Pseudospectral method for Fisher equation in a disk

Abstract

In this paper, we develop a mixed Jacobi–Fourier pseudospectral method for solving the Fisher equation in a disc. Fisher’s equation always plays a large role in many fields, such as logistic population growth models, tissue engineering, nuclear reactions, and neurophysiology, etc.. It is very important to study how to solve these equations numerically. In this work, we employ the generalized Jacobi approximation to simulate the singularity of solutions at the regional center. Some mixed Jacobi–Fourier interpolation approximation results are established, which play important roles in numerical simulation of various problems defined in a disc. As an application, the mixed Jacobi–Fourier pseudospectral scheme is provided for the Fisher equation in a disk. The generalized stability and convergence of the proposed scheme are proved. Some numerical results are presented to demonstrate the efficiency of this new algorithm.