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Öğe Application of some nonclassical methods for p-defocusing complex Klein–Gordon equation(Springer, 2023) Yokuş, Asif; İskenderoğlu, Gülistan; Kaya, DoğanThis research studies the high-energy Klein–Gordon equation, related to the relativistic Schrödinger wave equation. Solutions are produced using the Kudryashov transform and conventional wave transform. The study’s most significant finding is the examination of the physical and mathematical differences in the traveling wave solutions, derived using two alternative transformations. The graphs of the intensity of the wave function and the conclusions at various velocity levels are also examined. The stability analysis of the moving waves, which coincides with Graham’s diffusion law, of the relationship between wave velocity and density is examined and its relationship with the propagation equation is investigated.Öğe Applications of a new expansion method for finding wave solutions of nonlinear differential equations(2012) Duran, Serbay; Kaya, DoğanIn this paper, we presented a new expansion method constructed by taking inspiration for the Kudryashov method. Bernoulli equation was chosen in the form of F' = FB n-AF and made some expansions on the auxiliary Bernoulli equation which used in this method. In this auxiliary Bernoulli equation, by taking n = 3, some wave solutions obtained from Burgers equation and the shallow water wave equation system. As a result, for special values, we concludedthree dimensional wave views for solutions of Burgers Equation and the shallow water wave equation system.To sum up, it is considered that this method can be applied to other nonlinear evolution equations in mathematics physics. © IDOSI Publications, 2012.Öğe Applications of the sub equation method for the high dimensional nonlinear evolution equation(Erzincan Binali Yıldırım Üniversitesi, Fen Bilimleri Enstitüsü, 2021) Duran, Serbay; Kaya, DoğanIn this article, the Generalized (3+1)-dimensional Shallow Water-Like (SWL) equation is taken into consideration and exact solutions have been constructed of the SWL equation using sub equation method. This method is an easier and efficient method for finding analytic solutions of the nonlinear evolution differential equation. The method appears to be easier and faster for symbolic computation. Moreover, 2D, 3D and contour graphical representation of the obtained results of the specified equation is made using a ready-made package program for certain values and thus the conformity of the founded results has been demonstrated.Öğe Breaking analysis of solitary waves for the shallow water wave system in fluid dynamics(Springer, 2021) Duran, Serbay; Kaya, DoğanAnalytical solutions and physical interpretations for the shallow water wave system, which is the modeling of a physical phenomenon in applied mathematics, are presented in this study. The solutions of the shallow water wave system, which models the formation, interaction and breaking of shallow water waves on the ocean surface as a result of external effects (e.g., wind), are examined using the modified expansion method. Besides, physically, the differences between deep and shallow water waves and their transitions with each other are examined. Hyperbolic and trigonometric wave solutions are produced with the advantages of the modified expansion method over other traditional methods. In these wave solutions, the values that cause the breaking of the wave are calculated by considering the velocity parameter. The behavior of the wave at these values is presented with the help of simulations. Finally, the states of solitary wave solutions concerning each other in the same system are compared and analyzed according to the velocity parameter.Öğe Conservation laws and a new expansion method for sixth order Boussinesq equation(American Institute of Physics Inc., 2015) Yokuş, Asif; Kaya, DoğanIn this study, we analyze the conservation laws of a sixth order Boussinesq equation by using variational derivative. We get sixth order Boussinesq equation's traveling wave solutions with (1/G)-expansion method which we constitute newly by being inspired with (G/G)-expansion method which is suggested in [1]. We investigate conservation laws of the analytical solutions which we obtained by the new constructed method. The analytical solution's conductions which we get according to new expansion method are given graphically. © 2015 AIP Publishing LLC.