, π] interval, with boundary conditions which has λ complex eigenparameter. 

We have shown symmetric of appropriate operator to the problem. We have obtained 

asymptotic of solution functions and asymtotic of wronskian of the solution functions by 

using them. Moreover, we have examined Green function and asymtotic expansion of 

eigenvalues. 

References 

[1] Birkhoff, G. D. On The Asymptotic Character of The Solution of The Certain Linear 

Differential Equations Containing Parameter, Trans. Amer. Math. Soc., Vol. 9 1908, 

pp. 219-231. 

[2] Birkhoff, G. D. Boundary Value and Expantion Problems of Ordinary Linear Differ- 

ential Equations, Trans. Amer. Math. Soc., Vol. 9 1908, pp. 373-395. 

[3] Boyce, W. E.; Diprima, R. C. Elementary Differential Equations and Boundary Value 

Problems, John Willey and Sons, New York, 1977, pp. 544-554. 

[4] Fulton, C. T. Two-point Boundary Value Problems with Eigenvalue Parameter Con- 

tained in The Boundary Condition, Proceedings of the Royal Society of Edinburgh. 

Section A 77, 1977, p. 293-308. 

[5] Hinton, D. B. An Expansion Theorem for Eigenvalue Problem with Eigenvalue Pa- 

rameter in The Boundary Condition, Quart. J. Math. Oxford, vol. 30, No;2, 1979, 

33-42. 

[6] Kerimov, N. B.; Mamedov, Kh. K. On a Boundary Value Problem with a Spectral 

Parameter in The Boundary Conditions, Sibirsk. Math. J. 40, No:2, 1999, 281-290. 

[7] Levitan, B. M., Sarqsyan, I.S. Sturm-Liouville and Direct Operators, Moskov, Nauka, 

1988. 

[8] Mukhtarov, O. Sh., Kadakal, M and Muhtarov, F. S. On discontinuous Sturm- 

Liouville problems with transmission conditions, J. Math. Kyoto Univ. 44-4 (2004), 

779798. 

1 

Page 120

[9] Naimark, M. A. Linear Differantial Operators, Ungar, New York, 1967. 

[10] Schneider, A. A Note Eigenvalue Problems with Eigenvalue Parameter in The Bound- 

ary Conditions, Math. Z. 136, 1974, 163-167. 

[11] Shkalikov, A. A. Boundary Value Problems for Ordinary Differential Equations with 

a Parameter in Boundary Conditions, Trudy., Sem., Imeny, I. G. Petrovsgo, 9, 1983, 

190-229. 

[12] Titchmarsh, E. C. Eigenfunction Expansions Associated with Second Order Differen- 

tial Equations, 2nd end, Oxford Univ. Pres, London, 1962. 

[13] Walter, J. Regular Eigenvalue Problems with Eigenvalue Parameter in The Boundary 

Conditions, Math. Z., 133, 1973, 301-312. 

[14] Zayed, E.M.E. and Ibrahim, S.F.M., Regular Eigenvalue Problem with Eigenparam- 

eter in the Boundary Conditions, Bull. Cal. Math. Soc. 84 379-393, 1992. 

2 

Page 121


Real Time 3D Palmprint Pose Estimation and Feature Extraction Using Multiple View 

Geometry Techniques 

Ö. Bingöl 1 , M. Ekinci 2 

1 Department of Software Engineering, Gumushane University, Gumushane, Turkey 

2 Department of Computer Engineering, Karadeniz Technical University, Trabzon, Turkey 

In this paper, it was aimed to develop a system that works in real time for to obtain palmprint pose 

(point of view) of a fully opened hand towards the camera. This system will be both a platform independent 

model (non-touchable) and arising from the hand movement rotations, translations and scaling independet 

model. For this purpose, pointed at the same direction two cameras (stereo) is used instead of single-camera 

vision systems system. Palmprint informations carried to 3D space using Mutliple View Geometry 

techniques from the obtained images. Thus, the problems are eliminated in previous studies as rotation, 

translation, scaling and platform dependecy. 

Common points must be identified and mapped for capture of 3D palmprint on obtained images from 

two cameras. SURF algorithm based on Hessian matrix is determined common interest points on real-time 

snapshots of each cameras. The Levenberg-Marquardt optimization algorithm is used to minimize deviations 

from the characteristics of the cameras. Paired interest points of palmprint was considered to be 

approximately on a plane. Normal of 3D plane will give palmprint pose (point of view) according to the 

cameras. Finally, the palmrint image were transferred to the 2D surface with affine transformation. As a 

result, palmprint patterns have been obtained for strong 2D recognition palmprint systems. 

References 

Page 122 

[1] Bay H., Tuytelaars T. and Van Gool L., SURF: Speeded Up Robust Features, in: ECCV, 2006. 

[2] Hartley R. and Zisserman, A., Multiple View Geometry in Computer Vision, Cambridge University 

Press: Cambridge, UK, 2000 

[3] Trucco E. and Verri A., Introductory Techniques for 3-D Computer Vision. N.J.: Prentice Hall, 

1998. 

[4] B.D. Lucas and T. Kanade, An Iterative Image Registration Technique With an Application to 

Stereo Vision, Proc. Int"l Joint Conf. Artificial Intelligence, pp. 674-679, 1981. 

[5] Schweighofer, G. and Pinz, A. Robust Pose Estimation From a Planar Target. IEEE Transactions on 

Pattern Analysis and Machine Intelligence, 28(12), 2024–2030, 2006. 

[6] Zhang D., Kong A., You J. and Wong M., Online Palmprint Identification,IEEE Trans. Pattern Anal. 

Mach. Intell., 25 (9), pp. 1041–1050, 2003. 

[7] Han C.C., Cheng H.L., Lin C.L. and Fan K.C., Personal Authentication Using Palmprint Features, 

Pattern Recognition, 36 (2), 2003. 

[8] T. Connie, A.T.B. Jin, M.G.K. On, D.N.C. Ling, An Automated Palmprint Recognition System, 

Image Vision Comput., 23 (5), pp. 501–515, 2005. 

[9] Ekinci M., Aykut M., Palmprint Recognition by Applying Wavelet Subband Representation And 

Kernel PCA, Lecture Notes in Artificial Intelligence, pp. 628–642, 2007. 

[10] Ekinci M., Aykut M., Palmprint Recognition by Applying Wavelet-Based Kernel PCA, J. Comput. 

Sci. Technol., 23, pp. 851–861, 2008.


Chaos in cubic-quintic nonlinear oscillator 

Patanjali Sharma 1 and V. G. Gupta 2 

1 Dept. of Mathematics, Banasthali University, Banasthali 304 022 

2 Dept. of Mathematics, University of Rajasthan, Jaipur 302 004 

In this paper, we used Hamiltonian formulation and Lie transform to investigate a strongly nonlinear 

oscillator. Using Chirikovâ€ s overlap criterion we find the value of εcr at which the chaos loses its local 

character and becomes global. The results of Lie transformation analysis and Chirikovâ€ s criteria for 

the oscillator are compared with numerically generated Poincare Maps. 

References 

[1] Chirikov, B.V., A Universal Instability of Many-Dimensional Oscillator Systems, Physics Reports 

52 1979, 265-376. 

[2] Deprit, A., Canonical Transformations Depending on a Small Parameter, Celestial Mechanics, 1 

1969, 12-30. 

[3] Goldstein, H., Poole, C., and Safko, J., Classical Mechanics, Third Edition, Pearson Education, 

Inc., 2004. 

[4] Kamel, A. A., Perturbation Theory Based on Lie Transforms, NASA Contractor Report CR-1622 

(1970). 

[5] N. Abouhazim, B. Mohamed and R. H. Rand, Two models for the parametric forcing of a nonlinear 

oscillator, Nonlinear Dynamics, 50, 2007, 147-160. 

[6] Rand, R. H., Topics in Nonlinear Dynamics with Computer Algebra, Gordon and Breach, Lang- 

horne, PA, 1994. 

[7] Zounes, R. S. and Rand, R. H., Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation, 

Nonlinear Dynamics, 27, 2002, 87-105. 

Page 123

The Numerical Solution of Boundary Value Problems by using Galerkin 

Method 

S. Alkan 1 , T. Yeloğlu 2 and D. Yılmaz 2 

1 Department of Management Information Systems, Bartın University, Bartın, Turkey 

2 Department of Mathematics, Mustafa Kemal University, Hatay, Turkey 

duygu.yilmazz@yandex.com 


In this study, we obtain approximate solutions of some boundary value problems by the Galerkin 

method. To demonstrate the effectiveness of the Galerkin method, we give some examples. Also, 

we compare the obtained solutions and their exact solutions by using Mathematica. 

