Abstracts 11 International Workshop on Dynamical Systems and
Transkript
Abstracts 11 International Workshop on Dynamical Systems and
Abstracts 11th International Workshop on Dynamical Systems and Applications Çankaya University Ankara, TURKEY June 26–28, 2012 Preface The 11th International Workshop on Dynamical Systems and Applications is held at Çankaya University, Ankara, Turkey, during June 26-28, 2012. These workshops constitute the annual meetings of the series of dynamical systems seminars traditionally organized at Middle East Technical University throughout each academic year. The theme of this current workshop will be ”Fractional Differential Equations and Dynamic Equations with Applications”. However, talks are not restricted to these subjects only. The workshop brings together about 120 mathematicians from 14 countries. We would like to express our gratitude to Çankaya University, Turkish Mathematical Society Ankara Branch, and Casio-Mersa system administration for their support and sponsorship of the meeting. In addition, the staff at the office of Public Relations at Çankaya University, the Cultural Affairs office staff and the secretary at the Department of Mathematics and Computer Science, the Dean’s office secretary at the Faculty of Arts and Sciences, as well as many faculty and students of the Department of Mathematics and Computer Science deserve heartfelt thanks. Organizing committee co-chairs: Billur Kaymakçalan and Ağacık Zafer i Scientific Committee Ağacık Zafer, Middle East Technical University Allaberen Ashyralyev, Fatih University Aydın Tiryaki, İzmir University Azer Khanmamedov, Hacettepe University Etibar Panakhov, Fırat University Gusein Guseinov, Atılım University Okay Çelebi, Yeditepe University Ömer Akın, TOBB Economy and Technology University Varga Kalantarov, Koç University Organizing Committee Ağacık Zafer, Middle East Technical University (co-chair) Billur Kaymakçalan, Çankaya University (co-chair) Özlem Defterli, Çankaya University Dumitru Baleanu, Çankaya University Fahd Jarad, Çankaya University Raziye Mert, Çankaya University Adnan Bilgen, Çankaya University Feza Güvenilir, Ankara University Fatma Karakoç, Ankara University Aytekin Enver, Gazi University Mustafa Fahri Aktaş, Gazi University Abdullah Özbekler, Atılım University Erdal Karapınar, Atılım University Türker Ertem, Middle East Technical University ii Contents Preface . . . . . . . . . . . . . . . . . . . Scientific Committee . . . . . . . . . . . Organazing Committee . . . . . . . . . . Plenary Talks . . . . . . . . . . . . . . . Ravi P. AGARWAL . . . . . . . . . Ravi P. AGARWAL . . . . . . . . Delfim F. M. TORRES . . . . . . . Invited Talks . . . . . . . . . . . . . . . Murat ADIVAR . . . . . . . . . . . Alemdar HASANOĞLU . . . . . . Contributed Talks . . . . . . . . . . . . Thabet ABDELJAWAD . . . . . . Nihan ACAR . . . . . . . . . . . . Ali AKGÜL . . . . . . . . . . . . . Ömer AKIN . . . . . . . . . . . . . Elvan AKIN-BOHNER . . . . . . . Jehad O. ALZABUT . . . . . . . . Sakri AMINE . . . . . . . . . . . . Muhammad ARSHAD . . . . . . . Ola A. ASHOUR . . . . . . . . . . Allaberen ASHYRALYEV . . . . . Muzaffer ATEŞ . . . . . . . . . . . Ferhan M. ATICI . . . . . . . . . . Akbar AZAM . . . . . . . . . . . . Dumitru BALEANU . . . . . . . . Ayşe Hümeyra BİLGE . . . . . . . Zeyneb BOUDERBALA . . . . . . Artur M. C. BRITO da CRUZ . . . Özlem DEFTERLİ . . . . . . . . . Alireza Khalili GOLMANKHANEH Gusein Sh. GUSEINOV . . . . . . Tuba GÜLŞEN . . . . . . . . . . . Mehmet GÜMÜŞ . . . . . . . . . . Alaa E. HAMZA . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i ii ii 1 1 1 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 8 8 9 9 9 10 10 11 11 12 12 12 13 Taher HASSAN . . . . . . . . . Betül HİÇDURMAZ . . . . . . Derrardjia ISHAK . . . . . . . Niyaz İSMAGİLOV . . . . . . . Fatma KARAKOÇ . . . . . . . Erdal KARAPINAR . . . . . . Afshin KHASSEKHAN . . . . . Atul KUMAR . . . . . . . . . . Gholam Reza Rokni LAMOOKI Alaeddin MALEK . . . . . . . . Raziye MERT . . . . . . . . . . Karima M. ORABY . . . . . . Süleyman ÖĞREKÇİ . . . . . . Abdullah ÖZBEKLER . . . . . Elif ÖZTÜRK . . . . . . . . . . M. Mine ÖZYETKİN . . . . . . Erhan PİŞKİN . . . . . . . . . Abolhassan RAZMINIA . . . . Safia SLIMANI . . . . . . . . . Yeter ŞAHİNER . . . . . . . . Murat ŞAT . . . . . . . . . . . Aydın TİRYAKİ . . . . . . . . Fatma TOKMAK . . . . . . . . Deniz UÇAR . . . . . . . . . . Paolo VETTORI . . . . . . . . Ali YAKAR . . . . . . . . . . . Ahmet YANTIR . . . . . . . . Burcu Silindir YANTIR . . . . Fikriye Nuray YILMAZ . . . . Tuğba YILMAZ . . . . . . . . . Uğur YÜKSEL . . . . . . . . . Ağacık ZAFER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 14 14 15 16 16 16 17 17 18 18 19 19 19 20 20 21 21 21 22 22 22 23 23 23 24 24 25 25 25 26 26 Plenary Talks Lidstone and Complementary Lidstone Polynomials and Interpolation I Ravi P. AGARWAL Texas A&M University - Kingsville, TX, USA Ravi.Agarwal@tamuk.edu In this lecture we shall: • Define Lidstone polynomials and provide its several different representations which involve Bernoulli polynomials, Bernoulli numbers, Euler polynomials and Euler numbers. • Establish several equalities and inequalities (most of these are the best possible). • Present an explicit representation of the Lidstone interpolating polynomial, and then for the error function give Peano’s and Cauchy’s representations. • Provide best possible error inequalities, best possible criterion for the convergence of Lidstone series, and a quadrature formula with best possible error bound. • Construct complementary Lidstone interpolating polynomial, provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Lidstone and Complementary Lidstone Boundary Value Problems II Ravi P. AGARWAL Texas A&M University - Kingsville, TX, USA Ravi.Agarwal@tamuk.edu The equalities and inequalities established in Lecture 1 will be used here to study Lidstone and Complementary Lidstone boundary value problems. We shall: • Provide necessary and sufficient conditions for the existence and uniqueness of solutions. • Establish sufficient conditions for the convergence of Picard’s and Approximate Picard’s iterative methods. • Develop sufficient conditions for the convergence of Quasilinearization and Approximate Quasilinearization. • Define Lidstone Disconjugacy and give a best possible criterion. • Obtain sufficient conditions for the existence of positive solutions. 