A new hybrid computational method based on Legendre wavelets and modiﬁed hat functions for solving nonlinear stochastic Itó–volterra integral equations

Sharafi, Fatemeh; Basirat, Behrooz; Baloch Shahriary, Mohammad Reza

doi:10.22034/jfmt.2026.586637.1034

A new hybrid computational method based on Legendre wavelets and modiﬁed hat functions for solving nonlinear stochastic Itó–volterra integral equations

Document Type : Original Article

Authors

Fatemeh Sharafi ¹

Behrooz Basirat ²

Mohammad Reza Baloch Shahriary ¹

¹ Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran

² Department of Mathematics, Birjand Branch, Islamic Azad University, Birjand, Iran

10.22034/jfmt.2026.586637.1034

Abstract

This paper develops an eﬃcient hybrid method combining Legendre wavelets and modiﬁed hat functions to solve nonlinear stochastic Itó–Volterra integral equations. By exploiting the global orthogonality of Legendre wavelets and local quadratic interpolation of modiﬁed hat functions, the approach overcomes single-basis limitations. A new stochastic operational matrix for Legendre wavelets is derived from the modiﬁed hat function basis, which is exact for polynomial degrees up to two and an O(h3) approximation otherwise. Its existence, uniqueness, invertibility, and numerical stability are rigorously proven. Mean-square convergence analysis, using Itô isometry and the Burkholder–Davis–Gundy inequality, conﬁrms a third-order rate. Four numerical examples, including a self-convergence test, demonstrate high accuracy and eﬃciency. Comparisons with Wilson wavelet, Legendre wavelet, hat function, and Monte Carlo methods conﬁrm 10−10 mean-square errors with lower cost for mˆ ≥ 96.

Keywords

Stochastic Itó–Volterra integral equation

Legendre wavelets

Modiﬁed hat functions

Stochastic operational matrix

Brownian motion

Mean-square convergence

Subjects