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

Ghosh Uttam 1*, Sarkar Susmita 1**, Das Shantanu 2†

1 Department of Applied Mathematics, University of Calcutta, Kolkata, India

∗E-mail: uttam_math@yahoo.co.in, **susmita62@yahoo.co.in

2 Reactor Control Systems Design Section E & I Group BARC Mumbai India

†E-mail: shantanu@barc.gov.in

### Received:

Received: 05 April 2016; accepted: 10 May 2016; published online: 24 August 2016

### DOI: 10.12921/cmst.2016.0000016

### Abstract:

Development of new analytical and numerical methods and their applications for solving non-linear partial differential equations (both classical and fractional) is a rising field of Applied Mathematical research because of its applications in Physical, Biological and Social Sciences. In this paper we have used a 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 the fractional sub-equation method reduces to classical solution when the order of fractional derivative tends to one. Finally numerical simulation has been done. The numerical simulation justifies that the solutions of two fractional differential equations reduce to shock solution for KP-Burger equation and soliton solution for coupled KdV equations when the order of derivative tends to one.

### Key words:

coupled KdV equation, fractional differential equation, fractional sub-equation method, generalized tanh-method, Jumarie fractional derivative, KP-Burger equation

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