1 Combined Derivatives on Time Scales and Fractional Variational Calculus Delfim F. M. TORRES University of Aveiro, PORTUGAL delfim@ua.pt Combined derivatives appear naturally in the context of mechanics and the calculus of variations, both on time scale and fractional settings. In this talk we give a personal view to the subject, and review some recent results on delta-nabla time scale derivatives and right-left fractional derivatives. 2 Invited Talks Convex Analysis on Multidimensional Mixed Domains Murat ADIVAR İzmir University of Economics, İzmir, TURKEY murat.adivar@ieu.edu.tr Coauthors: Shu-Cherng FANG In this study, we establish a platform which enables us to study convexity and duality properties of the sets, cones and functions on multidimensional discrete, continuous and mixed domains. Nonlinear Diffusion and Monotonicity of Differential Operators Alemdar HASANOĞLU İzmir University, İzmir, TURKEY alemdar.hasanoglu@izmir.edu.tr This study paper deals with nonlinear differential operators Au := −div(D(|∇u|2 )∇u) in Ω ∈ R, arising in PDE-based image processing (J. Weickert, in: Scale-Space Theory in Computer Vision, B. ter Haar Romeny, L. Florack, J. Koenderink, M. Viergever (eds.), Lecture Notes in Computer Science, Springer, Berlin, 1997.), computational material science (A. Hasanov, Int. J. Non-Linear Mechanics, 46(5)(2011)), diffusion and nonlinear heat transfer (E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B Nonlinear Monotone Operators, New York: Springer, 1990). We show that, the in all these (quite different) physical models, defined to be as nonlinear isotropic diffusion models, the properties of nonlinear differential operators are almost the same. Using these properties we formulate some new coefficient inverse problems, with nonlocal measured output data (i.e. in the form of integral operator) for the steady state (Au := −div(D(|∇u|2 )∇u)) and evolution (Au := −div(D(|∇u|2 )∇u)) equations. Then we derive a quasisolution approach based on weak solution theory for PDEs, which permits one to prove existence of a solution of the considered coefficient inverse problems. Important questions related to instability of a solution are also discussed. 3 Contributed Talks On Delta and Nabla Riemann and Caputo Fractional Differences Thabet ABDELJAWAD Çankaya University, Ankara, TURKEY thabet@cankaya.edu.tr We investigate two types of dual identities for Riemann and Caputo fractional sums and differences. The first type relates nabla and delta type fractional sums and differences. The second type, represented by the Q-operator, relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences we have to mix both the nabla and delta operators. The Sequential Fractional Difference Equations Nihan ACAR Western Kentucky University, Kentucky, USA nihan.acar515@topper.wku.edu Coauthors: Ferhan M. ATICI In this talk, the extended table for discrete Laplace transform known as N -transform will be presented for some basic functions of discrete fractional calculus with nabla operator. N -transform is a great tool to solve α-th order nabla fractional difference equations as demonstrated in the paper [F. M. Atici and P. W. Eloe, The Rocky Mountain Journal of Mathematics, Special issue honoring Prof. Lloyd Jackson 41(2011), 353-370]. We will also demonstrate how to use N -transform to solve some classes of fractional difference equations with initial conditions. Next, we give the definition of Casoration for the set of solutions up to n-th order nabla fractional difference equations. By calculating the Casoration of the solutions we will classify the solutions of the sequential nabla fractional difference equations as linearly independent or linearly dependent. Finally, we concentrate on the solutions of up to second order nabla fractional difference equations. We will examine characteristic roots in three cases, namely real and distinct, real and same, and complex. Riemann-Liouville definition of fractional difference will be used throughout our work. 4 Numerical Solution of the Second-Order One-Dimensional Telegraph Equation Based on Reproducing Kernel Hilbert Space Method Ali AKGÜL Dicle University, Diyarbakır, TURKEY aliakgul00727@gmail.com Coauthors: Mustafa İNÇ In this paper, we proposed a reproducing kernel method for solving the telegraph equation with initial and boundary conditions based on the reproducing kernel theory. Its exact solution is represented in the form of series in the reproducing kernel Hilbert space. Some numerical examples have been studied to demonstrate the accuracy of the present method. Results of numerical examples show that the presented method is simple and effective. Possible Fuzzy Solutions For Second Order Initial Value Problems Ömer AKIN TOBB University of Economics and Technology, Ankara, TURKEY omerakin@etu.edu.tr Coauthors: Burhan TÜRKŞEN In this study, we state a fuzzy initial value problem of the second order fuzzy differential equations. Here we investigate problems with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions. We propose an algorithm based on alpha-cut of a fuzzy set. Finally we present some examples by using our proposed algorithm. Following these we try to extend the results for low as well as high fuzzy sets. Oscillation Criteria for Second Order Strongly Superlinear and Strongly Sublinear Dynamic Inclusions Elvan AKIN-BOHNER Missouri University S&T, Rolla, MS, USA akine@mst.edu Coauthors: Shurong SUN In this