Meshless RBF Method for Linear and Nonlinear Sobolev Equations

Subject Areas : Numerical Analysis

Mehran Nemati ¹ , Mahmoud Shafiee ² , Hamideh Ebrahimi ³

1 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
2 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
3 - Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran

Received: 2020-11-07 Accepted : 2020-12-24 Published : 2020-12-01

Keywords: Energy method, Crank-Nicolson method, Radial basis functions (RBFs), Finite differences, Method of lines, Sobolev equations,

Abstract :

Radial Basis Functions are considered as important tools for scattered data interpolation. Collocation procedure is a powerful technique in meshless methods which is developed on the assumption of radial basis functions to solve partial differential equations in high dimensional domains having complex shapes. In this study, a numerical method, implementing the RBF collocation method and finite differences, is employed for solving not only 2-D linear, but also nonlinear Sobolev equations. First order finite differences and Crank-Nicolson method are applied to discretize the temporal part. Using the energy method, it is shown that the applied time-discrete approach is convergent in terms of time variable with order . The spatial parts are approximated by implementation of two-dimensional MQ-RBF interpolation resulting in a linear system of algebraic equations. By solving the linear system, approximate solutions are determined. The proposed scheme is verified by solving different problems and error norms and are computed. Computations accurately demonstrated the efficiency of the suggested method.