Efficient Implicit Numerical Methods for Nonlinear Fisher Equation

Kumari, Richa; Vimal, Vikash; ., Amrita

doi:10.22034/cmde.2025.66534.3114

Efficient Implicit Numerical Methods for Nonlinear Fisher Equation

Articles in Press

Document Type : Research Paper

Authors

¹ Department of Physics, Patna University, Ashok Rajpath, Patna-800005, India.

² Department of Mathematics, Nalanda University, Rajgir, Nalanda-803116, Bihar, India.

10.22034/cmde.2025.66534.3114

Abstract

This paper explores various numerical methods for solving the one-dimensional nonlinear Fisher equation using the finite difference and Newton methods. The study focuses on achieving higher accuracy in numerical solutions, the proposed approach being first-order accurate in time and second-order accurate in space. The numerical results for different values of $\alpha$ closely match the exact solutions. Several examples are presented, comparing the $L_2$ and $L_{\infty}$ errors with the exact solution and the existing methods from the literature and leading to high accuracy. These types of equations arise in various fields of sciences and engineering, the main application of this equation has been found in the biomedical sciences. The solution of this equation helps to determine the size of the brain tumor.

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