A local meshless method for the one-dimensional Fisher’s equation

This study presents a novel local meshless approach for solving one-dimensional Fisher’s equation, combining a local scheme, Gaussian radial basis functions (G-RBF), and a collocation technique. The method leverages the Gaussian basis’s nonlinear fitting capability, the sparsity of the local scheme to avoid ill-conditioned matrices, and the simplicity of collocation. After time discretization using a finite difference scheme, the method constructs local approximations at each collocation point using G-RBFs over small subsets of neighboring nodes. Numerical experiments confirm its effectiveness in solving Fisher-type problems, with errors decreasing smoothly as collocation points increase and maintaining stable accuracy over time. The proposed method demonstrates computational efficiency, robustness, and potential for handling large-scale reaction-diffusion systems.