Numerical solution of stochastic integral equations by using Bernoulli operational matrix

Abstract

In this paper, a new computational method based on stochastic operational matrix for integration of Bernoulli polynomials is proposed for solving nonlinear Volterra–Fredholm–Hammerstein stochastic integral equations. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra–Fredholm–Hammerstein stochastic integral equation is reduced to nonlinear system of algebraic equations with unknown Bernoulli coefficients. This work is inspired by Bazm (2015), where the authors study the deterministic integral equations. In order to show the rate of convergence of the suggested approach, we present theorems on convergence analysis and error estimation. Some illustrative error estimations and examples are provided and included to demonstrate applicability and accuracy of the technique.