Shifted Chebyshev spectral Galerkin method to solve stochastic Itô–Volterra integral equations driven by fractional Brownian motion appearing in mathematical physics

The main aim of this article is to provide a spectral Galerkin method based on the shifted Chebyshev polynomial of the first kind to solve stochastic Itô–Volterra integral equations driven by fractional Brownian motion. The presented method uses the Gauss–Legendre quadrature rule and Itô approximation to reduce stochastic Itô–Volterra integral equations driven by fractional Brownian motion into the system of algebraic equations. Then Newton’s method is used to solve them numerically. Also, convergence and error analysis of the discussed scheme has been investigated in the Sobolev space. A few illustrative examples are discussed to show the applicability and reliability of the proposed method. Finally, the numerical results of the spectral Galerkin technique based on the shifted Chebyshev polynomial and shifted Chebyshev cardinal functions are compared to the actual solution.