Keywords: Boundary value problems (BVPs), Galerkin method, Mathematica 

References 

Page 124 

[1] Alkan, S., Sınır Değer Problemlerinin Nümerik Çözümleri, Yüksek Lisans Tezi, Muğla 

Üniversitesi, Muğla, 2011. 

[2] Evans, G., Blackledge, J., Yardley, P., Numerical Methods for Partial Differential Equations, 

Springer, New York, 290p., 2000. 

[3] Bhatti, M.I., Bracken, P., Solutions of differential equations in a Bernstein polynomial basis, 

J. Comput. Appl. Math., 205, 272-280, 2007.

Semismooth Newton method for gradient constrained minimization problem 

S.Anyyeva 1 and K.Kunisch 1 

1 Institute of Mathematics and Scientific Computing, Karl Franzens University, Graz, Austria 


In this paper we treat a gradient constrained minimization problem which has applications in me- 

chanics and superconductivity [1, 2, 5]: 

Find a solution y ∈ K such that (1) 

J(y) = min 

v∈K J(v), 

where J(v) = 1 

� 

2 

Ω 

|∇v| 2 � 

dx − fvdx, 

Ω 

K = {v ∈ H 1 0 (Ω)| |∇v(x)| ≤ 1 a.e. in Ω}. 

Here Ω ⊂ R n , n ≤ 3, is bounded Lipschitz domain and f ∈ L 2 (Ω) is given. A particular case of this 

problem is the elasto-plastic torsion problem. 

In order to get the numerical approximation to the solution we have developed an algorithm in an in- 

finite dimensional space framework using the concept of the generalized, so called, Newton differentiation 

[3,4,6]. At first we regularize the problem in order to approximate it with the unconstrained minimiza- 

tion problem and to make the pointwise maximum function Newton differentiable. Afterwards, using 

semismooth Newton method, we obtain continuation method in function space. For the numerical im- 

plementation the variational equations at Newton steps are discretized using finite elements method. We 

compare the numerical results in two-dimensional case obtained using C 1 -conforming and non-conforming 

finite elements discretization. 

References 

[1] Duvaut G. and Lions J., Inequalities in mechanics and physics, Berlin : Springer, 1976. 

[2] Ekeland I. and Temam R., Convex analysis and variational problems, SIAM, Amsterdam, 1987. 

[3] Ito K. and Kunisch K., Lagrange Multiplier Approach to Variational Problems and Applications, 

vol. 15 of Advances in Design and Control, Society for Industrial and Applied Mathematics, U.S., 2008. 

[4] Hintermüller M., Ito K. and Kunisch K., The primal-dual active set strategy as a semismooth 

Newton method, SIAM J. Optimization 13, 865–888 (2003). 

[5] Glowinski R., Lions J. and Trémolièrs R., Numerical analysis of variational inequalities, North 

Holland publishing company - Amsterdam - New York - Oxford, 1981, ISBN 0444861998. 

[6] Kummer B., Generalized Newton and NCP methods: Convergence, regularity, actions, Discuss. 

Math. Differ. Incl. Control Optim., 2000, p.209-244 

Page 125

The Finite Element Method Solution of Variable Diffusion Coefficient 

Convection-Diffusion Equations 


S.H Aydın 1 and C. Çiftçi 2 


2 Department of Mathematics, Ordu University, Ordu, Turkey 

Mathematical modeling of many physical and engineering problems is defined with convection- 

diffusion equation. Therefore, there are many analytic and numeric studies about convection- 

diffusion equation in literature. The finite element method is the most preferred numerical 

method in these studies since it can be applied to many problems easily. But, most of the studies 

in literature are about constant coefficient case of the convection-diffusion equation. In this 

study, the finite element formulation of the variable coefficient case of the convection-diffusion 

equation is given in both one and two dimensional cases. Accuracy of the obtained formulations 

are tested on some problems in one and two dimensions. 

References 

Page 126 

[1] Linβ T., Analysis of a Galerkin Finite Element Method on a Bakhvalov-Shishkin Mesh for a 

Linear Convection-Diffusion Problem, IMA J. Numer. Anal., 20(4) 621-623, 2000. 

[2] Linβ T., Layer-adapted Meshes for Convection-Diffusion Problems, Comput. Meth Appl. 

M., 192(9-10), 1061-1105, 2003 

[3] Reddy J.N., An Introduction to the Finite Element Method, The McGraw-Hill Companies, 

2006.

Multiple solutions for quasilinear equations depending 

on a parameter 

Shapour Heidarkhani 

a Department of Mathematics, Faculty of Sciences, 

Razi University, 67149 Kermanshah, Iran 

e-mail addresses: s.heidarkhani@razi.ac.ir 


The purpose of this talk is to use a very recent three critical points theorem due to Bonanno 

and Marano [1] to establish the existence of at least three solutions for quasilinear 

second order differential equations on a compact interval [a, b] ⊂ R under appropriate hypotheses. 

We exhibit the existence of at least three (weak) solutions and, and the results are 

illustrated by examples. 

Keywords- Dirichlet problem; Critical point; Three solutions; Multiplicity results. 

AMS subject classification: 34B15; 47J10. 

1 Main results 

Consider the following quasilinear two-point boundary value problem 

� 

−u ′′ = (λf(x, u) + g(u))h(u ′ ) in (a, b), 

u(a) = u(b) = 0 

where [a, b] ⊂ R is a compact interval, f : [a, b] × R → R is an L 1 -Caratéodory function, 

g : R → R is a Lipschitz continuous function with g(0) = 0, i.e., there exists a constant 

L ≥ 0 provided 

|g(t1) − g(t2)| ≤ L|t1 − t2| 

for all t1, t2 ∈ R, h : R →]0, +∞[ is a bounded and continuous function with m := inf h > 0 

and λ is a positive parameter. 

Employing Theorem 3.6 of [1], we establish the existence of at least three distinct (weak) 

solutions in W 1,2 

0 ([a, b]) to the problem (1) for any fixed positive parameter λ belonging to 

an exact interval which will be observed in the main results. 

1 

Page 127 

(1)

We mean by a (weak) solution of problem (1), any u ∈ W 1,2 

0 ([a, b]) such that 

� b 

u 

a 

′ (x)v ′ � b 

(x)dx − [λf(x, u(x)) + g(u(x))]h(u 

a 

′ (x))v(x)dx = 0 

for every v ∈ W 1,2 

0 ([a, b]). Denote M := sup h and suppose that the constant L ≥ 0 satisfies 

LM(b − a) 2 < 4. 

We introduce the functions F : [a, b] × R → R, H : R → R and G : R → R respectively, 

as follows 

and 

� t 

F (x, t) = f(x, ξ)dξ for all (x, t) ∈ [a, b] × R, 

0 

� t � τ 

H(t) = 

0 

0 

1 

dδdτ for all t ∈ R 

h(δ) 

� t 

G(t) = − g(ξ)dξ for all t ∈ R. 

0 

We now formulate our main result. 

Theorem 1. Assume that there exist a positive constant r and a function w ∈ W 1,2 

0 ([a, b]) 

such that 

(α1) � b 

a [G(w(x)) + H(w′ (x))]dx > r, 

� 

2M(b−a)r 

(α2) 

� b 

a sup 

t∈[− 

(α3) lim sup |t|→+∞ 

respect to x ∈ [a, b]. 

Then, for each 

4−LM(b−a) 2 

, 

F (x,t) 

t 2 

r 

� 

2M(b−a)r 

4−LM(b−a) 2 

] 

< 4−LM(b−a)2 

2M(b−a) 2 r 

⎤ 

⎥ 

� b 

⎥ a 

λ ∈ Λ1 := ⎥ 

⎦ 

[G(w(x)) + H(w′ (x))]dx 

, 

F (x, w(x))dx 

� b 

a 

F (x,t)dx 

� b 

F (x,w(x))dx 

a < � b 

a [G(w(x))+H(w′ (x))]dx , 

� b 

a sup � 

2M(b−a)r 

t∈[− 

4−LM(b−a) 2 , 

� b 

a sup t∈[− 

� 2M(b−a)r 

4−LM(b−a) 2 , 

� F (x, t)dx uniformly 

2M(b−a)r 

4−LM(b−a) 2 ] 

r 

� F (x, t)dx 

2M(b−a)r 

4−LM(b−a) 2 ] 

the problem (1) admits at least three distinct weak solutions in W 1,2 

0 ([a, b]). 