paper, we establish some oscillation criteria for strongly superlinear and strongly sublinear dynamic inclusions. Oscillation problems are in differential and difference equations have become very attractive recently. These areas have started to be unified and extended for more powerful general theory, so called dynamic equations on time scales. Results in this paper even are new in continuous case. 5 Oscillation of Solutions for Third–Order Half-Linear Neutral Difference Equations Jehad O. ALZABUT Prince Sultan University, Riyadh, SAUDI ARABIA jalzabut@psu.edu.sa Coauthors: Ömer AKIN, Yaşar BOLAT, N. DOĞAN In this article, we study the oscillation of solutions for third order neutral difference equations of the form α ∆ a (n) ∆2 [x (n) ± p (n) x (δ (n))] + q (n) xα (τ (n)) = 0. (1) Sufficient conditions are established to prove that every solution of (1) either oscillates or converges to zero. We support the main results by numerical examples. Asymptotic Dynamics of the Slow-Fast Hindmarsh-Rose Neuronal System Sakri AMINE Cheffia Eltarf, ALGERIA sakriamine@hotmail.fr Coauthors: Benchetteh AZDINE This work addresses the asymptotic dynamics of a neuronal mathematical model. The first aim is the understanding of the biological meaning of existing mathematical systems concerning neurons such as Hodgkin-Huxley or Hindmarsh-Rose models. The local stability and the numerical asymptotic analysis of Hindmarsh-Rose model are then developed in order to comprehend bifurcations and dynamics evolution of a single Hindmarsh-Rose neuron. This has been performed using numerical tools borrowed from the nonlinear dynamical system theory. Fixed points of Kannan and Chatterjea mappings on a closed ball without the assumption of continuity Muhammad ARSHAD International Islamic University, Islamabad, PAKISTAN marshad zia@yahoo.com Coauthors: Saqib HUSSAIN After Banach contraction principal, a variety of generalizations in the setting of point to point mappings have been obtained. In 1969, Kannan [R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60(1968), 71-76] a premier Indian Mathematician proved a contraction theorem for a complete metric space proved that a mapping T : X → X satisfying a contraction condition d(T x, T y) < k[d(x, T x) + d(y, T y)] for all x, y ∈ X where 0 < k < (1/2) has a unique fixed point in X. Chatterjea [S. K. Chatterjea, Fixed 6 point theorems, C.R. Acad. Bulgare Sci., 25(1972), 727–730] followed Kannan and in 1972 proved a fixed point theorem for a complete metric space (X, d), states that a mapping T : X → X satisfying a contraction condition d(T x, T y) < k[d(x, T y) + d(y, T x)] for all x, y ∈ X where 0 < k < (1/2) has a unique fixed point in X. It is impartant to note that these three theorems are independent of each other and have laid down the foundation of modern fixed points theory. In this study we establish some fixed point theorems for mappings satisfying Kannan and Chatterjea locally contractive conditions on a closed ball in a complete metric space. Our results generalize/improve some well-known classical results of the literature. On q-dual Integral Equations Ola A. ASHOUR Cairo University, Cairo, EGYPT oa ashour@hotmail.com Our aim in this talk is to introduce q-analogues of some known Dual Integral Equations and to obtain their solutions in different techniques. Finite Difference Method for Stochastic Parabolic and Hyperbolic Equations Allaberen ASHYRALYEV Fatih University, İstanbul, TURKEY aashyr@fatih.edu.tr It is known that most problems in heat flow, fusion process, model financial instruments like options, bonds and interest rates and other areas which are involved with uncertainty lead to stochastic differential equation with parabolic type. These equations can be derived as models of indeterministic systems and consider as methods for solving boundary value problems. The method of operators as a tool for investigation of the solution to stochastic partial differential equations in Hilbert and Banach spaces, has been systematically developed by several authors. Numerical analytic methods for stochastic differential equations have been studied extensively by many researchers. However, finite difference method for stochastic partial differential equations was not well-investigated. The main goal of this study is to construct and investigate the difference schemes for stochastic parabolic and hyperbolic equations. The single step and two step difference schemes for the numerical solution of stochastic parabolic and hyperbolic equations are presented. The convergence estimates for the solution of these difference schemes are established. For the numerical study, procedure of modified Gauss elimination method is used to solve these difference schemes. 7 Global Asymptotic Stability and Ultimate Boundednes of a Class of Third Order Nonlinear Systems Muzaffer ATEŞ Yüzüncü Yıl University, Van, TURKEY ates.muzaffer65@gmail.com In this paper, we study the problems of the global stability and the boundedness results of the second order vector differential equations, ... . .. .. . . . .. X +F (X, X , X ) X +G(X, X ) X +H(X) = P (t, X, X , , X ) (∗) in two cases (i) P = 0 and (ii) P (6= 0), ||P (t, X, Y )|| ≤ (A + ||Y ||)q(t), where δ0 , δ1 are constants. For case (i) we obtain some sufficient conditions which ensure that the solution x = 0 of Eq. (*) is globally asymptotically stable. For case (ii) the ultimate boundedness results of Eq. (*) is obtained. Our results include a well-known result in the literature. Finally, a concrete example is given to check our results. Models of Inventory with Deteriorating Items on Non-periodic Time Domains Ferhan M. ATICI Western Kentucky University, Kentucky, USA ferhan.atici@wku.edu Coauthors: Alex LEBEDINSKY, Fahriye M. UYSAL In this talk, we first introduce deterministic model of inventory with deteriorating items on complex time domains which may be periodic as discrete or non periodic as unevenly distributed collection of time. Then we show how the dynamic model can be optimized using techniques of time scale calculus. After we demonstrate that both the conventional discrete-time and time scale calculus model yield the same results, we show an example based on this model that could not be solved using conventional discrete-time techniques because the points on time scales may be spaced at uneven intervals. 