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.Conference Object Citation - WoS: 1Citation - Scopus: 1Exact Quantization of Cauchy-Euler Type Forced Parametric Oscillator(IOP Publishing Ltd., 2016) Atılgan Büyükaşık, Şirin; Çayiç, ZehraDriven and damped parametric quantum oscillator is solved by Wei-Norman Lie algebraic approach, which gives the exact form of the evolution operator. This allows us to obtain explicitly the probability densities, time-evolution of initially Glauber coherent states, expectation values and uncertainty relations. Then, as an exactly solvable model, we introduce the driven Cauchy-Euler type quantum parametric oscillator, which appears as self-adjoint quantization of the classical Cauchy-Euler differential equation. We discuss some typical behavior of this oscillator under the influence of external terms and give a concrete example.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.Article Citation - WoS: 9Citation - Scopus: 10Application of the Exp-Function Method To Nonlinear Lattice Differential Equations for Multi-Wave and Rational Solutions(John Wiley and Sons Inc., 2011) Aslan, İsmailIn this paper, we extend the basic Exp-function method to nonlinear lattice differential equations for constructing multi-wave and rational solutions for the first time. We consider a differential-difference analogue of the Korteweg-de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant.Article Citation - WoS: 6Citation - Scopus: 6Some Exact Solutions for Toda Type Lattice Differential Equations Using the Improved (g'/g)-expansion Method(John Wiley and Sons Inc., 2012) Aslan, İsmailNonlinear lattice differential equations (also known as differential-difference equations) appear in many applications. They can be thought of as hybrid systems for the inclusion of both discrete and continuous variables. On the basis of an improved version of the basic (G′/G)- expansion method, we focus our attention towards some Toda type lattice differential systems for constructing further exact traveling wave solutions. Our method provides not only solitary and periodic wave profiles but also rational solutions with more arbitrary parameters. © 2012 John Wiley & Sons, Ltd.Article Citation - WoS: 26Citation - Scopus: 27Analytic Solutions To Nonlinear Differential-Difference Equations by Means of the Extended (g'/g)-expansion Method(IOP Publishing Ltd., 2010) Aslan, İsmailIn this paper, a discrete extension of the (G′/G)-expansion method is applied to a relativistic Toda lattice system and a discrete nonlinear Schrödinger equation in order to obtain discrete traveling wave solutions. Closed form solutions with more arbitrary parameters, which reduce to solitary and periodic waves, are exhibited. New rational solutions are also obtained. The method is straightforward and concise, and its applications in physical sciences are promising. © 2010 IOP Publishing Ltd.Conference Object Citation - WoS: 1Citation - Scopus: 1The Extended Discrete (g'/g)-expansion Method and Its Application To the Relativistic Toda Lattice System(American Institute of Physics, 2009) Aslan, İsmailWe propose the extended discrete (G′/G)-expansion method for directly solving nonlinear differentialdifference equations. For illustration, we choose the relativistic Toda lattice system. We derive further discrete hyperbolic and trigonometric function traveling wave solutions, as well as discrete rational function solutions.