References 

[1]G. Bonanno, S. A. Marano, On the structure of the critical set of non-differentiable 

functions with a weak compactness condition, Appl. Anal. 89 (2010) 1-10. 

2 

Page 128 

⎡ 

⎢ 

⎣

The Modified Bi-quintic B-spline base functions: An Application to Diffusion Equation 

S. Kutluay 1 and N.M. Ya˘gmurlu 1 

1 Department of Mathematics, Faculty of Arts and Sciences, ˙ Inönü University, Malatya, Turkey 


In this paper, the bi-quintic B-spline base functions are modified on a general 2-dimensional problem 

and then they are applied to two-dimensional Diffusion problem in order to obtain its numerical solutions. 

The computed results are compared with the results given in the literature. The error norms L2 and L∞ 

are computed and found to be marginally accurate and efficient. 

References 

[1] B. S. Moon, D. S. Yoo,Y.H. Lee, I.S. Oh, J.W. Lee, D.Y. Lee and K.C. Kwon, A non-separable 

solution of the diffusion equation based on the Galerkin’s method using cubic splines , Appl. Math. and 

Comput., 217 (2010) 1831–1837. 

[2] K.N.S. Kasi Viswanadham, S.R. Koneru, Finite element method for one-dimensional and two- 

dimensional time dependent problems with B-splines, Comput. Methods Appl. Mech. Engrg. 108, 

(1993) 201-222. 

[3] L.R.T. Gardner, G.A. Gardner, A two dimensional bi-cubic B-spline finite element: used in a study 

of MHD-duct flow, Comput. Methods Appl. Mech. Engrg. 124 (1995) 365-375. 

[4] J. Wu, X. Zhang, Finite Element Method by Using Quartic B-Splines, Numerical Methods for 

Partial Differential Equations, 10 (2011) 818-828 

Page 129

Numerical Solutions of the Modified Burgers’ Equation by Cubic B-spline Collocation 

Method 

S. Kutluay 1 , Y. Ucar 1 and N.M. Ya˘gmurlu 1 

1 Department of Mathematics, Faculty of Arts and Sciences, ˙ Inönü University, Malatya, Turkey 


In this paper, a numerical solution of the modified Burgers’ equation is obtained by a cubic B-spline 

collocation method. In the solution process, a linearization technique has been applied to deal with the 

non-linear term appearing in the equation. The computed results are compared with the results given in 

the literature. The error norms L2 and L∞ are also computed and found to be sufficiently small. 

References 

[1] D. Irk, Sextic B-spline collocation method for the modified Burgers’ equation, Kybernetes , 38 

(2009) 1599–1620. 

[2] M. A. Ramadan and T. S. El-Danaf, Numerical treatment for the modified burgers equation, 

Mathematics and Computers in Simulation 70, (2005) 90-98. 

[3] T. Roshan and K.S. Bhamra, Numerical solutions of the modified Burgers’ equation by Petrov- 

Galerkin method, Applied Mathematics and Computation 218 (2011) 3673-3679. 

[4] A. G. Brastos and L. A. Petrakis, An explicit numerical scheme for the modified Burgers’ equation, 

International Journal for Numerical Methods in Biomedical Engineering, 27 (2011) 232-237. 

Page 130

The Modified Kudryashov Method for Solving Some Evolution Equations 


S.M. Ege 1 and E. Misirli 1 


The study of numerical methods for solving partial differential equations and the travelling wave 

solutions of these equations have significant roles in physical science over the last decades 

from both theoretical and the practical points of view. Mathematical physics consist of many 

mathematical models which described by the nonlinear partial differential equations. The 

investigation of the travelling wave solutions of nonlinear evolution equations appears in various 

scientific fields, such as plasma physics, fluid mechanics, hydrodynamic, optical fibers, chemical 

physics. Many powerful and effective methods are used for investigating the explicit travelling 

wave solutions. 

In this paper, we have applied the modified Kudryashov method for solving some nonlinear 

evolution equations by the help of commutative algebra. This method is applicable for the other 

nonlinear partial differential equations. 

We consider the general nonlinear partial differential equation for a function u of two variables, space x 

and time t : 

P( u, ut , ux, uxx, utt , uxt ,...) � 0 

(1) 

It is useful to summarize the steps of modified Kudryashov method as follows[5]: 

Step 1. We investigate the travelling wave solutions of Eq.(1) of the form: 

u( x, t) � u( �), 

� �kx� wt, 

(2) 

where k and w are arbitrary constants. Then Eq.(1) reduces to a nonlinear ordinary differential equation 

of the form: 

G( u, u� , u�� , u�� 

,...) � 0 

(3) 

Step 2. We suppose that the exact solutions of Eq.(3) can be obtained in the following form: 

Page 131

N 

i 

u( �) � y( �) 

�� a Q , 

(4) 

i 

i�0 

1 

where Q 

1 e � � and the function Q is the solution of equation 

� 

2 

Q� Q Q 

� � (5) 

Step 3. According to the method, we assume that the solution of Eq.(3) can be expressed in the form 

N 

u( �) �aQ� ... 

(6) 

N 

Calculation of value N in formula (6) that is the pole order for the general solution of Eq. (3). In order to 

determine the value of N we balance the highest order nonlinear terms in Eq. (3) analogously as in the 

classical Kudryashov method. Supposing 

of Eq. (3) and balancing the highest order nonlinear terms we have: 

Step 4. Substituting Eq.(4) into Eq.(3) and equating the coefficients of 

l ( s) 

p r 

u ( �) u ( � ) and ( u ( � )) are the highest order nonlinear terms 

s � rp 

N � , 

(7) 

r�l� 1 

algebraic equations. By solving this system, we obtain the exact solutions of Eq.(1). 

References 

i 

Q to zero, we get a system of 

[1] Kudryashov N A 2012 One method for finding exact solutions of nonlinear differential equations 

Commun. Nonlinear Sci. 17 2248–2253 

[2] Kudryashov N A 1990 Exact solutions of the generalized Kuramoto–Sivashinsky equation Phys. 

Lett. A 147 287–91 

[3] Kabir M M, Khajeh A, Aghdam E A, Koma A Y 2011 Modified Kudryashov method for finding exact 

solitary wave solutions of hihher-order nonlinear equations Math. Methods Appl. Sci. 35 213-219 

[4] Kabir M M 2011 Modified Kudryashov method for generalized forms of the nonlinear heat 

conduction equation Int. J. Phys. Sci. 6 6061-6064 

[5] Pandir Y, Gurefe Y, Misirli E 2012 A new approach to Kudryashov's method for solving some 

nonlinear physical models Int. J. Phys. Sci.7 2860-2866 

[6] Ryabov P N, Sinelshchikov D I, Kochanov M B 2011 Application of Kudryashov method for finding 

exact solutions of higher order nonlinear evolution equations Appl. Math.Comput. 218 3965-3972 

Page 132

Study of an inverse problem that models the detection of corrosion in metalic plate whose 

lower part is embedded 

SAIDMohamed Said 

Laboratoire LMA University of Kasdi Merbah Faculty of Sciences and technology Ouargla, 

30000 Ouargla Algeria 


In this work, we will study an inverse problem to determine corrosion in an inaccessible location 

of a metalic plate. Our study area is inside the plate metalic plate whose lower part is embedded 

therefore inacssecible. We will perform measurements on the upper part of the plate, which is 

not in contact with the ground. For this, we will send an electric field on this part and take 

measurements. This problem is modeled by a Laplace problem with mixed presence of an 

unknown term in the boundary conditions this term is an unknown function which can take 

several forms. It is this function that we will detect the presence or absence of corrosion inside 

the tube and we will then follow our steps to the top edge of the field information on the 

evolution of this corrosion. We will first formulate our problem which is an inverse problem and 

we will make a theoretical study and we will that this problem has a unique solution also this 

solution is stable. After, we will solve this problem by constructing an iterative algorithm which 

gives problems that will cross a series of impedance functions which determines the rate of 

corrosion. Finally we study the convergence and we will then make a numerical application 

References 

Page 133 

[1]P. Grisvard, Alternative of Fredholm relating to the problem of Dirichlet in a polygon, Boll. 