8 Coincidence Points of Fuzzy Mappings Akbar AZAM COMSATS Institute of Information Technology, Islamabad, PAKISTAN akbarazam@yahoo.com A large variety of the most important problems of applied mathematics reduced to finding solutions of nonlinear functional equation, which can be formulated in terms of finding the fixed points of a nonlinear operator. Since the appearance of celebrated Banach contraction principle in 1932, several generalizations and improvements of this theorem have been obtained. Heilpern [S. Heilpern, Fuzzy mappings and fixed point theorems, J. Math. Anal.Appl. 83(1981), 566-569.] generalized the Banach Contraction Principle by introducing a contraction condition for fuzzy mappings and established a fixed point theorem for fuzzy mappings in complete metric linear spaces. Subsequently several originators studied the existence of fixed points and common fixed points of fuzzy mappings satisfying a Banach type contractive condition. In the present paper, we establish a coincidence point theorem for a pair of fuzzy mappings under a contractive type condition and obtain a significant extension of Heilpern result. About a Nonlinear Fractional Differential Equation Dumitru BALEANU Çankaya University, Ankara, TURKEY and Institute of Space Sciences, Magurele-Bucharest, ROMANIA dumitru@cankaya.edu.tr In this paper we study the existence and uniqueness of a nonlinear fractional differential equation with periodic boundary condition. The Classification of Integrable Evolution Equation in 1+1 Dimensions Ayşe Hümeyra BİLGE Kadir Has University, İstanbul, TURKEY ayse.bilge@khas.edu.tr All integrable polynomial evolution equations in 1+1 dimensions are known to be symmetries of the Korteweg deVries, the Sawada-Kotera and the Kaup equations. We classify the non-polynomial equations by the existence of a formal symmetry and we show that there are non-polynomial hierarchies which are nevertheless expected to be related to the equations above. 9 Periodic Solutions for a class of Autonomous Newton Differential Equations Zeyneb BOUDERBALA Badji Mokhtar University, Annaba, ALGERIA zeynebbouderbala@yahoo.fr Coauthors: Amar MAKHLOUF In this work, we provide sufficient conditions for the existence of periodic solutions of the second order autonomous differential equation x00 = M (x) with M(x) a 2π periodic function (with a small parameter of perturbation ). Note that this is a particular class of autonomous Newton differential equations. Moreover we provide some applications. A Symmetric Dynamic Calculus on Time Scales Artur M. C. BRITO da CRUZ The Polytechnic Institute of Setubal and University of Aveiro, PORTUGAL artur.cruz@estsetubal.ips.pt Coauthors: Natalia MARTINS and Delfim F. M. TORRES We define a symmetric derivative on time scales and derive some of its properties. We introduce partial symmetric derivatives for two-variable functions. A generalized diamond integral, which is a refined version of the diamond-α integral, is also introduced. A mean value theorem is proved for the generalized diamond integral as well as versions of Holder’s, Cauchy-Schwarz’s and Minkowski’s inequalities. 10 Anticipation of the Dynamics of Genetic Regulatory Networks Özlem DEFTERLİ Çankaya University, Ankara, TURKEY. defterli@cankaya.edu.tr Coauthors: Armin FUGENSCHUH, Gerhard-Wilhelm WEBER Inferring and anticipation of genetic networks based on experimental data and environmental measurements is a challenging research problem of mathematical modeling. In this study, we discuss the models of genetic regulatory systems, so-called geneenvironment networks. The dynamics of such kind of systems are described by a class of time-continuous systems of ordinary differential equations containing unknown parameters to be optimized. Accordingly, time-discrete version of that model class is studied and improved by using higher-order numerical methods. The presented time-continuous and time-discrete dynamical models are identified based on given data, as an illustrative example, by solving the constrained and regularized nonlinear mixed-integer problem. By using this solution and applying both the new and existing discretization schemes, we generate corresponding time-series of gene-expressions for each numerical method with a comparative study with respect to various criteas. Chaos On New System With Fractional Order Alireza Khalili GOLMANKHANEH Islamic Azad University, Urmia, IRAN alireza@physics.unipune.ac.in Coauthors: R. AREFI, D. BALEANU In this paper, the fractional version of a new system which is similar to Liu system has been studied. We have shown that this chaotic system again will be chaotic when the order of system is less than 3. We use the Adams-Bash forth algorithm to solve the system and show strange attractors in trajectories of solutions. Fixed points and Lyapunov exponent of this system have been found and their stability have been investigated for existence of chaos. 