On the Solutions of Fractional Burgers-Fisher and Generalized Fisher’s Equations Using Two Reliable Methods

Two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), are presented. Haar wavelet method is an efficient numerical method for the numerical solution of arbitrary order partial differential equations like Burgers-Fisher and generalized Fisher equations. The approximate solutions thus obtained for the fractional Burgers-Fisher and generalized Fisher equations are compared with the optimal homotopy asymptotic method as well as with the exact solutions. Comparison between the obtained solutions with the exact solutions exhibits that both the featured methods are effective and efficient in solving nonlinear problems. The obtained results justify the applicability of the proposed methods for fractional order Burgers-Fisher and generalized Fisher’s equations. 1. Introduction Fractional calculus is a field of applied mathematics which deals with derivatives and integrals of arbitrary orders. In the last few decades, fractional calculus has been extensively investigated due to its broad applications in mathematics, physics, and engineering such as viscoelasticity, diffusion of biological population, signal processing, electromagnetism, fluid mechanics, electrochemistry and so on. Fractional differential equations are extensively used in modeling of physical phenomena in various fields of science and engineering. For this we need a reliable and efficient technique for the solution of fractional differential equations. Recently, orthogonal wavelets bases are becoming more popular for numerical solutions of partial differential equations due to their excellent properties such as ability to detect singularities, orthogonality, flexibility to represent a function at different levels of resolution, and compact support. In recent years, there has been a growing interest in developing wavelet based numerical algorithms for solution of fractional order partial differential equations. Among them, the Haar wavelet method is the simplest and is easy to use. Haar wavelets have been successfully applied for the solutions of ordinary and partial differential equations, integral equations, and integrodifferential equations. Therefore, the main focus of the present paper is the application of Haar wavelet technique for solving the problem of Burgers-Fisher and generalized Fisher’s equations. The obtained numerical approximation results of this method are then also compared with the optimal homotopy asymptotic method. Consider the generalized one-dimensional Burgers-Fisher equation of fractional order: where , and are parameters and . This