02. Fen Fakültesi / Faculty of Science
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Browsing 02. Fen Fakültesi / Faculty of Science by Subject "(G′/G)-expansion method"
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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., 2015-01) 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: 14Citation - Scopus: 16Analytic Investigation of a Reaction-Diffusion Brusselator Model With the Time-Space Fractional Derivative(Walter de Gruyter GmbH, 2014-04) Aslan, İsmailIt is well known that many models in nonlinear science are described by fractional differential equations in which an unknown function appears under the operation of a derivative of fractional order. In this study, we propose a reaction-diffusion Brusselator model from the viewpoint of the Jumarie's modified Riemann-Liouville fractional derivative. Based on the (G'/G)-expansion method, various kinds of exact solutions are obtained. Our results could be used as a starting point for numerical procedures as well.Article Citation - WoS: 77Citation - Scopus: 110Analytic Study on Two Nonlinear Evolution Equations by Using the (g'/g)-expansion Method(Elsevier Ltd., 2009-03) Aslan, İsmail; Öziş, TurgutThe validity and reliability of the so-called (G′/G)-expansion method is tested by applying it to two nonlinear evolutionary equations. Solutions in more general forms are obtained. When the parameters are taken as special values, it is observed that the previously known solutions can be recovered. New rational function solutions are also presented. Being concise and less restrictive, the method can be applied to many nonlinear partial differential equations.Article Comment On: the (g'/g)-expansion Method for the Nonlinear Lattice Equations [commun Nonlinear Sci Numer Simulat 17 (2012) 3490-3498](Elsevier, 2012-12) Aslan, İsmailWe show that two of the nonlinear lattice equations studied by Ayhan & Bekir [Commun Nonlinear Sci Numer Simulat 17 (2012) 3490-3498] have already been investigated by Aslan [Commun Nonlinear Sci Numer Simulat 15 (2010) 1967-1973] using an improved version of the same method. The solutions obtained by the latter one include the solutions obtained by the former one. © 2012 Elsevier B.V.Article Citation - WoS: 10Citation - Scopus: 11The Discrete (g'/g)-expansion Method Applied To the Differential-Difference Burgers Equation and the Relativistic Toda Lattice System(John Wiley and Sons Inc., 2012-01) Aslan, İsmailWe introduce the discrete (G′/G)-expansion method for solving nonlinear differential-difference equations (NDDEs). As illustrative examples, we consider the differential-difference Burgers equation and the relativistic Toda lattice system. Discrete solitary, periodic, and rational solutions are obtained in a concise manner. The method is also applicable to other types of NDDEs. © 2010 Wiley Periodicals, Inc.Article Citation - WoS: 30Citation - Scopus: 36Discrete Exact Solutions To Some Nonlinear Differential-Difference Equations Via the (g'/g)-expansion Method(Elsevier Ltd., 2009-12) Aslan, İsmailWe extended the (G′/G)-expansion method to two well-known nonlinear differential-difference equations, the discrete nonlinear Schrödinger equation and the Toda lattice equation, for constructing traveling wave solutions. Discrete soliton and periodic wave solutions with more arbitrary parameters, as well as discrete rational wave solutions, are revealed. It seems that the utilized method can provide highly accurate discrete exact solutions to NDDEs arising in applied mathematical and physical sciences.Article Citation - WoS: 40Citation - Scopus: 56Exact and Explicit Solutions To Some Nonlinear Evolution Equations by Utilizing the (g'/g)-expansion Method(Elsevier Ltd., 2009) Aslan, İsmailIn this paper, we demonstrate the effectiveness of the so-called (G′/G)-expansion method by examining some nonlinear evolution equations with physical interest. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.Article Citation - WoS: 25Citation - Scopus: 25Exact and Explicit Solutions To the Discrete Nonlinear Schrödinger Equation With a Saturable Nonlinearity(Elsevier Ltd., 2011-11-14) Aslan, İsmailWe analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G′/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before. © 2011 Elsevier B.V. All rights reserved.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., 2016-12) 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.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.Article Citation - WoS: 27Citation - Scopus: 28A Note on the (g'/g)-expansion Method Again(Elsevier Ltd., 2010-09) Aslan, İsmailWe report an observation on two recent analytic methods; the (G′/G)-expansion method and the simplest equation method. © 2010 Elsevier Inc. All rights reserved.Article Citation - WoS: 50Citation - Scopus: 56On the Validity and Reliability of the (g'/g)-expansion Method by Using Higher-Order Nonlinear Equations(Elsevier Ltd., 2009-05) Aslan, İsmail; Öziş, TurgutIn this study, we demonstrate the validity and reliability of the so-called (G′/G)-expansion method via symbolic computation. For illustrative examples, we choose the sixth-order Boussinesq equation and the ninth-order Korteweg-de-Vries equation. As a result, the power of the employed method is confirmed.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., 2012-03) 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: 31Citation - Scopus: 36Symbolic Computation and Construction of New Exact Traveling Wave Solutions To Fitzhugh-Nagumo and Klein-Gordon Equations(Walter de Gruyter GmbH, 2009) Öziş, Turgut; Aslan, İsmailWith the aid of the symbolic computation system Mathematica, many exact solutions for the Fitzhugh-Nagumo equation and the Klein-Gordon equation with a quadratic nonlinearity are constructed by an auxiliary equation method, the so-called (G'/G)-expansion method, where the new and more general forms of solutions are also obtained. Periodic and solitary traveling wave solutions capable of moving in both directions are observed.Article Citation - WoS: 18Citation - Scopus: 17Symbolic Computation of Exact Solutions for Fractional Differential-Difference Equation Models(Vilnius University Press, 2014-11) Aslan, İsmailThe aim of the present study is to extend the (G′=G)-expansion method to fractional differential-difference equations of rational type. Particular time-fractional models are considered to show the strength of the method. Three types of exact solutions are observed: hyperbolic, trigonometric and rational. Exact solutions in terms of topological solitons and singular periodic functions are also obtained. As far as we are aware, our results have not been published elsewhere previously.