Öğe Düzenli Uzun Dalga Denkleminin Hiperbolik Tip Yürüyen Dalga Çözümleri(2020) , Hülya; Yokuş, Asıf; Kaya, DoğanBu çalışmanın temel amacı (1 / G') -açılım yöntemi kullanılarak Düzenli Uzun Dalga (RLW) denklemi için yürüyen dalga çözümlerini elde etmektir. Elde edilen çözümlerde sabitlere özel değerler verilerek 3 boyutlu, 2 boyutlu ve kontur grafikleri sunulmuştur. Bu grafikler Düzenli Uzun Dalga denkleminin özel bir çözümüdür vedenklemin durağan bir dalgasını temsil etmektedir. Bu makalede sunulan çözümleri ve grafikleri bulmak için bilgisayar paket programı kullanılmaktadır.Öğe Exploring the influence of layer and neuron configurations on Boussinesq equation solutions via a bilinear neural network framework(Springer Science and Business Media B.V., 2024) Isah, Muhammad Abubakar; Yokus, Asif; Kaya, DoğanThis study examines the Boussinesq equation, which is a nonlinear partial differential equation used to describe long wave propagation in shallow water and has broader applications, including nonlinear lattice waves, vibrations in nonlinear strings, and ion sound waves in plasma. The Boussinesq equation provides an insight into the nonlinear long wave propagation behavior in shallow water by taking wave phase into account. Its versatility extends its utility beyond fluid dynamics to various physical phenomena. By providing specific activation functions in the ``2-3-1?? and ``2-5-1?? neural network models, respectively, the generalized lump solution and the precise analytical solutions are produced using the bilinear neural network approach. These analytical solutions, together with the related rogue waves, dark soliton, and bright soliton, are derived using symbolic computation. These findings fill in the gaps in the current research about the Boussinesq equation. The dynamical properties of these waves are displayed on three-dimensional, contour, density, and two-dimensional graphs. The response of the wave solution to different values of wave speed in relation to the wave phase it contains has been described with the help of wave intensity. In addition, the advantages and disadvantages of the layers used in the analytical technique to generate solutions have been discussed. The efficient techniques employed in this research are useful for studying the nonlinear differential equations in one-dimensional nonlinear lattice waves, vibrations in a nonlinear string, and ion sound waves in plasma.Öğe Finding the traveling wave solutions of some nonlinear partial differential equations by an expansion method(İstanbul Ticaret Üniversitesi, 2016) Kaya, DoğanIn this study, we construct an expansion method. We have implement this method for finding traveling wave solutions of nonlinear Klein-Gordon equation, Benjamin-Bona-Mahony equation, sixth-order Boussinesq equation and Konopelchenko-Dubrovky system.Öğe Lie group analysis for initial and boundary value problem of time-fractional nonlinear generalized KdV partial differential equation(TUBITAK, 2019) Kaya, Doğan; İskandarova, GülistanThe Lie group analysis or in other word the symmetry analysis method is extended to deal with a timefractional order derivative nonlinear generalized KdV equation. Our research in this work aims to use transformation methods and their analysis to search for exact solutions to the nonlinear generalized KdV differential equation. It is shown that this equation can be reduced to an equation with Erdelyi-Kober fractional derivative. In this study, we research the initial and boundary conditions, considering them invariant, and so we get two ordinary initial value problems, i.e. two Cauchy problems. Conservation laws for the given equation are also investigated. In this work, we introduce symmetry analysis and find conservation laws for the nonlinear generalized time-fractional KdV equation by the Lie groups method using fractional derivatives in the Riemann-Liouville sense. © Tübitak.Öğe Lie symmetry analysis of Caputo time-fractional K(m,n) model equations with variable coefficients(Yildiz Technical University, 2024) İSkenderoğlu, Gülistan; Kaya, DoğanIn this study, we consider model equations K(m,n) with fractional Caputo time derivatives. By applying the Lie group symmetry method, we determine all symmetries for these equations and present the reduced symmetric equations for the equation K(m,n) with fractional Caputo time derivatives. Furthermore, we obtain the exact solution for K(1,1) with the fractional Caputo time derivative and provide graphs depicting the behavior at different orders of the fractional time derivative. Additionally, by considering the symmetries of the equation, we establish the conservation laws for K(m,m) with the fractional Caputo time derivative. Copyright 2021, Yıldız Technical University.