Solution of time-fractional stochastic nonlinear sine-Gordon equation via finite difference and meshfree techniques

In this article, we introduce a numerical procedure to solve time-fractional stochastic sine-Gordon equation. The suggested technique is based on finite difference method and radial basis functions interpolation. By using this algorithm, first time-fractional stochastic nonlinear sine-Gordon equation is converted to elliptic stochastic differential equations. Then, the meshfree method based on radial basis functions (RBFs) is used to approximate the obtained equation. In fact, the finite difference method is used to approximate the unknown function in the time direction and generalized Gaussian RBF is applied to estimate the obtained equation in the space direction. The most important advantage of this method is that the noise terms are simulated directly at the collocation points at each time step. By employing this method, the equation decreased to a nonlinear system of algebraic equations which can be solved simply. The obtained results of solving three examples confirm the validity and capability of the proposed solution.

Automated summary

This article introduces a numerical procedure for solving the time-fractional stochastic sine-Gordon equation, utilizing a combination of finite difference method and radial basis functions interpolation. The method converts the equation to elliptic stochastic differential equations and uses a mesh-free approach based on RBFs to approximate the obtained equation. The technique simplifies the equation to a nonlinear system of algebraic equations which can be easily solved, as demonstrated by solving three examples.