On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions

Abstract

The main intention of the present work is to develop a numerical scheme based on radial basis functions (RBFs) to solve fractional stochastic integro-differential equations. In this paper, the solution of fractional stochastic integro-differential equation is approximated by using strictly positive definite RBFs such as Gaussian and strictly conditionally positive definite RBFs such as thin plate spline. Then, the quadrature methods are used to approximate the integrals which are appeared in this scheme. When we use thin plate spline to approximate the solution of mentioned equation, we encounter logarithm-like singular integrals which cannot be computed by common quadrature formula. To overcome this difficulty, we introduce the non-uniform composite Gauss–Legendre integration rule and employ it to estimate the singular logarithm integral appeared in this case. This method transforms the solution of linear fractional stochastic integro-differential equations to the solution of linear system of algebraic equations which can be easily solved. We also discuss the error analysis of the proposed method and demonstrate that the rate of convergence of this approach is arbitrary high for infinitely smooth RBFs. Finally, the efficiency and accuracy of the proposed method are checked by some numerical examples.