11 Description of the structure of arbitrary functions of the Laplace-Beltrami operator Gusein Sh. GUSEINOV Atılım University, Ankara, TURKEY guseinov@atilim.edu.tr One of the fundamental methods of investigation in the theory of operators is to examine the functions of the operator and get in this way informations about the operator itself. In this work, we describe the structure of arbitrary rapidly decreasing function of the Laplace-Beltrami operator in n-dimensional hyperbolic space showing that the function of the Laplace-Beltrami operator is an integral operator and giving an explicit formula for its kernel. Isospectral Problems for Differential Operators Tuba GÜLŞEN Fırat University, Elazığ, TURKEY tubayalcin87@hotmail.com Coauthors: Etibar PENAHLI In this talk, we give a relatively proof of the Gelfand-Levitan equation and a proof of the existence of transmutation operator and investigate isospectral problem of Dirac operator. In particular, research of inverse problems for Sturm-Liouville and Dirac operators have been investigated by many mathematicians: J.Poschel and E.Trubowitz, JR.Max Jodeit and B.M. Levitan, H-H.Chern, etc. On The Dynamics of the recursive sequences xn+1 = a + xpn−k /xqn Mehmet GÜMÜŞ Bülent Ecevit University, Zonguldak, TURKEY m.gumus@karaelmas.edu.tr Coauthors: Özkan ÖCALAN, Nilüfer B. FELAH In this paper, we investigate the boundedness character, the oscillatory and the periodic character of positive solutions of the difference equation xn+1 = a + ((xpn−k )/(xqn )), n = 0, 1, ... where k = 2, 3, ... and a, p, q are positive constans and the initial conditions x−k , ..., x0 are arbitrary positive numbers. We investigate the existence of a prime two periodic solution for k is odd and we find solutions which converge to this periodic solution. Moreover, when k is even, we prove that there are no prime two periodic solutions of the equation above. 12 Semigroups of Operators on Time Scales and Applications Alaa E. HAMZA Cairo University, Giza, EGYPT hamzaaeg2003@yahoo.com Coauthors: Karima ORABY In this paper we define the generator A of a strongly continuous semigroup (C0 semigroup) {T (t) : t ∈ T} of bounded linear operators from a Banach space X into itself, where T ⊆ R≥0 is a time scale, which is an additive semigroup. Many properties of {T (t) : t ∈ T} and its generator A are established. We also prove that the dynamic equations of the form x∆ (t) = Ax(t), x(0) = x0 ∈ D(A), t ∈ T, has the unique solution x(t) = T (t)x0 , t ∈ T, where D(A) is the domain of A. Finally, some of well-known results, like Hille-Yosida Theorem, are generalized. Oscillation Criteria for Second Order Nonlinear Dynamic Equations with pLaplacian and Damping Taher HASSAN Mansoura University, Mansoura, EGYPT (Present Address: Hail University, KSA) tshassan@mans.edu.eg Coauthors: Qingkai KONG This paper concerns the oscillation of solutions of the second order nonlinear dynamic equation with p-Laplacian and damping ∆ r(t) ϕα x∆ (t) + p (t) ϕα x∆σ (t) + q(t)f (xσ (t)) = 0 on a time scale T which is unbounded above. Sign changes are allowed for the coefficient functions r (t), p (t), and q (t). Several examples are given to illustrate the main results. 13 A Study on Difference Schemes of a Fractional Schrodinger Differential Equation Betül HİÇDURMAZ İstanbul Medeniyet University, İstanbul, TURKEY betulhicdurmaz@gmail.com Coauthors: Allaberen ASHYRALYEV The study is on difference schemes of a special type of fractional Schrodinger differential equation. The first and second orders of accuracy difference schemes for numerical solution of the fractional Schrodinger differential equation are considered. Then, stability estimates for solutions of these difference schemes are obtained. Some applications of these stability theorems on different problems are presented. Fixed Points and Stability in Neutral Nonlinear Differential Equations with Variable Delays Derrardjia ISHAK University of Annaba, ALGERIA iderrardjia@hotmail.fr Coauthors: Ahcene DJOUDI By means of Krasnoselskii’s fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend and improve the work of C.H. Jin and J.W. Luo [Nonlinear Anal. 68 (2008), 3307–3315], and also those of T.A. Burton [Fixed Point Theory 4 (2003), 15-32; Dynam. Systems Appl. 11 (2002), 499–519] and B. Zhang [Nonlinear Anal. 63 (2005), e233–e242]. In the end we provide an example to illustrate our claim. 14 On Pathwise Optimality for Controlled Diffusion type Processes Niyaz ISMAGILOV Ufa State Aviation Technical University, Ufa, RUSSIA niyaz.ismagilov@gmail.com In the work, we consider a stochastic optimal control problem of diffusion type processes with pathwise cost functional, that is, the problem of finding a control function such that it minimizes cost for every single trajectory of state variable. More precisely, consider a stochastic differential equation describing dynamics of some system dxt = b(t, xt , ut ) dt + σ(t, xt , ut )dWt , (1) with initial value x(0) = x0 . In the above equation xt is state variable function, ut is control function and Wt is a standard Wiener process. To measure performance of control we introduce a cost functional of the following form Z T f (t, xt , ut )dt. (2) J= 0 The problem is to find control that minimizes the functional (2) subject to dynamics equation (1). Previous work on optimal control of diffusion processes has mainly been concerned with problem of finding control that minimizes “mean” value of cost Z T ¯ f (t, xt , ut )dt → min, J =E 0 where E denotes expectation (see [Krylov, N.V., Controlled Diffusion Processes. SpringerVerlag, Berlin, 2009], [Gichman, I.I., Skorochod,A.V., Controlled stochastic processes. Springer Verlag, New York, 1979.]). In contrast, stated above problem is concerned with minimization of functional (2), which represents cost for every single path of state variable. Hence pathwise optimality is the distinguishing feature of this work. In the work we introduce new method of solving problems of pathwise cost minimization. The main idea of the method is that original stochastic control problem can be reduced to deterministic control problem (Nasyrov, F.S., Local times, symmetric integrals and stochastic analysis. Fizmatlit, Moscow, 2011 (in Russian).). Solution to the latter gives pathwise optimal solution to the original problem. 