Un. Mat. Ital. 5(4) (1972), 132-164 

[2]P. Grisvard, Singularities in boundary values problem, Dunod Paris (1992) 

[3] BRESIS. H . Analyse Fonctionnelle Masson Paris 1983 

[4] J. Cheng M Chouli X Yang An iterative BEM method for the inverse problem of detecting 

corrosion in a pipe 

numerical Mathematics A journal of Chineese universities, Vol 14 N°3 Ang 2005 

[5] A Tveito , R Winter Introduction to partial differential equations a computitionnal approach 

springer 2008- 

[6] M Chouli introduction aux problèmes inverses elliptiques et paraboliques Springer Vering 

Berlin Heidlberg 2009 

[7] M Chouli Stability estimates for an inverse elliptic problem Journal of inverse problems 3 

posed Prob,(10), 2002 

N°6, 601-610.


Commuting nilpotent operators with maximal rank 

Semra Öztürk Kaptano˘glu 

Department of Mathematics, Orta Do˘gu Teknik Üniversitesi, Ankara, T¨rkiye 

Let X, ˆ X be commuting nilpotent matrices over a field k with nilpotency p t . We show that if X − ˆ X 

is a certain linear combination of products of commuting nilpotent matrices, then X is of maximal rank 

if and only if ˆ X is of maximal rank. In the case, k is an algebraically closed field of positive characteristic 

p there is an interpretation about module over group algebras. 

References 

[1] J. F. Carlson, E. Friedlander, J. Pevtsova, Modules of constant Jordan type, J. Reine Angew. 

Math., 614, 191234, 2008. 

[2] S. Ö. Kaptano˘glu, Commuting Nilpotent Operators and Maximal Rank, Complex Anal. Oper. 

Theory, 4, 901–904, 2010. 

Page 134


Finding Global minima with a new class of …lled function 

T. Hamaizia 

Department of Mathematics, Larbi Ben M’hidi University, Oum Elbouaghi, Algeria. 

Global Optimization problems arise in many …elds of science and technology [2-4]. Filled function 

method is a type of e¢ cient methods to obtain the global solution of a multivariable function. The key 

idea of the …lled function method is to leave from a current local minimizer x to a lower minimizer x of 

the original objective function f(x) with the auxiliary function P (x) constructed at the local minimizer. 

This method was introduced in Ge’s paper [1] for continuous global optimization problem, the …rst …lled 

has the form 

p(x; r; ) = 

where r and are two adjustable parametres. 

1 

r + f(x) exp 

kx x k k 2 

This paper gives a new de…nition of the …lled function. It shows that the …lled function given in some 

paper are the special forms of this …lled functions. 

References 

[1] Ge Renpu, A …lled function method for …nding global minimizer of a function of several variables[J]. 

mathematical programming. 46(1990) 191–204.. 

[2] Ge R.P.and Qin Y.F., A class of …lled functions for …nding a global minimizer of a function of 

several variables[J]. Journal of optimization theory and applications. 54(1987) 241–252. 

[3] Xian Liu, Finding global minima with a computable …lled function. Journal of global optimization. 

19(2001) 151–161. 

[4] Xian Liu, Wilsun Xu.: A new …lled function applied to global optimization. Computer and 

Operation Research. 31(2004) 61–80. 

2 

! 

Page 135

Weak Convergence Theorem For A Semi-Markovian Random Walk With 

Delay And Pareto Distributed Interference Of Chance 

T. Kesemen 1 and F. Yetim 2 

1 Karadeniz Technical University, Faculty of Sciences, Department of Mathematics, Trabzon, 

Turkey 

2 Avrasya University, Faculty of Art and Science, Department of Mathematics, Trabzon, Turkey 


In this study, a semi-Markovian random walk with delay and a discrete interference of 

chance �X(t) � is constructed. The weak convergence theorem is proved for the ergodic 

distribution of the process X(t) and the limit form of the ergodic distribution is found, when the 

random variables ��n �, 

n � 0 have Pareto distribution with parameters ( �� , ) where the random 

variables � describe the discrete interference of chance. 

n 

References 

Page 136 

[1] Aliyev R.T, Khaniyev T.A., Kesemen T., Asymptotic expansions for the moments of a semi- 

Markovian random walk with gamma distributed interference of chance, Communications in 

Statistics-Theory and Methods, 39, 130-143, 2010. 

[2] Feller W., Introduction to Probability Theory and Its Applications II, J. Wiley, New York, 

1971. 

[3] Khaniyev T.A., Atalay K.D., On the weak convergence theorem for the ergodic distribution 

for an inventory model of Type (s,S), Hacettepe Journal of Mathematics and Statistics, 39(4), 

599-611, 2010.

PARAMETER DEPENDENT NOVIER-STOKES LIKE 

PROBLEMS 

VELI B. SHAKHMUROV 

Department of Mechanical Engineering, Okan University, Ak…rat, Tuzla 34959 

Istanbul, Turkey, 

E-mail: veli.sahmurov@okan.edu.tr 


In this talk, the following nonstationary Novier-Stokes like equation with 

variable coe¢ cients 

@u 

@t A" (x) u + (u:r) u + r' = f (x; t) ; div u = 0; x 2 G; t 2 (0; T ) ; 

L1"u = X 

" i 

is considered, where 

R n + = 

i=0 

@ i u 

i 

@xi n 

x 0 

; 0; t = 0, 2 f0; 1g ; 

u (x; 0) = a (x) ; x 2 R n +; t 2 (0; T ) ; 

n 

x 2 R n ; xn > 0; x = x 0 

; xn ; x 0 

= (x1; x2; :::; xn 1) 

A" (x) u = " 

nX 

k=1 

ak (x) @2u @x2 ; i = 

k 

1 

2 

i + 1 

q 

, q 2 (1; 1) , 

" is a small positive parameter, i are complex numbers, ak are continious 

functions on R n n; 

u = u" (x) = (u1" (x; t) ; u2" (x; t) ; :::; un" (x; t)) 

are represent the unknown velocity, f = (f1 (x; t) ; f2 (x; t) ; :::; fn (x; t)) represents 

a given external force and a denotes the initial velocity. 

The existence, uniqueness and L p estimates of solution the above problem 

is derived. 

1 

o 

; 

Page 137

A New Spline Approximation for the Solution of One-space 

Dimensional Second Order Non-linear Wave Equations 

With Variable Coefficients 

VENU GOPAL and R. K. MOHANTY 

Department of Mathematics 

Faculty of Mathematical Sciences 

University of Delhi 

Delhi-110 007, INDIA 

Abstract: In this paper, we propose a new three-level implicit nine point 

compact finite difference formulation of order two in time and four in space 

directions, based on non-polynomial spline in compression for the solution of 

one-space dimensional second order non-linear hyperbolic partial differential 

equations with variable coefficients and significant first order space derivative 

term. We describe the Mathematical formulation procedure in details and also 

discussed the stability. Numerical results are provided to justify the usefulness 

of the proposed method. 

Keywords: Non-polynomial spline in compression; Non-linear Wave equation; 

Maximum absolute errors 

REFERENCES 

1. Mohanty, R. K.; Gopal, Venu. High accuracy cubic spline finite difference 

approximation for the solution of one-space dimensional non-linear wave equations. 

Appl. Math. Comput. 218 (2011), no. 8, 4234–4244. 

2. Jain, M. K.; Aziz, Tariq. Spline function approximation for differential equations. 

Comput. Methods Appl. Mech. Engrg. 26 (1981), no. 2, 129–143. 

Page 138 

3. Jain, M. K.; Aziz, Tariq. Cubic spline solution of two-point boundary value 

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Engrg. 38 (1983), no. 2, 137–151. 

5. Jain, M. K. Numerical solution of differential equations. Second edition. A Halsted 

Press Book. John Wiley & Sons, Inc., New York, 1984. 

6. W.G. Bickley, Piecewise cubic interpolation and two point boundary value problems, 

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7. D. J. Fyfe, The use of cubic splines in the solution of two point boundary value 

problems, Comput. J. 12 (1969) 188–192. 

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second order boundary value problems, J. Optim. Theory Appl., 118 (2003) 45-54. 

9. Khan, Arshad; Khan, Islam; Aziz, Tariq A survey on parametric spline function 

approximation. Appl. Math. Comput. 171 (2005), no. 2, 983–1003. 

10. J. Rashidinia ; Mohammadi, R.; Jalilian, R. Spline methods for the solution of 

hyperbolic equation with variable coefficients. Numer. Methods Partial Differential 

Equations 23 (2007), no. 6, 1411–1419. 

11. J. Rashidinia, R. Jalilian, V. Kazemi, Spline methods for the solutions of hyperbolic 

equations, Appl. Math. Comput., 190 (2007) 882-886. 

