Two reliable methods for numerical solution of nonlinear stochastic Itô–Volterra integral equation

Abstract

This article proposes two efficient methods to solve nonlinear stochastic Itô–Volterra integral equations. The shifted Jacobi spectral Galerkin method and shifted Jacobi operational matrix method have been applied to solve these equations. The presented methods convert this equation to a system of nonlinear algebraic equations and then Newton’s method have been implemented to solve the obtained algebraic equations numerically. Convergence analysis for both presented methods have been investigated. Also, the results obtained by proposed methods have been compared. The accuracy and reliability of the presented methods have been proved by some numerical instances.

Keywords:

• Nonlinear stochastic Itô–Volterra integral equations
• shifted Jacobi polynomial
• shifted Jacobi spectral Galerkin method
• shifted Jacobi operational matrix method
• Itô approximation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 60H05
• 60H20
• 45D05