Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 11Citation - Scopus: 12Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types(IOP Publishing Ltd., 2016) Aslan, İsmailDifferential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous. Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations, the majority of the results deal with polynomial types. Limited research has been reported regarding such equations of rational type. In this paper we present an adaptation of the (G′/G)-expansion method to solve nonlinear rational differential-difference equations. The procedure is demonstrated using two distinct equations. Our approach allows one to construct three types of exact traveling wave solutions (hyperbolic, trigonometric, and rational) by means of the simplified form of the auxiliary equation method with reduced parameters. Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.Article Citation - WoS: 36Citation - Scopus: 40Exact Solutions for a Local Fractional Dde Associated With a Nonlinear Transmission Line(IOP Publishing Ltd., 2016) Aslan, İsmailOf recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation which is related to a nonlinear electrical transmission line. Explicit traveling wave solutions (kink/antikink solitons, singular, periodic, rational) are obtained via the discrete tanh method coupled with the fractional complex transform.Article Citation - WoS: 22Citation - Scopus: 26Exact Solutions for Fractional Ddes Via Auxiliary Equation Method Coupled With the Fractional Complex Transform(John Wiley and Sons Inc., 2016) Aslan, İsmailDynamical behavior of many nonlinear systems can be described by fractional-order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)-expansion method coupled with the so-called fractional complex transform. The solution procedure is elucidated through two generalized time-fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.Article Citation - WoS: 15Citation - Scopus: 15Exact Solutions of a Fractional-Type Differential-Difference Equation Related To Discrete Mkdv Equation(IOP Publishing Ltd., 2014) Aslan, İsmailThe extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 (2011) 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters: hyperbolic, trigonometric and rational, which have not been reported before.Article Citation - WoS: 21Citation - Scopus: 21An Analytic Approach To a Class of Fractional Differential-Difference Equations of Rational Type Via Symbolic Computation(John Wiley and Sons Inc., 2015) Aslan, İsmailFractional derivatives are powerful tools in solving the problems of science and engineering. In this paper, an analytical algorithm for solving fractional differential-difference equations in the sense of Jumarie's modified Riemann-Liouville derivative has been described and demonstrated. The algorithm has been tested against time-fractional differentialdifference equations of rational type via symbolic computation. Three examples are given to elucidate the solution procedure. Our analyses lead to closed form exact solutions in terms of hyperbolic, trigonometric, and rational functions, which might be subject to some adequate physical interpretations in the future. Copyright © 2013 JohnWiley & Sons, Ltd.Conference Object Citation - WoS: 8Citation - Scopus: 11Two-Circles Theorem, Q-Periodic Functions and Entangled Qubit States(IOP Publishing Ltd., 2014) Pashaev, OktayFor arbitrary hydrodynamic flow in circular annulus we introduce the two circle theorem, allowing us to construct the flow from a given one in infinite plane. Our construction is based on q-periodic analytic functions for complex potential, leading to fixed scale-invariant complex velocity, where q is determined by geometry of the region. Self-similar fractal structure of the flow with q-periodic modulation as solution of q-difference equation is studied. For one point vortex problem in circular annulus by fixing singular points we find solution in terms of q-elementary functions. Considering image points in complex plane as a phase space for qubit coherent states we construct Fibonacci and Lucas type entangled N-qubit states. Complex Fibonacci curve related to this construction shows reach set of geometric patterns.Article Citation - WoS: 4Citation - Scopus: 4On the Application of the Exp-Function Method To Nonlinear Differential-Difference Equations(Elsevier Ltd., 2012) Aslan, İsmailWhen applying the Exp-function method to nonlinear-differential difference equations, Bekir (2010) [1] reported incorrect results. © 2012 Elsevier Inc. All rights reserved.Article Citation - WoS: 18Citation - Scopus: 22A Discrete Generalization of the Extended Simplest Equation Method(Elsevier Ltd., 2010) Aslan, İsmailWe modified the so-called extended simplest equation method to obtain discrete traveling wave solutions for nonlinear differential-difference equations. The Wadati lattice equation is chosen to illustrate the method in detail. Further discrete soliton/periodic solutions with more arbitrary parameters, as well as discrete rational solutions, are revealed. We note that using our approach one can also find in principal highly accurate exact discrete solutions for other lattice equations arising in the applied sciences. © 2009 Elsevier B.V. All rights reserved.