A Space–Time Collocation Method Using the Method of Particular Solutions with Polynomial Basis Functions for Solving the Fisher–KPP Equation

The method of particular solutions (MPS) has been widely applied for solving various types of partial differential equations. In this paper, the space–time collocation technique is implemented using MPS with polynomial basis functions (MPS-PBF) to solve the nonlinear Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation in both one and two dimensions. The Picard iteration method is used to deal with the nonlinearity of the problem. Four numerical examples are provided, and their results are compared with established methods to demonstrate the effectiveness of the proposed scheme.