Approximate solution of two dimensional linear and nonlinear stochastic Itô–Volterra integral equations via meshless scheme

In this paper, the authors propose a practical and easy numerical scheme using two-dimensional moving least squares (2D-MLS) and spectral-collocation method to solve two-dimensional linear stochastic Itô–Volterra integral equations (2D-LSIVIEs) and nonlinear stochastic Itô–Volterra integral equations (2D-NSIVIEs). The 2D-NSIVIE is solved that its unknown function is also nonlinear in the stochastic part, and these problems are rarely discussed in the existing literature. By using the proposed scheme, the problem becomes a system of algebraic equations that can be solved by the appropriate method. We presented an error bound to be sure of the convergence and reliability of the method. Finally, the efficiency and the applicability of the present scheme are investigated through several examples. The CPU time reported for these examples confirms the convenience and speed of the proposed method.