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Öğe ANALYTICAL AND NUMERICAL SOLUTIONS FOR TIME-FRACTIONAL NEW COUPLED MKDV EQUATION ARISING IN INTERACTION OF TWO LONG WAVES(Asia Pacific Academic, 2019) Tasbozan, Orkun; Şenol, Mehmet; Kurt, Ali; Baleanu, DumitruThe aim of this paper is to present new exact solution sets of nonlinear conformable time-fractional new coupled mKdV equations which arise in interaction of two long waves with different dispersion relations by means of sub-equation method. In addition, we also propose an analytical-approximate method namely residual power series method (RPSM) for the system. The fractional derivatives have been explained in newly defined conformable type, during the solution procedure. The exact solutions of the system obtained by the sub-equation method have been compared to approximate solutions derived by RPSM. The results showed that both methods are robust, dependable, easy to apply and a good alternative for seeking solutions of fractional partial differential equations. © 2019 Asia Pacific Journal of Mathematics.Öğe Analytical Approximation for Cahn-Hillard Phase-Field Model for Spinodal Decomposition of a Binary System(2021) Tozar, Ali; Tasbozan, Orkun; Kurt, AliPhase transformations which lead to dramatical property change are\rvery important for engineering materials. Phase-eld methods are one of the most\rsuccessful and practical methods for modelling phase transformations in materials.\rThe Cahn-Hillard phase-eld model is among the most promising phase-eld models.\rThe most successful aspect of the model is that it can predict spinodal decomposition\r(which is essential to determining the microstructure of an alloy) in a binary system.\rIt is used in both materials science and many other elds, such as polymer science,\rastrophysics, and computer science. In this study, the Cahn-Hillard phase-eld model\ris evaluated by an analytical approach using the (1=G0)-expansion method. The so-\rlutions obtained are tested for certain thermodynamic conditions, and their accuracy\rof predicting the spidonal decomposition of a binary system is conrmed.Öğe Analytical solutions of Cahn-Hillard phase-field model for spinodal decomposition of a binary system(Iop Publishing Ltd, 2020) Tozar, Ali; Tasbozan, Orkun; Kurt, AliSpinodal decomposition is a very important and challenging issue not for only materials science but for also many other fields in science. Phase-field models, which have become very popular in recent years, are very promising for the evaluation of phase transformations such as spinodal decomposition. In this study, the Cahn-Hillard equation which is one of the most trending phase-field models is analytically solved by the double exp-function and the modified extended exp-function methods. Spatio-temporal energy and concentration distributions were evaluated by applying two of the ten obtained analytical solutions to the free energy equation. The results obtained were found to be successful in predicting spinodal decomposition as is expected from the Cahn-Hillard model.Öğe Applying the New Extended Direct Algebraic Method to Solve the Equation of Obliquely Interacting Waves in Shallow Waters(Ocean Univ China, 2020) Kurt, Ali; Tozar, Ali; Tasbozan, OrkunIn this study, the potential Kadomtsev-Petviashvili (pKP) equation, which describes the oblique interaction of surface waves in shallow waters, is solved by the new extended direct algebraic method. The results of the study show that by applying the new direct algebraic method to the pKP equation, the behavior of the obliquely interacting surface waves in two dimensions can be analyzed. This article fairly clarifies the behaviors of surface waves in shallow waters. In the literature, several mathematical models have been developed in attempt to study these behaviors, with nonlinear mathematics being one of the most important steps; however, the investigations are still at a level that can be called 'baby steps'. Therefore, every study to be carried out in this context is of great importance. Thus, this study will serve as a reference to guide scientists working in this field.Öğe Approximate analytical solution of the time fractional Whitham-Broer-Kaup equation using the homotopy analysis method(Academic Press, 2015) Kurt, Ali; Tasbozan, OrkunIn this paper, the homotopy analysis method (HAM) is applied to the time fractional Whitham-Broer-Kaup equation to obtain its approximate analytical solutions. The HAM solution includes an auxiliary parameter h which provides a suitable way of adjusting and controlling the convergence region of solution series. © 2015 Academic Publications, Ltd.Öğe Approximate Analytical Solutions of Conformable Time Fractional Clannish Random Walker’s Parabolic(CRWP) Equation and Modified Benjamin-Bona-Mahony(BBM) equation(Emrah Evren KARA, 2020) Atilgan, Emrah; Tasbozan, Orkun; Kurt, Ali; Mohyud-Din, Syed TauseefIn this paper, we propose the approximate analytical solutions of conformable time fractional Clannish Random Walker’s Parabolic(CRWP) equation and Modified Benjamin-Bona-Mahony(BBM) equation with the aid of generalized homotopy analysis method (q-HAM). The h curves of approximate solutions for both equations are illustrated by graphics to deter-mine the convergence interval. h values obtained from these graphics are used to compare approximate solutions with the analytical solutions. The results show that approximate solutions are consistent with the analytical solutions. Also it is understood that the method is reliable, applicable and efficient technique to get the exact solutions of fractional partial differential equations. © 2020, Emrah Evren KARA. All rights reserved.