A Robust Least-Squares Scheme Based on Orthonormal Bernstein Operational Matrices for Stochastic Itô-Volterra Integral Equations with Abel Kernels

Authors

  • Atena Ghiasi Department of Mathematics, Lorestan University
  • Bahman Ghazanfari Lorestan University

Abstract

This paper presents a robust and computationally efficient numerical framework for solving a class of linear stochastic It\^{o}-Volterra integral equations with Abel-type weakly singular kernels. The primary challenge, arising from the dual complexity of kernel singularity and the non-differentiable nature of Brownian motion, is addressed through a least-squares minimization strategy in a finite-dimensional space spanned by orthonormal Bernstein polynomials (OBPs). A key innovation is the construction of an exact operational matrix for the singular integral term, which facilitates an analytical treatment of the singularity and bypasses the instabilities associated with numerical quadrature. The utilization of an orthonormal basis ensures a well-conditioned algebraic system, significantly enhancing numerical stability even for high-degree approximations. Convergence of the proposed scheme is rigorously established in the mean-square sense. Numerical experiments validate the theoretical findings, demonstrating that the method achieves high precision with absolute errors of order $10^{-4}$ using as few as 4 to 8 basis functions. Comparative analysis confirms that the OBP-based framework outperforms traditional non-orthogonal schemes in terms of both accuracy and robustness.

Author Biography

Bahman Ghazanfari, Lorestan University

 

Associate Professor of Applied Mathematic, Department of Mathematics, Faculty of science, Lorestan University, P.C.: 68137-17133, Khorramabad, Iran

Published

2026-08-24

Issue

Vol. 20

Section

Vol. 20, No. 6, (2026)

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