The Exact Solution of the Fractional Burger’s Equation using the Modified Homogeneous Balance Method

Abstract

In this paper, the Modified Homogeneous Balance Method, which is embedded with a fractional Riccati equation, is used to find exact solutions to the fractional Burger’s equation. Thus, this method incorporates a fractional Riccati equation whose right-hand side is a polynomial of degree two, accounting for the bright and dark solitons of the fractional Burger’s equation. In addition, exact solutions of the fractional Burger’s equation revealed kink-antikink interactions, and variations in the amplitudes of kink and antikink solutions reflect the nonlinear dispersive of the fractional Burger’s equation. The unique feature about the Modified Homogeneous Balance Method is that the embedded Riccati equation facilitates the speed of convergence of the solution of the Burger’s equation in any functional space. Interestingly, the speed of convergence of the series to the exact solution of the Burger’s equation using the Modified Homogeneous Balance Method was observed to be best as compared with the Variational Iteration Method, Fractional Reduced Differential Transform Method, Generalized Differential Transform Method and the Homotopy Perturbation Method. Thus, the number of series needed to converge to the exact solution of the fractional Burger’s equation using the Modified Homogeneous Balance Method is fewer than that of the Variational Iteration Method, Fractional Reduced Differential Transform Method, Generalized Differential Transform Method and the Homotopy Perturbation Method.

Keywords

• Modified Homogeneous Balance Method (MHBM)
• Fractional Burger’s equation
• Nonlinear partial differential equations
• Fractional calculus
• Analytical methods
• Exact solutions