Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation

Abstract

In this paper, the local boundary integral equation (LBIE) method based on generalized moving least squares (GMLS) is proposed for solving extended Fisher---Kolmogorov (EFK) equation. Since the local weak forms are directly approximated from nodal data using a GMLS approximation, this method is called direct local boundary integral equation (DLBIE) method. In addition to DLBIE method, three other meshless local boundary integral equation methods such as LBIE method based on moving least squares, LBIE method based on radial point interpolation, and LBIE method based on moving Kriging are applied to find the numerical solution of EFK equation. A comparison between these methods is done from the perspective of accuracy and computational efficiency. The computational efficiency is the most significant advantage of the DLBIE method in comparison with the other local boundary integral equation methods. DLBIE shifts the numerical integrations over low-degree polynomials instead of over complicated shape functions and this reduces the computational costs, significantly. The main purpose of this paper is to show that the local boundary integral equation methods can be used for solving the fourth-order non-linear partial differential equations especially EFK equation. The numerical results confirm the good efficiency of the proposed methods for solving our model.