Öğe APPROXIMATE ANALYTICAL SOLUTIONS TO CONFORMABLE MODIFIED BURGERS EQUATION USING HOMOTOPY ANALYSIS METHOD(Sciendo, 2019) Kurt, Ali; Tasbozan, OrkunIn this paper the authors aspire to obtain the approximate analytical solution of Modified Burgers Equation with newly defined conformable derivative by employing homotopy analysis method (HAM).Öğe Comparison of two reliable methods to solve fractional Rosenau-Hyman equation(Wiley, 2021) Senol, Mehmet; Tasbozan, Orkun; Kurt, AliIn this study, we examine the numerical solutions of the time-fractional Rosenau-Hyman equation, which is a KdV-like model. This model demonstrates the formation of patterns in liquid drops. For this purpose, two reliable methods, residual power series method (RPSM) and perturbation-iteration algorithm (PIA), are used to obtain approximate solutions of the model. The fractional derivative is taken in the Caputo sense. Obtained results are compared with each other and the exact solutions both numerically and graphically. The outcome shows that both methods are easy to implement, powerful, and reliable. So they are ready to implement for a variety of partial fractional differential equations.Öğe Functional Variable Method for conformable fractional modified KdV-ZK equation and Maccari system(Tbilisi Centre Math Sci, 2017) Cenesiz, Yucel; Tasbozan, Orkun; Kurt, AliModeling the motion and propagation characteristics of waves have importance in coastal, ocean and maritime engineering. Especially, waves are the major source of environmental actions on beaches or on man-made fixed or floating structures in most, geographical areas. So Maccari system has great application in mentioned areas. The modified KdV is ion acoustic perturbations evolution model in a plasma with two negative ion components which have different temperatures. As for the KdV equation, the modified ZK (mZK) equation arises naturally as weakly two-dimensional variations of the mKdV equation. hi this paper authors used functional variable method(FVM) for the first time to obtain exact travelling wave and soliton solutions of conformable fractional modified KdV-Zakharov-Kuznetsov(mKdv-ZK) equation and Maccari system. As a consequence, new solutions are obtained and it is seen that FVM is an valuable and efficient tool for solving nonlinear equations and systems where the derivatives defined by means of conformable fractional derivative.Öğe Generalized Sub-Equation Method for the (1+1)- Dimensional Resonant Nonlinear Schrodinger’s Equation(2021) Tasbozan, Orkun; Tozar, Ali; Kurt, AliInterest in studying nonlinear models has been increasing in recent years. Dynamical systems, in which the state of the system changes continuously over time, have nonlinear interactions. The use of unique nonlinear differential equations is inescapable in the evaluation of such systems. In mathematical point of view, for obtaining analytical solutions of nonlinear differential equations, it must be fully integrable. Consequently, the importance of fully integrable nonlinear differential equations for nonlinear science has become indisputable. Among these equations, one of the most studied by physicists and mathematicians is the nonlinear Schrödinger equation. This equation has undergone many modifications to evaluate different phenomena. In this study, the resonant nonlinear Schrödinger equation, which is the most important of these physical equations in terms of explaining many physical phenomena, is solved analytically with the generalized sub-equation method.Öğe Multiple-solitons for generalized (2+1)-dimensional conformable Korteweg-de Vries-Kadomtsev-Petviashvili equation(Elsevier, 2022) Akinyemi, Lanre; Senol, Mehmet; Tasbozan, Orkun; Kurt, AliThis paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili (KdV-KP) equation. This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation. The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric, hyperbolic, and rational solutions. The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations. Furthermore, the obtained solutions have not been reported in the previous literature and might have significant impact on future research. (c) 2021 Shanghai Jiaotong University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)Öğe New analytical solutions and approximate solution of the space-time conformable Sharma-Tasso-Olver equation(Natural Sciences Publishing, 2019) Tasbozan, Orkun; Çenesiz, Yucel; Kurt, Ali; Iyiola, Olaniyi S.The main purpose of this article is to find the exact and approximate solutions of space-time conformable Sharma-Tasso- Olver equation using first integral method (FIM) and q-homotopy analysis method (q-HAM) respectively. The obtained exact and numerical solutions are compared with each other. Also, the numerical results obtained by q-HAM are compatible with the exact solutions obtained by FIM; hence, it is clearly seen that these techniques are powerful and efficient in finding approximate and exact solutions for nonlinear conformable partial differential equations. © 2018 NSP.