________________________________________ *Corresponding author: Email: haniye_hajinezhad@yahoo.com; Asian Research Journal of Mathematics 18(5): 22-30, 2022; Article no.ARJOM.85584 ISSN: 2456-477X _______________________________________________________________________________________________________________________________________ A numerical Approximation for the One-dimensional Burger–Fisher Equation Haniye Hajinezhad a* a Department of Mathematics, Payame Noor University, Tehran, Iran. Author’s contribution The sole author designed, analyzed, interpreted and prepared the manuscript. Article Information DOI: 10.9734/ARJOM/2022/v18i530375 Open Peer Review History: This journal follows the Advanced Open Peer Review policy. Identity of the Reviewers, Editor(s) and additional Reviewers, peer review comments, different versions of the manuscript, comments of the editors, etc are available here: https://www.sdiarticle5.com/review-history/85584 Received 13 February 2022 Accepted 16 April 2022 Published 04 May 2022 __________________________________________________________________________________ Abstract In this paper, an implicit finite difference method based on the Crank–Nicolson method is proposed for the numerical solution of the one-dimensional Burger–Fisher equation. The Crank–Nicolson scheme provides a system of nonlinear difference equations, which is solved by an integration of the Jacobian-Free-Newton- Krylov (JFNK) and GMRES methods. Various numerical examples are given to demonstrate the efficiency of the proposed scheme. Comparison of the computed solutions with the analytical ones demonstrates the accuracy of this proposed method. Keywords: Crank–Nicolson scheme; burger–fisher equation; jacobian-free-newton-krylov method; GMRES method. 1 Introduction Being a combination of convection, diffusion, and reaction mechanisms, the Burger –Fisher equation is highly nonlinear. This equation has many applications in many scientific fields such as gas dynamics, number theory, elasticity, heat conduction, etc. [1]. Recently, several methods have been proposed to solve it. Here, we briefly discuss the methods of some researchers. Pirdawood and Sabawi [2] proposed an accurate scheme using the compact finite difference method and the Runge Kutta method. Gürbüz and Sezer [3] applied a modified Laguerre matrix-collocation method. Singh et al. Original Research Article