12. Rashidinia, J.; Mohammadi, R. Non-polynomial cubic spline methods for the 

solution of parabolic equations. Int. J. Comput. Math. 85 (2008), no. 5, 843–850. 

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problems. Appl. Comput. Math. 9 (2010), no. 2, 258–266. 

Page 139 

14. Hengfei Ding, Yuxin Zhang, Parametric spline methods for the solution of 

hyperbolic equations, Appl. Math. Comput., 204 (2008) 938-941. 

15. C.T. Kelly, Iterative Methods for Linear and Non-linear Equations, SIAM 

Publications, Philadelphia, 1995. 

16. L.A. Hageman and D.M. Young, Applied Iterative Methods, Dover Publication, New 

York, 2004.

An error correction method for solving stiff initial value problems based on a 

cubic C 1 -spline collocation method 


Xiangfan Piao a , Sang Dong Kim a , Philsu Kim a,1,∗ 

a Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea 

For solving nonlinear stiff initial value problems, we develop an improved error correction method (IECM) which 

originates from the error corrected Euler methods (ECEM) recently developed by the authors (see [17, 18]) and 

reduces the computational cost and further enhances the stability for the ECEM. We use the stabilized cubic C 1 -spline 

collocation method instead of the Chebyshev collocation method used in ECEM for solving the asymptotic linear 

ODE for the difference between the Euler polygon and the true solution. It is proved that IECM is A-stable, a semiimplicit 

one-step method, and of order 4 with only one evaluation of the Jacobian at each integration step. Also, 

we use the iteration process of the Lobatto IIIA method developed by [13] for solving the induced matrix system. 

It is shown that this iteration process does not require such the nonlinear function evaluation as the implicit method 

does and hence it reduces the numerical computational cost efficiently. Numerical evidence is provided to support the 

theoretical results with several stiff problems. 

Keywords: Euler polygon, Cubic C 1 -spline collocation method, Lobatto IIIA method, Error correction method, Stiff 

initial value problem 

References 

[1] C.A. Addisonand I. Gladwell, Second derivative methods applied to implicit first and second order systems, Internat. J. Numer. Methods 

Engng. 20 (1984) pp. 1211–1231. 

[2] P. Amodioand L. Brugnano, A note on the efficient implementation of implicit methods for ODEs, J. Comput. Appl. Math. 87 (1997) pp. 1–9. 

[3] T.A. Bickart, An efficient solution process for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 14 (1977) pp. 1022–1027. 

[4] L. Brugnanoand C. Magherini, Blended implementation of block implicit methods for ODEs, Appl. Numer. Math. 42 (2002) pp. 29–45. 

[5] J.C. Butcher, On the implementation of implicit Runge-Kutta methods, BIT 16 (1976) pp. 237–240. 

[6] J.C. Butcherand G. Wanner, Runge-Kutta methods: some historical notes, Appl. Numer. Math. 22 (1996) pp. 113–151. 

[7] G.J. Cooperand J.C. Butcher, An iteration scheme for implicit Runge-Kutta methods, IMA J. Numer. Anal. 3 (1983) pp. 127–140. 

[8] G.J. Cooperand R. Vignesvaran, A scheme for the implementation of implicit Runge-Kutta methods, Computing 45 (1990) pp. 321–332. 

[9] G.J. Cooperand R. Vignesvaran, Some schemes for the implementation of implicit Runge-Kutta methods, J. Comput. Appl. Math. 45 (1993) 

pp. 213–225. 

[10] G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963) pp. 27–43. 

[11] S. González-Pinto, C. GonzÁlezand J.I. Montijano, Iterative schemes for Gauss methods, Comput. Math. Appl. 27 (1994) 67–81. 

[12] S. González-Pinto, J.I. Montijanoand L. Rández, Iterative schemes for three-stage implicit Runge-Kutta methods, Comput. Math. Appl. 27 

(1994) 67–81. 

[13] S. González-Pinto, S. Pérez Rodríguezand J.I. Montijano, On the numerical solution of stiff IVPs by Lobatto IIIA Runge-Kutta methods, J. 

Comput. Appl. Math. 82 (1997) pp. 129–148. 

[14] S. GonzÁlez-Pintoand R. Rojas-Bello, Speeding up Newton-type iterations for stiff problems, J. Comp. Appl. Math. 181 (2005) pp. 266–279. 

[15] E. Hairer, G. Wanner, Solving Ordinary Differential Equations, II Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991 

pp. 5-8. 

[16] E. Hairer, G. Wanner, Solving ordinary differential equations. II Stiff and Differential-Algebraic Problems, Springer Series in Computational 

Mathematics, Springer (1996). 

∗ Corresponding author 

Email addresses: piaoxf76@hanmail.net (Xiangfan Piao), skim@knu.ac.kr (Sang Dong Kim),kimps@knu.ac.kr (Philsu Kim) 

1 This work was supported by basic science research program through the National Research Foundation of Korea(NRF) funded by the ministry 

of education, science and technology (grant number 2011-0029013). 

1 

Page 140

Page 141 

An error correction method for stiff initial value problems 2 

[17] P. Kim, X. Piaoand S.D. Kim, An error corrected Euler method for solving stiff problems based on Chebyshev collocation, SIAM J. Numer. 

Anal. 49 (2011) pp. 2211–2230. 

[18] S.D. Kim, X. Piao, D.H. Kimand P. Kim, Convergence on Error correction methods for solving initial value problems, J. Comp. Appl. Math., 

236 (2012) pp. 4448–4461. 

[19] J. Kweon, S.D. Kim, X. Piaoand P. Kim, A Chebyshev collocation method for stiff initial values and its stability, Kyungpook Mathematical 

Journal 51 (2011) pp. 435–456. 

[20] H. Ramos, A non-standard explicit integration scheme for initial-value problems, Appl. Math. Comp. 189(1) (2007) pp. 710–718. 

[21] H. Ramos, J. Vigo-Aguiar, A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comp. Appl. Numer. 204 

(2007) pp. 124–136. 

[22] S. Sallamand M. Naim Anwar, Stabilized cubic C 1 -spline collocation method for solving first-order ordinary initial value problems, Intern. 

J. Computer Math. 74 (2000) pp. 87–96. 

[23] E. Schäfer, A new approach to explain the ‘high irradiance responses’ of photomorphogenesis on the basis of phytochrome, J. Math. Biology. 

2 (1975) 41–56. 

[24] J.G. Verwer, Gauss-Seidel iteration for stiff odes from chemical kinetics, SIAM J. Sci. Comput. 15 (1994) pp. 1243–1250. 

[25] X.Y. Wu, J.L. Xia, Two low accuracy methods for stiff systems, Appl. Math. Comput. 123 (2001) pp. 141-153.


On Numerical Solution of Multipoint NBVP 

for Hyperbolic-Parabolic Equations with Neumann Condition 

A. Ashyralyev 1 and Y. Ozdemir 2 


2 Department of Mathematics, Duzce University, Duzce, Turkey 

Certain problems of modern physics and technology can be effectively described in terms of nonlocal 

problem for partial differential equations. These nonlocal conditions arise mainly when the data on the 

boundary cannot be measured directly. Methods of solutions of nonlocal boundary value problems for 

partial differential equations and partial differential equations of mixed type have been studied extensively 

by many researchers in [1-5]. 

In this paper, numerical solutions of difference schemes of multipoint nonlocal boundary value problem 

for multidimensional hyperbolic-parabolic equation with Neumann condition are considered. The first 

and second orders of accuracy difference schemes are established. The theoretical statements for the 

solution of these difference schemes are supported by results of numerical experiments. 

References 

[1] Ashyralyev A. and Aggez N., A note on difference schemes of the nonlocal boundary value problems 

for hyperbolic equations, Num. Func. Anal. & Opt., 25(5-6), 439-462, 2004. 

[2] Ashyralyev A. and Gercek O., Nonlocal boundary value problems for elliptic-parabolic differential 

and difference equations, Dis. Dyn. in Nat. & Soc., 2008(2008), 1-16, 2008. 

[3] Ashyralyev A. and Ozdemir Y., On stable implicit difference scheme for hyperbolic-parabolic 

equations in a Hilbert space, Num. Math. for Par. Diff. Eqn., 25(5), 1110-1118, 2009. 


differential and difference equations, Tai. Jour. of Math.., 14(1), 165-164, 2010. 

[5] Koksal M. E., Recent developments on operator-difference schemes for solving nonlocal BVPs for 

the wave equation, Dis. Dyn. in Nat. & Soc., 2011(2011), 1-14, 2011. 

