Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods

Abstract

The nonlinear Sine-Gordon equation is one of the widely used partial differential equations that appears in various sciences and engineering. The main purpose of writing this article is providing an efficient numerical method for solving two-dimensional (2D) time-fractional stochastic Sine–Gordon equation on non-rectangular domains. In this method, radial basis functions (RBFs) and finite difference scheme are used to calculate the approximate solution of the mentioned problem. The complexity of solving this problem arises from its high dimension, irregular area, stochastic and fractional terms. Finite difference technique is applied to overcome on the problem dimension, whereas interpolation method based on RBFs is the best idea for solving problems defined in irregular domains. The stochastic Sine–Gordon equation is transformed into elliptic stochastic differential equations using the finite difference method and meshfree method based on RBFs are used to approximate the obtained stochastic differential equation. Some numerical examples are included to investigate the efficiency and accuracy of the presented method.