15 Oscillatory and Periodic Solutions of Impulsive Differential Equations with Piecewise Constant Argument Fatma KARAKOÇ Ankara University, Ankara, TURKEY fkarakoc@ankara.edu.tr Coauthors: Hüseyin Bereketoğlu, Gizem Seyhan In this talk we give some results for the existence of oscillatory and periodic solutions of a class of impulsive differential equations with piecewise constant argument. Remarks on Coupled Fixed Point Theorems Erdal KARAPINAR Atılım University, Ankara, TURKEY erdalkarapinar@yahoo.com In this talk, we prove new coupled fixed point theorems extending some recent results in the literature on this topic. We also present applications of these new results through a number of examples. Approximate Solution of a class of Fredholm Integral Equation of second kind with hypersingular kernel Afshin KHASSEKHAN Tarbiat Moallem Center, Salmas, IRAN khassekhan@gmail.com In this work a method is offered for solving a class of hyper singular integral equation of the second kind. Chebyshev polynomials of first kind are used to approximate the kernel function. The integrals are computed in terms of chebyshev polynomials too. In addition, numerical examples that illustrate the pertinent features of the method are presented. 16 One-Dimensional Solute Transport For Uniform And Varying Pulse Type Input Point Source Through Inhomgeneous Medium Atul KUMAR Lucknow University, Uttar Pradesh, INDIA atul.tusaar@gmail.com Coauthors: Dilip Kumar JAISLAW and R. R. YADAV To solves the analytically, conservative solute transport equation for a solute undergoing convection, dispersion, retardation in a one-dimensional inhomogeneous porous medium. In the present study, the solute dispersion parameter is considered uniform while the velocity of the flow is considered spatially dependent. Retardation factor is also considered. The velocity of the flow is considered inversely proportional to the spatially dependent function while retardation factor is considered inversely proportional to square of the velocity of flow. Analytical approaches introduced for two cases: former one is for uniform input point source and latter case is for varying input point source where the solute transport is considered initially solute free from the domain. The variable coefficients in the advection-diffusion equation are reduced into constant coefficients with the help of the transformations which introduce by new space variables, respectively. The Laplace transformation technique is used to get the analytical solutions. Figures are presented illustrating the dependence of the solute transport upon velocity, dispersion and adsorption coefficient. A Dynamical Systems Approach in Mathematical Modeling of Thyroid Gholam Reza Rokni LAMOOKI University of Tehran and Institute for Research in Fundamental Sciences (IPM), Tehran, IRAN rokni@khayam.ut.ac.ir Coauthors: Alireza MANI, Amirhossein SHIRAZI In this paper we utilize the reaction kinetics of chemicals involved in thyroid gland to obtain a system of differential equations. The system has a large scale with one input and is highly nonlinear. We specifically focus on relaxation oscillations of the system. See Berman (1961) and Danziger (1956) for early applications of differential equations to study thyroid, Rosenfeld (2000) for a historic review, and Leow (2007) for underlying feedback mechanism. Mentioned relaxation oscillations is under the influence of the input and the negative feedback regulator and can be destroyed by various phenomena including input bandwidth, input amplitude and the rates of reactions. Various thyroid malfunctioning will be discussed including inhibition of reactions via minerals. The results will be illustrated by a series of analysis as well as graphs describing the normal behavior and malfunctioning. The analysis above opens an aperture into the understanding of the largest gland in body. 17 Novel Formulation for the Two Phase Immiscible Flow in Petroleum Reservoir Alaeddin MALEK Tarbiat Modares University, Tehran, IRAN mala@modares.ac.ir Coauthors: Samaneh KHODAYARI-SAMGHABADI The cognition of the fluid flow in porous media is an important task for engineers and mathematicians. One of the complicated kinds of this phenomenon is the fluid of two phase flow in porous media. In this case, two immiscible fluids that are supposed to be incompressible, contact with each other through a porous media. The problem is modeled with a set of governing equations (PDE initial boundary value problem); involving the Darcy law and mass conservation equation for each phase that pressure and saturation are unknown. In this paper we propose new formulation for the two phase immiscible flow in petroleum reservoir, when we have one injection and one production well that are existed in the Five-spot pattern. We split the coupled system into pressure and saturation equations. We solve pressure equation with second order implicit method and for the saturation if there is not capillary pressure equation, we can get closed analytical form of the solution for the water saturation equation; otherwise saturation equation is numerically solved by using the Runge-Kutta method. A control volume finite element method on unstructured grid is applied for spatial discretization of the corresponding non-linear partial differential system. Numerical results for pressure, saturation after 2000 days for a reservoir containing injection and production wells are presented. A Halanay-type Inequality on Time Scales in Higher Dimensional Spaces Raziye MERT Çankaya University, Ankara, TURKEY raziyemert@cankaya.edu.tr Coauthors: Jia BAOGUO, Lynn ERBE In this study, we investigate a certain class of Halanay-type inequalities on time scales in higher dimensional spaces. By means of the obtained inequality, we derive some new global stability conditions for linear delay dynamic systems on time scales. An example is given to illustrate the results. 