Öğe Modeling of dark solitons for nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod(Tech Science Press, 2021) Durur, Hülya; Yokuş, Asıf; Kaya, Doğan; Ahmad, HijazIn this paper, sub equation and ? ? 1=G’ expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation (LWE) in a magneto-electro-elastic circular rod. The proposed methods have been used to construct hyperbolic, rational, dark soliton and trigonometric solutions of the LWE in the magnetoelectro-elastic circular rod. Arbitrary values are given to the parameters in the solutions obtained. 3D, 2D and contour graphs are presented with the help of a computer package program. Solutions attained by symbolic calculations revealed that these methods are effective, reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.Öğe A new approach for Painleve analysis of the generalized Kawahara equation(Amer Inst Physics, 2015) Kutlu, Büşra; Kaya, Doğan; Ashyralyev, A; Malkowsky, E; Lukashov, AIn this paper, generalized Kawahara equation with a source is considered. It is shown that this equation is nonintegrable by using Painleve test. An exact solution is also obtained with the help of simplified Hirota's method for the considered equation.Öğe Numerical and exact solutions for time fractional Burgers' equation(Int Scientific Research Publications, 2017) Yokus, Asif; Kaya, DoğanThe main purpose of this paper is to find an exact solution of the traveling wave equation of a nonlinear time fractional Burgers' equation using the expansion method and the Cole-Hopf transformation. For this purpose, a nonlinear time fractional Burgers' equation with the initial conditions considered. The finite difference method (FDM for short) which is based on the Caputo formula is used and some fractional differentials are introduced. The Burgers' equation is linearized by using the Cole-Hopf transformation for a stability of the FDM. It shows that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in L-2 and L-infinity. All of obtained results are discussed with an example of the Burgers' equation including numerical solutions for different situations of the fractional order and the behavior of potentials u is investigated with graphically. All the obtained numerical results in this study are presented in tables. We used the Mathematica software package in performing this numerical study. (C) 2017 All rights reserved.Öğe Numerical comparison of Caputo and Conformable derivatives of time fractional Burgers-Fisher equation(Elsevier, 2021) Yokuş, Asıf; Durur, Hülya; Kaya, Doğan; Ahmad, Hijaz; Nofal, Taher A.In this paper, the sub-equation method is used to obtain new types of complex traveling wave solutions of the time-fractional Burgers-Fisher equation. In this work is to compare the exact complex traveling wave solutions of new types and the numerical solutions obtained by suitable transformations of Caputo and Conformable de-rivatives. Also, to discuss the advantages and disadvantages of those derivatives and a new initial condition was created by using the obtained solution and the numerical solutions of the equation were obtained by the finite difference method. A comparison of the numerical solutions with the obtained exact solution is made. L2 and L? norm errors, absolute error values, Von Neumann stability analysis supporting this comparison are investigated. To consolidate the accuracy of the numerical results some tables and graphs are presented. For drawing complex mathematical operations and graphs, computer package programs are usedÖğe Numerical solutions of Fisher's equation with collocation method(American Institute of Physics Inc., 2015) Gülbahar, Sema; Yokuş, Asıf; Kaya, DoğanIn this paper, the collocation method using radial basis functions is presented for a numerical solution of Fisher's equation. In the solution process, we use linearization techniques for non-linear term existing in the equation. Furthermore, the methods which are analyzed by the error norms L2 and Lâ? and by matrix stability analysis are tested for numerical examples and results obtained by four radial basis functions from computer runs are compared. © 2015 AIP Publishing LLC.