Solving Burger-Fisher Equation (BFE) by using Banach Contraction Method (BCM)

In this article, we propose a method for solving the Burger-Fisher equation. The proposed method is the Banach Contraction Method (BCM). The Burger-Fisher equation is a combination of the Burger equation and the Fisher equation. There have been many methods used to solve the problem associated with the Burger-Fisher Equation, such as the Adomian Decomposition Method (ADM) and the Local Discontinuous Galerkin (LDG) method. Therefore, this paper presents the results of the research findings with two main objectives: We used the BCM to solve the Burger-Fisher Equation, followed by validating the accuracy of the BCM with the exact solutions. Plus, calculations were mainly performed using the MAPLE 2018 software to facilitate the calculations and simplify the complicated results in this study. As for the scope of this study, we cover the partial differential equation specifically in terms of the Burger-Fisher Equation and the BCM. In summary, the method used in this study has introduced a straightforward mathematical method for solving various differential equations, which improves and expands our knowledge in the field of mathematics. Currently, all the articles related to Burger-Fisher Equation and BCM are applicable in this study. Also, a procedure for implementing the method is provided, along with an error analysis. As a result, we calculate the exact solution of Burger-Fisher Equation using BCM up to only. As for and continuously the solution was too long thus, we provide the error between the exact solution and BCM by using various value of α, β and σ as for x from 0 to 1 and for t = 1, 10, and 50. We prove that the new technique is more practical by comparing the outcomes of the exact solution with certain previously reported findings.