18 Stability of Abstract Dynamic Equations on Time Scales Karima M. ORABY Suez Canal University, EGYPT koraby83@yahoo.com Coauthors: Alaa E. HAMZA In this paper, we investigate many types of stability (uniform stability, asymptotic stability, uniform asymptotic stability, global stability, global asymptotic stability, exponential stability, and uniform exponential stability) of the homogeneous linear dynamic equations. Finally, we give an illustrative example for a non-regressive homogeneous first order linear dynamic equation and we investigate its stability. Oscillation Theorems For Second Order Nonlinear Differential Equations Süleyman ÖĞREKÇİ Gazi University, Ankara, TURKEY s.ogrekci@gmail.com Coauthors: Adil MISIR In this study, we give some oscillation criterions for non-linear differential equations of second-order by using Riccati technique. Our results improve the theorems given in [Fanwei Meng, Yan Huang, Interval oscillation criteria for a forced second-order nonlinear differential equations with damping, Applied Mathematics and Computation 218 (2011) 1857–1861] and some known results inthe literature. Oscillation Criterion for Half-Linear Differential Equations with Periodic Coefficients Abdullah ÖZBEKLER Atılım University aozbekler@gmail.com Coauthors: O. DOSLY, R. Simon HILSCHER We present an oscillation criterion for second order half-linear differential equations with periodic coefficients. The method is based on the nonexistence of a proper solution of the related modified Riccati equation. Our result can be regarded as an oscillatory counterpart to the nonoscillation criterion by Sugie and Matsumura (2008). These two theorems provide a complete half-linear extension of the oscillation criterion of Kwong and Wong (2003) dealing with the Hill’s equation. 19 Difference Schemes for Elliptic Equations Elif ÖZTÜRK Uludağ University, Bursa, TURKEY elifozturk16@hotmail.com Coauthors: Allaberen ASHYRALYEV The nonlocal Bitsadze-Samarskii type nonlocal boundary value problems d2 u(t) − + Au(t) = f (t), 0 < t < 1, dt2 J P u(0) = ϕ, u(1) = αj u(λj ) + ψ, j=1 P J |αj | ≤ 1, 0 < λ1 < λ2 < · · · < λJ < 1 j=1 for the differential equations in a Hilbert space H with the self-adjoint positive definite operator A with a domain D(A) ⊂ H is considered. The second and fourth orders of accuracy difference schemes are presented. The stability and almost coercive stability of these difference schemes are established. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples. Integer Order Approximation of Uncertain Fractional Order Differentiations and Integrations M. Mine ÖZYETKİN İnönü University, Malatya, TURKEY munevver.ozyetkin@inonu.edu.tr Coauthors: Nusret TAN In this talk we deal with the robust stability of the fractional order system having interval order uncertainty. For this aim, integer order approximations of s[α,α] are computed. Using interval arithmetic rules, integer order equivalences of s[α,α] are obtained in terms of first, second, third and fourth order integer approximations dependent on α. It is shown that integer order equivalences of s[α,α] have interval transfer function (coefficients of equivalence transfer function have interval structure). Kharitonov stability criterion is applied to check the stability of the system. The proposed idea is supported through numerical examples. The details of the proposed method will be given in the final version of the presentation. 20 On the decay and blow up of solutions for coupled wave equations of Kirchhoff type with nonlinear damping and source terms Erhan PİŞKİN Dicle University, Diyarbakır, TURKEY episkin@dicle.edu.tr Coauthors: Necat POLAT In this work, we consider an initial boundary value problem for coupled wave equations of Kirchhoff type. For some restrictions on the initial data, we establish the exponential and polynomial decay. After that, we obtain the blow up of the solution with negative initial energy. Fractional Hyperchaotic Telecommunication Systems: A New Paradigm Abolhassan RAZMINIA Persian Gulf University, Bushehr, IRAN a.razminia@gmail.com Coauthors: Dumitru BALEANU The dynamics of hyperchaotic and fractional-order systems have increasingly attracted attention in recent years. In this paper, we mix two complex dynamics to construct a new telecommunication system. Using a hyperchaotic fractional order system, we propose a novel synchronization scheme between receiver and transmitter which increases the security of data transmission and communication. Indeed this is first work that can open a new way in secure communication system. Analysis of a predator-prey model with modified Leslie-Gower and Hollingtype II schemes with term refuge Safia SLIMANI Annaba, ALGERIA slimani safia@yahoo.fr Coauthors: Azzedine BENCHATTEH In this study we present a two-dimensional continuous time dynamical system modelling a predator-prey food chain based on a modified version of the Leslie-Gower scheme and on the Holling-type II scheme incorporating a term refuge. We show the boundedness of solutions, existence of an attracting set and global stability of the coexisting interior equilibrium 21 On Oscillation of Elliptic Inequalities Yeter ŞAHİNER Hacettepe University, Ankara, TURKEY ysahiner@hacettepe.edu.tr Coauthors: Ağacık ZAFER Sufficient conditions are obtained for the oscillation of solutions of half-linear elliptic inequalities with p(x)-Laplacian. The results obtained are new even for one-dimensional case. Inverse Problem For Interior Spectral Data of The Hydrogen Atom Equation Murat ŞAT Erzincan University, Erzincan, TURKEY murat sat24@hotmail.com Coauthors: Etibar S. PENAHLI We consider an inverse problem for the second order differential operators with a regular singularity, and show that the potential function can be uniquely determined by a set of values of eigenfunctions at some interior point and parts of two spectra. Boundedness Results of Certain Type Second Order Nonlinear Differential Equations Aydın TİRYAKİ İzmir University, İzmir, TURKEY aydin.tiryaki@izmir.edu.tr In this paper, by using standard methods we present some boundedness results for certain type second order nonlinear differential equations. The special cases of these results are valid for the well-known Emden-Fowler equation and half-linear equations. Also some examples illustrating the theory will be given. 