Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling

Abstract

In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.