Spectral semi-discretization algorithm for a class of nonlinear parabolic PDEs with applications

Abstract

This manuscript deals with a novel hybrid spectral collocation approach to find the approximate solutions of a class of nonlinear partial differential equations of parabolic type pertaining to various important physical models in mathematical biology. The problem under consideration arises in the study of propagation of gene and transmission of nerve impulses. A second-order time discretization algorithm based on Taylor series expansion is first employed to tackle the underlying nonlinearity of the problem. Then, the spectral collocation approach based on the alternative Laguerre polynomials (with positive coefficients) is adopted for approximation of the resulting semi-discrete ordinary differential equations. The convergence of the spectral technique is established. Three numerical test examples are given to demonstrate the applicability and efficiency of the proposed approach. The computed results are compared with the results obtained by other available methods in order to show the advantage of our method.