Page 142

Classification of exact solutions for the Pochhammer-Chree equations 


Y. Gurefe 1 , Y. Pandir 1 and E. Misirli 2 

1 Department of Mathematics, Bozok University, Yozgat, Turkey 


In this study, exact solutions to the Pochhammer-Chree equations are obtained by using complete 

discrimination system. These solutions can be reduced to soliton solution, rational and elliptic 

function solutions. Also, we propose a more general method for the generalized nonlinear partial 

differential equations. 

Keywords: Trial equation method, Soliton solutions, Elliptic function solutions. 

References 

Page 143 

[1] Malfliet W., Hereman W., The tanh method: exact solutions of nonlinear evolution and 

wave equations, Phys. Scr., 54, 563-568, 1996. 

[2] He J.H., Wu X.H., Exp-function method for nonlinear wave equations, Chaos Soliton. 

Fract., 30, 700-708, 2006. 

[3] Misirli E., Gurefe Y., Exp-function method for solving nonlinear evolution equations, 

Math. Comput. Appl. 16, 258-266, 2011. 

[4] Wazwaz A.M., The tanh-coth and the sine-cosine methods for kinks, solitons, and 

periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput., 195, 24-33, 

2008. 

[5] Jibin L., Lijun Z., Bifurcations of travelling wave solutions in generalized Pochhammer- 

Chree equation, Chaos Soliton. Fract., 14(4), 581-593, 2002. 

[6] Li B., Chen Y., Zhang H., Travelling wave solutions for generalized Pochhammer- 

Chree equations, Z. Naturforsch, 57(a), 874-882, 2002.

On the Density of Regular Functions in Variable Exponent Sobolev Spaces 


Yasin KAYA 

Dicle University, Diyarbakır,Turkey 

The talk will deal with when every function in a variable exponent Sobolev space can be 

approximated by a more regular function, such as a smooth or Lipschitz continuous function. 

Many researchers have made contributions , but still remain substantial gaps in our 

understanding of this intricate question. A discussion on methods also will take place. I will 

also give some my results for the density in variable exponent Sobolev spaces . 

References 

[ ] Cruz-Uribe, D., Fiorenza,A. Approximate identities in variable spaces. - Math. Nachr. 

280, 2007, 256–270. 

[ ] Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M., Lebesgue and Sobolev Spaces 

withVariable Exponents. Springer, 2011. 

Page 144 

[ ] Edmunds, D. E., Rakosnik, J. Density of smooth functions in ( ) ( ), Proc. Roy. Soc. 

London.Ser. A 437 (1992), 229,236. 

[ ] Fan, X. L., Wang, S., Zhao, D.. Density of ( ) in ( ) ( ) with discontinuous 

exponent ( ). Math. Nachr., 279:142–149, 2006 

[ ] Hastö, P. Counter-examples of regularity in variable exponent Sobolev spaces. InThe p-harmonic 

equation and recent advances in analysis, volume 370 of Contemp.Math., pages 133–143. Amer. 

Math. Soc., Providence, RI, 2005. 

[ ] Hastö, P. On the density of smooth functions in variable exponent Sobolev space.Rev. Mat. 

Iberoamericana, 23:215–237, 2007. 

[ ] Samko, S. Denseness of ( ) in the generalized Sobolev spaces ( ) ( ), pp. 333,342 in 

Directand inverse problems of mathematical physics (Newark, DE, 1997), Int. Soc. Anal. Appl. 

Comput. 5,Kluwer Acad. Publ., Dordrecht, 2000.


Numerical Solution of a Hyperbolic-Schrödinger 

Equation with Nonlocal Boundary Conditions 

Y. Ozdemir and M. Kucukunal 

Department of Mathematics, Duzce University, Duzce, Turkey 

A numerical method is proposed for solving hyperbolic-Schrödinger partial di¤erential equations with 

nonlocal boundary condition. The …rst and second orders of accuracy di¤erence schemes are presented. 

A procedure of modi…ed Gauss elimination method is used for solving these di¤erence schemes in the case 

of a one-dimensional hyperbolic-Schrödinger partial di¤erential equations. The method is illustrated by 

numerical examples. 

References 

[1] Ashyralyev A. and Aggez N., A note on di¤erence schemes of the nonlocal boundary value problems 

for hyperbolic equations, Num. Func. Anal. & Opt., 25(5-6), 439-462, 2004. 

[2] Ashyralyev A. and Gercek O., Nonlocal boundary value problems for elliptic-parabolic di¤erential 

and di¤erence equations, Dis. Dyn. in Nat. & Soc., 2008(2008), 1-16, 2008. 

[3] Ashyralyev A. and Ozdemir Y., On stable implicit di¤erence scheme for hyperbolic-parabolic 

equations in a Hilbert space, Num. Math. for Par. Di¤. Eqn., 25(5), 1110-1118, 2009. 


di¤erential and di¤erence equations, Tai. Jour. of Math.., 14(1), 165-164, 2010. 

[5] Ashyralyev A. and Sirma A., Modi…ed Crank-Nicholson di¤erence schemes for nonlocal boundary 

value problem for the Schrodinger equation, Dis. Dyn. in Nat. & Sci., 10.1155/2009/584718, 2009. 

[6] Simos T. E., Exponentially and trigonometrically …tted methods for the solution of the the 

Schrödinger equation, Acta App. Math., 110(3), 1331-1352, 2010. 

Page 145

New generalized hyperbolic functions to find exact solution of the nonlinear partial 


differential equation 

Y. Pandir 1 and H. Ulusoy 2 


In this article, we first time define new functions (called generalized hyperbolic functions) and devise 

new kinds of transformation (called generalized hyperbolic function transformation) to construct new 

exact solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function 

transformation of the generalized KdV equation. We obtain abundant families of new exact solutions of 

the equation and analyze the properties of this by taking different parameter values of the generalized 

hyperbolic functions. As a result, we find that these parameter values and the region size of the inde- 

pendent variables affect some solution structure. These solutions may be useful to explain some physical 

phenomena. 

1 Introduction 

To construct exact solutions to nonlinear partial differential equations, some important methods have 

been defined such as Hirota method, tanh-coth method, the exponential function method, (G ′ /G)- 

expansion method, the trial equation method, and so on [1-15]. There are a lot of nonlinear evolution 

equations that are integrated using the various mathematical methods. Soliton solutions, compactons, 

singular solitons and other solutions have been found by using these approaches. These types of solutions 

are very important and appear in various areas of applied mathematics. In Section 2, we give the defi- 

nition and properties of generalized hyperbolic functions. In Section 3, as applications, we obtain exact 

solution of the generalized KdV equation 

(u l )t + αu(u n )x + β[u(u n )xx]x + γu(u n )xxx = 0. (1) 

2 The definition and properties of the symmetrical hyperbolic 

Fibonacci and Lucas functions 

In this section, we will define new functions which named the symmetrical hyperbolic Fibonacci and 

Lucas functions for constructing new exact solutions of NPDEs, and then study the properties of these 

functions. 

Definition 2.1 Suppose that ξ is an independent variable, p, q and k are all constants. The generalized 

hyperbolic sine function is 

generalized hyperbolic cosine function is 

Page 146 

sinha(ξ) = pakξ − qa−kξ , (2) 

2 

cosha(ξ) = pakξ + qa−kξ , (3) 

2

generalized hyperbolic tangent function is 

generalized hyperbolic cotangent function is 

generalized hyperbolic secant function is 

generalized hyperbolic cosecant function is 

tanha(ξ) = pakξ − qa−kξ pakξ , (4) 

+ qa−kξ cotha(ξ) = pakξ + qa−kξ pakξ , (5) 

− qa−kξ secha(ξ) = 

cosecha(ξ) = 

2 

pakξ , (6) 

+ qa−kξ 2 

pakξ , (7) 

− qa−kξ the above six kinds of functions are said generalized new hyperbolic functions. Thus we can prove the 

following theory of generalized hyperbolic functions on the basis of Definition 2.1. 

Theorem 2.1. The generalized hyperbolic functions satisfy the following relations: 

cosh 2 a(ξ) − sinh 2 a(ξ) = pq, (8) 

1 − tanh 2 a(ξ) = pq.sech 2 a(ξ), (9) 

1 − coth 2 a(ξ) = −pq.cosech 2 a(ξ), (10) 

secha(ξ) = 

cosecha(ξ) = 

The following just part of them are proved here for simplification. 

