Numerical solution of stochastic mixed Volterra-Fredholm integral equations driven by space-time Brownian motion via two-dimensional block pulse functions

Abstract

The goal of this paper is to give a useful method for solving problems formulated by two dimensional stochastic integral equations driven by space-time white noise. Here, we consider two dimensional block pulse functions (2-D BPFs) and their operational matrix and stochastic operational matrix of integration. Since solution of parabolic partial differential equations driven by space-time white noise leads to a stochastic integral equations, we introduced a new method by 2-D BPFs and their operational integration matrix to transform a stochastic mixed Volterra-Fredholm integral equation to a system of algebraic equations. The benefit of this method is lower cost of setting up the system of equations without any integration. So, the computational cost of operations is low. The method is applied to two test examples to illustrate the accuracy and implementation of the method.