An iterative algorithm for solving two dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation

Abstract

The authors propose a numerical iterative algorithm based on a combination of the successive approximations method and the bilinear spline interpolation. This algorithm is used to obtain an approximate solution of two-dimensional nonlinear stochastic Ito^-Volterra integral equation. In fact, this algorithm is an attractive extension of the numerical iterative approach for a class of two-dimensional nonlinear stochastic Itô-Volterra integral equations. To reach this aim, the bilinear spline interpolation, Gauss-Legendre quadrature formulas for double integrals and two dimensional Ito^ approximation are presented. The effectiveness of the method is shown for three examples. The obtained results and the convergence analysis theorems reveal that the suggested algorithm is very efficient and the convergence rate is O(h2).