1 

, (11) 

cosha(ξ) 

1 

, (12) 

sinha(ξ) 

tanha(ξ) = sinha(ξ) 

, (13) 

cosha(ξ) 

cotha(ξ) = cosha(ξ) 

. (14) 


Theorem 2.2. The derivative formulae of generalized hyperbolic functions as following 

d(sinha(ξ)) 

dξ 

d(cosha(ξ)) 

dξ 

d(tanha(ξ)) 

dξ 

d(cotha(ξ)) 

dξ 

d(secha(ξ)) 

dξ 

d(cosecha(ξ)) 

dξ 

Proof of (17): According to (15) and (16), we can get 

d(tanha(ξ)) 

dξ 

= 

= k ln a cosha(ξ), (15) 

= k ln a sinha(ξ), (16) 

= kpq ln a sech 2 a(ξ), (17) 

= −kpq ln a cosech 2 a(ξ), (18) 

= −k ln a secha(ξ) tanha(ξ), (19) 

= −k ln a cosecha(ξ) cotha(ξ). (20) 

( ) ′ 


= 

cosha(ξ) 

(sinha(ξ)) ′ cosha(ξ) − (cosha(ξ)) ′ sinha(ξ) 

cosh 2 a(ξ) 

Page 147

= k ln a cosh2a(ξ) − k ln a sinh 2 a(ξ) 

cosh 2 = kpqsech 

a(ξ) 

2 ( ξ), (21) 

Similarly, we can prove other differential coefficient formulae in Theorem 2.2. 

Remark 2.1. We see that when p = 1, q = 1, k = 1 and a = e in (2)-(7), new generalized hyperbolic 

function sinha(ξ), cosha(ξ), tanha(ξ), cotha(ξ), secha(ξ) and cosecha(ξ), degenerate as hyperbolic func- 

tion sinh(ξ), cosh(ξ), tanh(ξ), coth(ξ), sech(ξ) and cosech(ξ), respectively. In addition, when p = 0 or 

q = 0 in (2)-(7), sinha(ξ), cosha(ξ), tanha(ξ), cotha(ξ), secha(ξ) and cosecha(ξ), degenerate as exponen- 

tial function 1 

2 pak(ξ) , ± 1 

2 qa−k(ξ) , 2pa −k(ξ) , ±2qa k(ξ) and ±1, respectively. 

References 

[1] Hirota R., Exact solutions of the Korteweg-de-Vries equation for multiple collisions of solitons, 

Phys. Lett. A, 27, 1192-1194, 1971. 

[2] Malfliet W., Hereman W., The tanh method: exact solutions of nonlinear evolution and wave 

equations, Phys. Scr., 54, 563-568, 1996. 

[3] Misirli E., Gurefe Y., Exp-function method for solving nonlinear evolution equations, Math. Com- 

put. Appl., 16, 258-266, 2011. 

[4] Ismail M.S., Biswas A., 1-Soliton solution of the generalized KdV equation with, Appl. Math. 

Comput., 216, 1673-1679, 2010. 

[4] Ren Y., Zhang H., New generalized hyperbolic functions and auto-Bcklund transformation to find 

new exact solutions of the (2+1)-dimensional NNV equation, Phys. Lett. A, 357, 438-448, 2006. 

Page 148

EQUIVALENCE OF AFFINE CURVES 

Yasemin SAĞIROĞLU 

Karadeniz Technical University 

Science Faculty 

Mathematics Department 

TRABZON/ TURKEY 

ysagiroglu@ktu.edu.tr 

ABSTRACT 

The definitions of affine curve, affine arclength and affine types of an affine curve are 

2 

given in R . Affine invariant parametrization of an affine curve which is invariant under the 

affine group is introduced. The complete system of affine differential invariants for affine 

plane curves is obtained and we show that these invariants are independent. The conditions of 

2 

equivalence of two affine curves is obtained in terms of affine differential invariants in R . 

Keywords: Affine geometry, affine curve, affine differential invariants, affine 

equivalence. 

MSC: 53A15, 53A55. 

References 

[1] E. De Angelis, T. Moons, L. Van Gool and P. Verstraelen, Complete system of affine 

semi-differential invariants for plane and space curves, In: Dillen, F. (ed.) et al., Geometry 

and topology of submanifolds, VIII, Proceedings of the international meeting on geometry of 

submanifolds, Brussel, Belgium, July 13-14 (1995) and Nordfjordeid, Norway, July 18- 

August 7 (1995). Singapore. World Scientific (1996), 85-94. 

[2] W. Barthel, Zur affinen Differentialgeometrie-Kurventheorie in der allgemeinen 

Affingeometrie, Proceedings of the Congress of Geometry, Thessaloniki (1987), 5-19. 

[3] W. Blaschke, Affine Differentialgeometrie, Berlin, 1923. 

[4] E. Cartan, La théorie des groupes finis et continus et la géometrie différentielle, Gauthier- 

Villars, Paris, 1951. 

[5] R.B. Gardner and G.R. Wilkens, The fundamental theorems of curves and hypersurfaces 

in centro-affine geometry, Bull. Belg. Math. Soc. 4 (1997), 379-401. 

[6] H.W. Guggenheimer, Differential Geometry, McGraw-Hill, New York, 1963. 

Page 149 

[7] S. Izumiya and T. Sano, Generic affine differential geometry of space curves, Proceedings 

of the Royal Society of Edinburg 128A (1998), 301-314. 

[8] D. Khadjiev, The Application of Invariant Theory to Differential Geometry of Curves, Fan 

Publ., Tashkent, 1988.

[9] D. Khadjiev and Ö. Pekşen, The complete system of global differential and integral 

invariants for equi-affine curves, Differential Geom. Appl. 20 (2004), 167-175. 

[10] Ö. Pekşen, D. Khadjiev, On invariants of curves in centro-affine geometry, J. Math. 

Kyoto Univ. 44 (2004), no.3, 603-613. 

[11] P.A. Schirokow, A.P. Schirokow, Affine Differentialgeometrie, Teubner, Leipzig, 1962. 

[12] W. Klingenberg, A Course in Differential Geometry, Springer-Verlag, New York, 1978. 

[13] K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge Univ. Pres, 1994. 

[14] H.P. Paukowitsch, Begleitfiguren und Invariantensystem minimaler 

Differentiationsordnung von Kurven im reellen n-dimensionalen affinen Raum, Mh. Math. 85 

(1978), no.2, 137-148. 

[15] J.P. Giblin, G. Sapiro, Affine-invariant distances, envelopes and symmetry sets, Geom. 

Dedicata 71 (1998), 237-261. 

[16] E.J.N. Looijenga, Invariants of quartic plane curves as automorphic forms, Contemp. 

Math. 422 (2007), 107-120. 

[17] Y. Sağıroğlu and Ö.Pekşen, The Equivalence of Equi-affine Curves, Turk. J. Math. 34 

(2010), 95-2011. 

[18] Y. Sağiroğlu, The equivalence of curves in SL(n,R) and its application to ruled surfaces, 

Appl. Math. Comput. 218 (2011), 1019-1024. 

[19] B. Su, Affine Differential Geometry, Science Pres, Beijing, Gordon and Breach, New 

York, 1983. 

[20] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, NJ, 1946. 

Page 150

Modified trial equation method for nonlinear differential equations 


Y. A Tandogan 1 , Y. Pandir 1 and Y. Gurefe 1,2 



In this research, we defined a new approach with respect to the trial equation method. This 

method is applied for constructing the soliton solutions, rational function solutions and elliptic 

function solutions. Also, we conclude that the modified trial equation method can be extended to 

solve other physical problems in nonlinear science. 

Keywords: Modified trial equation method, Soliton solutions, Elliptic function solutions. 

References 

Page 151 

[1] Gurefe Y., Sonmezoglu A., Misirli E., Application of the trial equation method for 

solving some nonlinear evolution equations arising in mathematical physics, Pramana-J. Phys., 

77, 1023-1029, 2011. 

[2] Gurefe Y., Sonmezoglu A., Misirli E., Application of an irrational trial equation method 

to high-dimensional nonlinear evolution equations, J. Adv. Math. Stud., 5, 41-47, 2012. 

[3] Jun C.Y., Classification of traveling wave solutions to the Vakhnenko equations, 

Comput. Math. Appl., 62, 3987-3996, 2011. 

[4] Liu C.S., Applications of complete discrimination system for polynomial for 

classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. 

Commun., 181, 317-324, 2010. 

[5] Pandir Y., Gurefe Y., Kadak U., Misirli E., Classifications of exact solutions for some 

nonlinear partial differential equations with generalized evolution, Abstr. Appl. Anal., 2012, 1- 

16, 2012. 