Öğe Numerical solutions of the Fractional Kdv-Burgers-Kuramoto equation(Serbian Society of Heat Transfer Engineers, 2018) Kaya, Doğan; Gülbahar, Sema; Yokuş, AsifNon-linear terms of the time-fractional KdV-Burgers-Kuramoto equation are linearized using by some linearization techniques. Numerical solutions of this equation are obtained with the help of the finite difference methods. Numerical solutions and corresponding analytical solutions are compared. The 2 L and L8 error norms are computed. Stability of given method is investigated by using the Von Neumann stability analysis. © 2018 Society of Thermal Engineers of Serbia.Öğe Refraction simulation of internal solitary waves for the fractional Benjamin–Ono equation in fluid dynamics(World Scientific, 2021) Duran, Serbay; Yokuş, Asıf; Durur, Hülya; Kaya, DoğanIn this study, the modified (1/G?)-expansion method and modified sub-equation method have been successfully applied to the fractional Benjamin–Ono equation that models the internal solitary wave event in the ocean or atmosphere. With both analytical methods, dark soliton, singular soliton, mixed dark-singular soliton, trigonometric, rational, hyperbolic, complex hyperbolic, complex type traveling wave solutions have been produced. In these applications, we consider the conformable operator to which the chain rule is applied. Special values were given to the constants in the solution while drawing graphs representing the stationary wave. By making changes of these constants at certain intervals, the refraction dynamics and physical interpretations of the obtained internal solitary waves were included. These physical comments were supported by simulation with 3D, 2D and contour graphics. These two analytical methods used to obtain analytical solutions of the fractional Benjamin–Ono equation have been analyzed in detail by comparing their respective states. By using symbolic calculation, these methods have been shown to be the powerful and reliable mathematical tools for the solution of fractional nonlinear partial differential equations.Öğe Solutions of the fractional combined KdV-mKdV equation with collocation method using radial basis function and their geometrical obstructions(Springeropen, 2018) Kaya, Doğan; Gülbahar, Sema; Yokuş, Asif; Gülbahar, MehmetThe exact solution of fractional combined Korteweg-de Vries and modified Korteweg-de Vries (KdV-mKdV) equation is obtained by using the expansion method. To investigate a geometrical surface of the exact solution, we choose . The collocation method is applied to the fractional combined KdV-mKdV equation with the help of radial basis for . and error norms are computed with the Mathematica program. Stability is investigated by the Von-Neumann analysis. Instable numerical solutions are obtained as the number of node points increases. It is shown that the reason for this situation is that the exact solution contains some degenerate points in the Lorentz-Minkowski space.Öğe Symmetry Analysis and Conservation Laws of the Boundary Value Problems for Time-Fractional Generalized Burgers’ Differential Equation(2019) Iskenderoglu, Gulistan; Kaya, DoğanMany physical phenomena in nature can be described or modeled via a differential equation or a system of differential equations. In this work, we restrict our attention to research a solution of fractional nonlinear generalized Burgers’ differential equations. Thereby we find some exact solutions for the nonlinear generalized Burgers’ differential equation with a fractional derivative, which has domain as $\mathbb{R}^2$ ×$\mathbb{R}^+$. Here we use the Lie groups method. After applying the Lie groups to the boundary value problem we get the partial differential equations on the domain $\mathbb{R}^2$ with reduced boundary and initial conditions. Also, we find conservation laws for the nonlinear generalized Burgers’ differential equation.Öğe Symmetry solution on fractional equation(2017) Iskandarova, Gulistan; Kaya, DoğanAs we know nearly all physical, chemical, and biological processes in naturecan be described or modeled by dint of a differential equation or a system ofdifferential equations, an integral equation or an integro-differential equation.The differential equations can be ordinary or partial, linear or nonlinear. So,we concentrate our attention in problem that can be presented in terms of adifferential equation with fractional derivative. Our research in this work is touse symmetry transformation method and its analysis to search exact solutionsto nonlinear fractional partial differential equations.
