Travelling Wave Solutions Of Nonlinear Time-Fractional Partial Differential Equations Via - Expansion Method

Abstract

Now a day’s fractional calculus concerns with differentiations and integrations of non-integer order has received great attentions of researchers. Fractional differential equations both ordinary and partial  play very important role in mathematical modeling in different areas of engineering and science like acoustics, electromagnetism, mathematical biology, fluid mechanics, solid state physics, signal processing quantum theory, diffusion and many other physical phenomenon. It is essential to obtain the closed-form solutions of these fractional differential equations (FDE’s) to be aware of the complete physical phenomenon of the governing mathematical model. In literature many mathematical methods has been represented for solving linear FDE’s, however it is still of great interest for researchers to achieve the closed-form solutions of non-linear time fractional partial differential equations (NTFPDE’s). In this, we shall derive the closed-form solutions of NTFPDE’s by -expansion method. This method will reduce the NTFPDE’s to non-linear ordinary differential equation (ODE), and then by using expansion series the non-linear ODE will be converted to nonlinear algebraic system of equations. By solving such system for required parameters, finally we will get closed-form solution of nonlinear FDE as to Trigonometric, Hyperbolic and Rational functions. Finally the graphical illustration of obtained solutions will be presented by using Maple.