[6] Ivanov R., On the integrability of a class of nonlinear dispersive wave equations, J. 

Nonlinear Math. Phys., 1294, 462-468, 2005.

Finite Difference Method for the Integral-Differential 

Equation of the Hyperbolic Type 

Zilal Direk ∗ and Maksat Ashyraliyev † 

∗ Department of Mathematics, Fatih University, Istanbul, Turkey, zdirek@fatih.edu.tr 

† Department of Mathematics, Bahcesehir University, 34353, Besiktas, Istanbul, Turkey, 

maksat.ashyralyyev@bahcesehir.edu.tr 

Abstract. 

Hyperbolic partial differential equations are used in many branches of physics, engineering and several areas of science, 

e.g. electromagnetic, electrodynamic, hydrodynamics, elasticity, fluid flow and wave propagation [1, 2, 3]. There is a great 

deal of work for solving these type of problems numerically and their stability in various functional spaces has received a 

great deal of importance. However, most of these works are studied in one-dimensional space (see [4, 5, 6, 7]). There are 

some studies about the numerical solution of two-dimensional hyperbolic equations with the collocation method or rational 

differential quadrature method [8, 9]. 

Let Ω be the unit open cube in the n-dimensional Euclidean space R n (0 < xk < 1, 1 ≤ k ≤ n) with the boundary S, 

¯Ω = Ω ∪ S. In [−1,1] × Ω the mixed problem for the multidimensional integral-differential equation of the hyperbolic type 

⎧ 

⎪⎨ 

⎪⎩ 

n 

vtt − ∑ (ar(x)Vxr 

r=1 

) xr = 

�t 

n 

∑ (br(p,x)vxr 

−t 

r=1 

) xr d p+ f(t,x), − 1 ≤ t ≤ 1, x = (x1,...,xn) ∈ Ω, 

v(t,x) = 0, x ∈ S, − 1 ≤ t ≤ 1, 

v(0,x) = ϕ(x), vt(0,x) = ψ(x), x ∈ ¯Ω 

is considered. In [10] it was proved that the problem (1) has a unique smooth solution v(t,x) for the smooth functions 

ar(x) ≥ δ > 0, r = 1,...,n, ϕ(x), ψ(x), x ∈ ¯Ω and f(t,x), b(t,x), t ∈ (−1,1), x ∈ Ω. Moreover, the first order of accuracy 

difference scheme was investigated. 

In the present paper the second order of accuracy difference scheme approximately solving the problem (1) is studied. The 

stability estimates for the solution of this difference scheme are established. Theoretical results are supported by numerical 

examples. 

Keywords: Finite Difference Method; Integral-Differential Equation of the Hyperbolic Type 

PACS: 02.60.Lj, 02.60.Nm, 02.70.Bf, 87.10.Ed 

REFERENCES 

Page 152 

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2. S. J. Weinsten, AICHE Journal 36, 1873–1889 (1990). 

3. K. R. Umashankar, Wave Motion 10, 493–525 (1988). 

4. A. Ashyralyev and N. Aggez, Numerical Functional Analysis and Optimization 25, 439–462 (2004). 

5. A. Ashyralyev and N. Aggez, Discrete Dynamics in Nature and Society 2011, 1–15 (2011). 

6. A. Ashyralyev, M. E. Koksal, and R. P. Agarwal, Computer Mathematics with Applications 61, 1855–1872 (2011). 

7. J. I. Ramos, Applied Mathematics and Computation 190, 804–832 (2007). 

8. M. Dehghan and A. Mohebbi, Numerical Methods for Partial Differential Equations 25, 232–243 (2009). 

9. M. Dehghan and A. Shokri, Numerical Methods for Partial Differential Equations25(2), 494–506 (2009). 

10. M. Ashyraliyev, Numerical Functional Analysis and Optimization 29(7–8), 750–769 (2008). 

11. P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdat. Voronezh. Gosud. Univ., 

Voronezh, Russia, 1975. 

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Normal Extensions of a Singular Differential Operator For First Order 


Z.I. Ismailov1 and R. Öztürk Mert2 


2 Department of Mathematics, Hitit University, Corum, Turkey 

In this work, in terms of boundary values all normal extensions of the minimal operator generated 

by a linear singular differential-operator expression for first order with operator coefficients in Hilbert 

space of vector-functions in a right half-infinite interval are described. Later on, a point spectrum of such 

extensions has been investigated. 

References 

[1] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, 

Springer, New-York, 1988. 

[2] A. Zettl, Sturm-Liouville Theory, Amer. Math. Soc., Mathematical Survey and Monographs, vol. 

121, Rhode Island, 2005. 

[3] J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktional operatoren, Math. Ann., 

102, 49-131, 1929-1930. 

[4] F. S. Rofe-Beketov, A. M. Kholkin, Spectral Analysis of Differential Operators, World Scientific 

Monograph Series in Mathematics, vol. 7, (World Scientific Publishing Co. Pte. Lad. Hanckensack, NJ), 

2005. 

[5] E. A. Coddington, Extension theory of formally normal and symmetric subspaces, Mem. Amer. 

Math. Soc., 134, 1-80, 1973. 

[6] N. Dunford and J. T. Schwartz, Linear Operators p.II, Second ed., Interscience, New York, 1963. 

[7] V. I. Gorbachuk and M. L. Gorbachuk, On boundary value problems for a first-order differential 

equation with operator coefficients and the expansion in eigenvectors of this equation, Soviet Math. Dokl., 

v.14(1), 244-248, 1973. 

[8] E. Bairamov, R. Öztürk Mert and Z. Ismailov, Selfadjoint extensions of a singular differential 

operator, J. Math. Chem., 50(5), 1100-1110, 2012. 

[9] Z. Ismailov and R. Öztürk Mert, Normal extensions of a singular multipoint differential operator 

of first order, Electronic Journal of Differential Equations, 36, 1-9, 2012. 

Page 153

Reproducing Kernel Hilbert Space Method for Solving the Pollution Problem of Lakes 

Z. Karabulut 1 and V. S. Ertürk 2 


1 Department of Mathematics, Ondokuz May¬s University, Samsun, Turkey 

2 Department of Matehmatics,Ondokuz May¬s University, Samsun, Turkey 

Pollution is a major threat for our environment. Monitoring pollution is the …rst step to save envi- 

ronment and has become possible with use of di¤erential equations. This study includes the problem of 

pollution of three lakes connected with pipes or channels [4]. Consider the following mathematical model 

describing the pollution of a system of lakes [1-3] : 

8 

>< 

>: 

dx1 

dt 

= F13 

V3 x3(t) + p(t) 

dx3 

dt 

dx2 

dt 

= F21 

V1 x1(t) 

F31 

V1 x1(t) 

= F31 

V1 x1(t) + F32 

V2 x2(t) 

F32 

V2 x2(t) 

F21 

V1 x1(t) 

F13 

V3 x3(t) 

The approximate solutions are obtained with Reproducing Kernel Hilbert Space Method [5-6] for 

three di¤erent models: impulse, step and sinusoidal. The absolute errors are calculated by comparing 

the numerical results to the analytic results. The errors are seen to be acceptable. All of the numerical 

computations have been calculated on a computer programme with MATHEMATICA . 

References 

[1] Yüzba¸s¬¸S., ¸Sahin N. and Sezer M., A Collocation Approach to Solving the Model of Pollution for 

a System of Lakes, Mathematical and Computer Modelling, 55, 330-341,2012. 

[2] Biazar J., Farrokhi L. and Islam M.R., Modeling the Pollution of a System of Lakes, Applied 

Mathematics and Computation, 178, 423-430, 2006. 

[3] Biazar J., Shahbala M. and Ebrahimi H., VIM for Solving the Pollution Problem of a System of 

Lakes, Journal Control Science and Engineering, Vol. 2010, 6 pages, 2010. 

[4] Aguirre J. and Tully D., Lake Pollution Model , 

hhttp : ==online:redwoods:cc:ca:us=instruct=darnold=deproj=Sp99=DarJoel=lakepollution:pdfi, 1999. 

[5] Geng F., Analytic Approximations of Solutions to Systems of Ordinary Di¤erential Equations with 

Variable Coe¢ cients, Mathematical Sciences,3(2), 133-146, 2009. 

[6] Li Y., Geng F. and Cui M., The Analytical Solution of a System of Nonlinear Di¤erential Equations, 

International Journal of Mathematical Analysis ,1(10), 451-462, 2007. 

Page 154 

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