A hybrid radial basis function–finite difference method for solving two-dimensional coupled nonlinear stochastic fractional sine–Gordon equations arising in telecommunications network

Abstract

In this paper, an efficient numerical technique based on the finite difference idea and radial basis functions (RBFs) is developed for approximating the solutions of two-dimensional stochastic fractional sine–Gordon equations over non-rectangular computational domains. In this technique, the Caputo fractional derivative is reformulated in an integral form, and the stochastic forcing is treated through a fractional Itô integral. The deterministic fractional integrals are approximated using a finite difference quadrature, while the stochastic fractional integrals are discretized by an Euler–Maruyama type approximation based on the Brownian increments. For the spatial discretization, RBF based estimation is employed to construct a meshfree approximation of the second derivative with respect to the spatial variable. The resulting fully discrete formulation leads, at each time level, to linear algebraic systems for the unknown RBF coefficients, which are solved directly. Moreover, the mean-square convergence of the proposed approximation for the stochastic fractional integrals is established using the Itô isometry property. Numerical experiments on non-rectangular computational domains are presented to investigate the accuracy and convergence behavior of the proposed method for different fractional orders and spatial discretizations. The numerical results demonstrate that the proposed hybrid RBF–finite difference approach provides accurate approximations for the considered coupled stochastic fractional sine–Gordon equations.