A new method for solving nonlinear fractional differential equations (original) (raw)
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Journal of Applied Mathematics and Physics, 2019
Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Burger (KdV-Burger) equation is solved using this method and we get some new travelling wave solutions. To acquire our purpose a complex transformation has been also used to reduce nonlinear fractional partial differential equations to nonlinear ordinary differential equations of integer order, in the sense of the Jumarie's modified Riemann-Liouville derivative. Afterwards, the improved Kudryashov method is implemented and we get our required reliable solutions where the results are justified by mathematical software Maple-13.
The modified Kudryashov method for solving some fractional-order nonlinear equations
Advances in Difference Equations, 2014
In this paper, the modified Kudryashov method is proposed to solve fractional differential equations, and Jumarie's modified Riemann-Liouville derivative is used to convert nonlinear partial fractional differential equation to nonlinear ordinary differential equations. The modified Kudryashov method is applied to compute an approximation to the solutions of the space-time fractional modified Benjamin-Bona-Mahony equation and the space-time fractional potential Kadomtsev-Petviashvili equation. As a result, many analytical exact solutions are obtained including symmetrical Fibonacci function solutions, hyperbolic function solutions, and rational solutions. This method is powerful, efficient, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.
Extended Kudryashov Method for Fractional Nonlinear Differential Equations
Mathematical Sciences and Applications E-Notes
In this study, we have propesed the extended Kudryashov method to obtain the exact solutions of nonlinear fractional differential equations. Definiton of modified Riemann Liouville sense fractional derivative is used and the proposed method is applied to two nonlinear fractional differential equations. Analytical solutions including hyperbolic functions are obtained.
Generalized Kudryashov Method for Time-Fractional Differential Equations
Abstract and Applied Analysis, 2014
In this study, the generalized Kudryashov method (GKM) is handled to find exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. These time-fractional equations can be turned into another nonlinear ordinary differantial equation by travelling wave transformation. Then, GKM has been implemented to attain exact solutions of time-fractional Burgers equation, time-fractional Cahn-Hilliard equation, and time-fractional generalized third-order KdV equation. Also, some new hyperbolic function solutions have been obtained by using this method. It can be said that this method is a generalized form of the classical Kudryashov method.
Mathematics and Statistics, 2020
In this article, the nonlinear partial fractional differential equation, namely the KdV equation is renewed with the help of modified Riemann-Liouville fractional derivative. The equation is transformed into the nonlinear ordinary differential equation by using the fractional complex transformation. The goal of this paper is to construct new analytical solutions of the space and time fractional nonlinear KdV equation through the extended () '/ G G-expansion method. The work produces abundant exact solutions in terms of hyperbolic, trigonometric, rational, exponential, and complex forms, which are new and more general than existing results in literature. The newly generated solutions show that the executed method is a well-organized and competent mathematical tool to investigate a class of nonlinear evolution fractional order equations.
Journal of Applied Mathematics and Physics, 2020
In this paper, the modified Kudryashov method is employed to find the traveling wave solutions of two well-known space-time fractional partial differential equations, namely the Zakharov Kuznetshov Benjamin Bona Mahony equation and Kolmogorov Petrovskii Piskunov equation, and as a helping tool, the sense of modified Riemann-Liouville derivative is also used. The propagation properties of obtained solutions are investigated where the graphical representations and justifications of the results are done by mathematical software Maple.
Contemporary Mathematics
In the current article, the generalized Kudryshov method is applied to determine exact solitary wave solutionsfor the time fractional generalized Hirota–Satsuma coupled KdV model. Here, fractional derivative is illustrated in the conformable derivative. Therefore, plentiful exact traveling wave solutions are achieved for this model, which encourage us to enlarge, a novel technique to gain unsteady solutions of autonomous nonlinear evolution models those occurs in physical and engineering branches. The obtained traveling wave solutions are expressed in terms of the exponential and rational functions. It is effortless to widen that this method is powerful and will be applied in further tasks to create advance exclusively innovative solutions to other higher-order nonlinear conformable fractional differential model in engineering problems.
Applied Mathematical Modelling, 2015
In this paper, a new method based on the Chebyshev wavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is proposed to solve time-fractional fifth-order Sawada-Kotera (SK) equation. Two-dimensional Chebyshev wavelet method is applied to compute the numerical solution of nonlinear time-fractional Sawada-Kotera equation. The approximate solutions of nonlinear time fractional Sawada-Kotera equation thus obtained by Chebyshev wavelet method are compared with the exact solutions as well as homotopy analysis method (HAM). The present scheme is very simple, effective and convenient for obtaining numerical solution of fractional Sawada-Kotera equation.
International Journal of Sciences: Basic and Applied Research, 2020
We have enucleated new and further exact general wave solutions, along with multiple exact traveling wave solutions of space-time nonlinear fractional Chan-Hillard equation, by applying a relatively renewed technique two variables -expansion method. Also, based on fractional complex transformation and the properties of the modified Riemann-Liouville fractional order operator, the fractional partial differential equations are transforming into the form of ordinary differential equation. This method can be rumination of as the commutation of well-appointed -expansion method introduced by M. Wang et al.. In this paper, it is mentioned that the two variables - expansion method is more legitimate, modest, sturdy and effective in the sense of theoretical and pragmatical point of view. Lastly, by treating computer symbolic program Mathematica, the uniqueness of our attained wave solutions are represented graphically and reveal a comparison in a submissive manner.
The Modified Simple Equation Method for Nonlinear Fractional Differential Equations
2015
In this study, the modified simple equation method is used to construct exact solutions of the space-time fractional modified Korteweg–de Vries equation, the spacetime fractional modified regularized long-wave equation and the space-time fractional coupled Burgers’ equations in mathematical physics. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. Also we can see that when the parameters are assigned special values, families of exact solitary wave solutions can be obtained by using this method.