Analytical Solutions of Classical and Fractional KP-Burger Equation and Coupled KdV equation

📝 Abstract

Evaluation of analytical solutions of non-linear partial differential equations (both classical and fractional) is a rising subject in Applied Mathematics because its applications in Physical biological and social sciences. In this paper we have used generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The exact solution obtained by fractional sub-equation method reduces to classical solution when order of fractional derivative tends to one. Finally numerical simulation has done. The numerical simulation justifies that the solutions of two fractional differential equations reduces to shock solution for KP-Burger equation and soliton solution for coupled KdV equations when order of derivative tends to one.

💡 Analysis

Evaluation of analytical solutions of non-linear partial differential equations (both classical and fractional) is a rising subject in Applied Mathematics because its applications in Physical biological and social sciences. In this paper we have used generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The exact solution obtained by fractional sub-equation method reduces to classical solution when order of fractional derivative tends to one. Finally numerical simulation has done. The numerical simulation justifies that the solutions of two fractional differential equations reduces to shock solution for KP-Burger equation and soliton solution for coupled KdV equations when order of derivative tends to one.

📄 Content

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Analytical Solutions of Classical and Fractional KP-Burger Equation and Coupled KdV equation Uttam Ghosh1a , Susmita Sarkar1b and Shantanu Das 3 1Department of Applied Mathematics, University of Calcutta, Kolkata, India
1aemail : uttam_math@yahoo.co.in 1bemail : susmita62@yahoo.co.in 3Reactor Control Systems Design Section E & I Group BARC Mumbai India email : shantanu@barc.gov.in Abstract Evaluation of analytical solutions of non-linear partial differential equations (both classical and fractional) is a rising subject in Applied Mathematics because its applications in Physical biological and social sciences. In this paper we have used generalized Tanh method to find the exact solution of KP-Burger equation and coupled KdV equation. The fractional Sub-equation method has been used to find the solution of fractional KP-Burger equation and fractional coupled KdV equations. The exact solution obtained by fractional sub-equation method reduces to classical solution when order of fractional derivative tends to one. Finally numerical simulation has done. The numerical simulation justifies that the solutions of two fractional differential equations reduces to shock solution for KP-Burger equation and soliton solution for coupled KdV equations when order of derivative tends to one.
Keywords: Generalized Tanh-method, Fractional Sub-Equation Method, KP-Burger equation, Coupled KdV equation, Fractional Differential Equation, Jumarie Fractional Derivative. 1.0 Introduction Exact solutions of non-linear differential equations give complete picture of physical systems which cannot be obtained from their linear approximation. But it is very difficult to find the exact solutions of non-linear differential equations. There are many approximate methods to find the solutions of the non- linear differential equations. The approximate methods are Adomain Decomposition Method[1-4], Homotopy Perturbation Method (HPM)[5-7 ], Differential Transform Method (DTM)[ 8] etc. Currently researcher in this field is developing new methods to find the exact solutions of non-linear differential equations. The Tanh method was introduced by Huiblin and Kelin[9] to find the travelling wave solutions of non-linear differential equations. Wazwaz [10] used this method to find soliton solutions of the Fisher equation in analytic form. Fan [11] modified this Tanh method to solve KdV-Burgers-Kuamoto equations and Boussinesq equation. Another growing field of applied science and engineering is the fractional calculus [12] where physical processes are studied in terms of the fractional differential equations. Zhang and Zhang [13] developed the fractional sub-equation method to find the travelling wave solutions of the Jumarie type fractional differential [14] equation in terms of the fractional tanh functions. The fractional sub-equation method and Generalized Tanh –method both are based on the Homogeneous balance principal [9] . The fractional sub-equations methods are used by authors to solve different non-linear fractional differential equations. Recently we have developed an algorithm to solve the linear fractional differential equations in terms of 2

one parameter Mittag-Leffler function[15]. In this paper we shall use Generalized Tanh method and Fractional Sub-equation method to exact solutions of KP-Burger and coupled KdV equations and the corresponding fractional differential equation. Using these methods we obtain the soliton solution and periodic solutions. Organization of the paper is as follows. In section 2.0 we describe the principle of
Tanh method and fractional sub-equation method. In section 3.0 we found the solutions of the KP- Burgers equation, in section 4.0 we found the solutions of the fractional order KP-Burgers equations. In section 5.0 we found the solutions of the coupled KdV equations, in section 6.0 we found the solutions of the fractional order coupled KdV equations. Finally numerical simulations are done for different values of the fractional order derivative.
2.0 Generalized Tanh Method and fractional Sub-equation Method (a) Generalized Tanh method In this method the solutions of the non-linear partial differential equations are expressed in terms of tanh and tan-functions. Consider the non-linear partial differential equation
( , , , , , , …) 0 t x y tt xx yy L u u u u u u u  (1) satisfied by ( , , ) u x y t . Use of the travelling wave transformation in the form
kx my ct    , where (k, m) are the wave vector and c is the velocity of propagating waves, equation (1) reduces to
( , , ,…) 0 L u u u    (2) This is an ordinary differential equation of ( ) u . The generalized tanh method of Fan and Hon[16 ] is based on the priori assumption that the travelling wave solutions can be expressed as the power series expansion which is the solution of non-linear Riccati dif

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