22 Positive Solutions For Second-Order Impulsive Boundary Value Problems on Time Scales Fatma TOKMAK Ege University, İzmir, TURKEY fatma.tokmakk@gmail.com Coauthors: İlkay Y. KARACA In this study, we consider second-order impulsive boundary value problems and eigenvalue problem on time scales. By using some fixed point theorems, we investigate the existence of positive solutions. Oscillatory Behaviour of a Higher Order Nonlinear Neutral Type Functional Dynamic Equation with Oscillating Coefficients Deniz UÇAR Uşak University, Uşak, TURKEY deniz.ucar@usak.edu.tr Coauthors: Yaşar BOLAT In this paper we are concerned with the oscillation of solutions of a certain more general higher order nonlinear neutral type functional dynamic equation with oscillating coefficients. We obtain some sufficient criteria for oscillatory behaviour of its solutions. Stability Conditions for Linear Fractional Difference Systems Paolo VETTORI University of Aveiro, PORTUGAL pvettori@ua.pt The well-known condition for the stability of fractional differential systems published by Matignon in 1996 [Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, pp. 963– 968] will be extended to discrete-time systems defined by fractional difference equations (nabla calculus). This will be accomplished using Laplace transforms on time scales, unifying the continuous and discrete-time cases. 23 On the Existence of Solutions for Fractional Differential Equations Ali YAKAR Gaziosmanpaşa University, Tokat, TURKEY ali.yakar@gop.edu.tr In this work, existence of solutions for fractional differential equations (FDEs) involving standart Riemann Liouville fractional derivative is established. Our main tool is the method of upper and lower solutions which is one of the effective ways to reveal the existence of solutions of nonlinear FDEs. Also we use functions satisfying Cp continuity assumption instead of imposing Holder continuity. In addition, it is shown that these results can be generalized to the existence of solutions for systems of fractional differential equations. Caratheodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces Ahmet YANTIR Yaşar University, İzmir, TURKEY ahmet.yantir@yasar.edu.tr Coauthors: Ireneusz Kubiaczyk, Aneta Sikorska-Nowak We prove the existence result for Carathéodory type solutions of the nonlinear SturmLiouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of Carathéodory solutions in terms of Kuratowski measure of noncompactness. We express the problem of finding the Carathéodory solutions of Sturm- Liouville boundary value problem as an integral operator on an appropriate set. The existence of fixed points this integral operator is proved by using Mönch’s fixed point theorem. We also remark that Kuratowski measure of noncompactness can be replaced by any axiomatic measure of noncompactness. 24 q-discrete Generalized Toda Equation Burcu Silindir YANTIR İzmir University of Economics, İzmir, TURKEY burcu.yantir@ieu.edu.tr We present the q-analogue of generalized Toda equation. We develop its three-qsoliton solutions, which are expressed in the form of polynomials in power functions. The q-analogue of generalized Toda equation is a unified equation as proper reductions of parameters give rise to various types of q-difference analogues of soliton equations such as Toda equation, KdV equation a nd sine-Gordon equation. Therefore, it comprises three-q-soliton solutions of these various types of q-difference soliton equations. All-at-once approach for the Optimal Control of Burgers Equation Fikriye Nuray YILMAZ Gazi University, Ankara, TURKEY yfikriye@gmail.com Coauthors: Bülent KARASÖZEN In this work, we apply the all-at-once method for the optimal control of unsteady Burgers equation. The all-at-once methods were applied in recent years for optimal control problems governed by linear elliptic and parabolic equations. The state equation is discretized and then the optimality system for the finite dimensional optimization problem is derived. This approach is also referred to as the black-box approach. In other words, an existing algorithm for the solution of the state equation is embedded into an optimization loop. For space discretization, we use the Galerkin finite element method. The nonlinear Burgers equation is discretized in time using the semi implicit discretization which results an effective linearization of the optimal control problem. Numerical results for distributed unconstrained and control constrained problems illustrate the performance of all-at-once approach with semi-implicit time discretization Fractional Davey-Stewartson Equations within Variational Iteration Method Tuğba YILMAZ Çankaya University, Ankara, TURKEY tugba87yilmaz@yahoo.com.tr Coauthors: Dumitru BALEANU The variational iteration method is applied for the fractional Davey-Stewartson equations in the Caputo sense and the approximate analytical solutions are obtained. 25 Initial Value Problems in Clifford Algebras Depending on Parameters Uğur YÜKSEL Atılım University, Ankara, TURKEY uyuksel@atilim.edu.tr This study deals with initial value problems of type ∂t u(t, x) = Lu(t, x), u(0, x) = u0 (x) in Clifford algebras depending on parameters. Lyapunov-type Inequalities for Planar Linear Dynamic Hamiltonian Systems Ağacık ZAFER Middle East Technical University, Ankara, TURKEY zafer@metu.edu.tr Coauthors: Martin Bohner Lyapunov-type inequalities are useful in studying the qualitative behavior of solutions such as oscillation, disconjugacy, and eigenvalue problems for differential and difference equations. In this talk we give new Lyapunov-type inequalities for linear Hamiltonian systems on arbitrary time scales, which improve recently published results and hence all the related ones in the literature. 26