Analytical and Numerical Solutions of Mathematical Biology Models: the Newell-Whitehead and Allen-Cahn Equations

Abstract

In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell-Whitehead-Segel-type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.