Öğe New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method(De Gruyter Poland Sp Zoo, 2017) Tasbozan, Orkun; Cenesiz, Yucel; Kurt, Ali; Baleanu, DumitruModelling of physical systems mathematically, produces nonlinear evolution equations. Most of the physical systems in nature are intrinsically nonlinear, therefore modelling such systems mathematically leads us to nonlinear evolution equations. The analysis of the wave solutions corresponding to the nonlinear partial differential equations (NPDEs), has a vital role for studying the nonlinear physical events. This article is written with the intention of finding the wave solutions of Nizhnik-Novikov-Veselov and Klein-Gordon equations. For this purpose, the exp-function method, which is based on a series of exponential functions, is employed as a tool. This method is an useful and suitable tool to obtain the analytical solutions of a considerable number of nonlinear FDEs within a conformable derivative.Öğe New Analytical Solutions for Time Fractional Benjamin-Ono Equation Arising Internal Waves in Deep Water(China Ocean Press, 2019) Tasbozan, OrkunIn this article, the author sets up the abundant traveling wave solutions for time fractional Benjamin-Ono equation which was introduced to describe internal waves in stratified fluids by using Jacobi elliptic function expansion method. The traveling wave solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. It can be seen that the obtained results are found to be important for the statement of some physical demonstrations of problems in mathematical physics and ocean engineering. In ocean engineering Benjamin-Ono equations are generally used in computer simulation for the water waves in deep water and open seas.Öğe New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations(Walter De Gruyter Gmbh, 2020) Durur, Hulya; Tasbozan, Orkun; Kurt, AliThe main purpose of this article is to obtain the new solutions of fractional bad and good modified Boussinesq equations with the aid of auxiliary equation method, which can be considered as a model of shallow water waves. By using the conformable wave transform and chain rule, nonlinear fractional partial differential equations are converted into nonlinear ordinary differential equations. This is an important impact because both Caputo definition and Riemann-Liouville definition do not satisfy the chain rule. By using conformable fractional derivatives, reliable solutions can be achieved for conformable fractional partial differential equations.Öğe NEW EXACT AND NUMERICAL SOLUTIONS OF FRACTIONAL KAUP-KUPERSHMIDT EQUATION(Asia Pacific Academic, 2019) Senol, Mehmet; Tasbozan, Orkun; Kurt, Ali; Ata, AyşeIn this article, the tanh method and the residual power series method (RPSM) are used to obtain new exact and numerical solutions of the time-fractional Kaup-Kupershmidt equation using the conformable fractional derivative definition. This definition is simple, effective and reliable in the solution procedure of the fractional differential equations that have complicated solutions with classical fractional derivative definitions like Caputo and Rieman-Liouville. © 2019 Asia Pacific Journal of Mathematics.Öğe New exact solutions of Burgers' type equations with conformable derivative(Taylor & Francis Ltd, 2017) Cenesiz, Yucel; Baleanu, Dumitru; Kurt, Ali; Tasbozan, OrkunIn this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers' equation, modified Burgers' equation, and Burgers-Korteweg-de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.Öğe New Exact Solutions of Fractional Fitzhugh-Nagumo Equation(2019) Tasbozan, Orkun; Kurt, AliThe main aim of this article is obtaining the travelling wave, solitary wave and periodicwave solutions for time fractional Fitzhugh-Nagumo equation which used as a model for reaction–diffusion, transmission of nerve impulses, circuit theory, biology and population genetics. The newextended direct algebraic method is employed for this aim. The fractional derivative is in conformablesense which is an applicable, well behaved and understandable definition.Öğe A New Method to Solve Time Fractional Diffusion Equations Arising in Chaos and Heat Conduction(Amer Scientific Publishers, 2018) Ozkan, Ozan; Kurt, Ali; Tasbozan, OrkunThe goal of the present paper is to construct a method to obtain the solution of conformable fractional partial differential equations (CFPDEs). Since these systems can be transformed to partial differential equations (PDEs) by using wave transform, the reduced system can be solved by using differential transform method (DTM) solution methods. Based on this idea, we build an efficient solution procedure that reduces CFPDEs to PDEs via wave transform, then approximate the solution of obtained system by using Differential Transform Method (DTM) which is a special procedure for solving PDEs. As an example, we implement the method to time fractional Diffusion Equation (TFDE).Öğe New optical solutions of complex Ginzburg-Landau equation arising in semiconductor lasers(Springer Heidelberg, 2019) Tasbozan, Orkun; Kurt, Ali; Tozar, AliNonlinear optics draws much attention by physicists and mathematicians due to its challenging mathematical structure. The study of non-hamiltonian and dissipative systems is one of the most complicated and challenging issues of nonlinear optics. Recent studies showed that there is a close relationship between superconductivity, Bose-Einstein condensation, and semiconductor lasers. Therefore, the cubic complex Ginzburg-Landau (CGLE) equation is thought to be a useful tool in investigating nonlinear optical events. On the other hand, the CGLE is a very general type of equation that governing a vast variety of bifurcations and nonlinear wave phenomena in spatiotemporally extended systems. In this article, we acquire the new wave solution of time fractional CGLE with the aid of Jacobi elliptic expansion method.
