A Numerical Method Based on Cubic Hat Functions to Solve a Class of Fractional Stochastic Integro-Differential Equations Driven by Multi-Dimensional Fractional Brownian Motion

Zarei, Zahra; Babaei, Afshin; Nemati, Somayeh; Ranjbar, Hassan

doi:10.22055/jamm.2026.50122.2383

A Numerical Method Based on Cubic Hat Functions to Solve a Class of Fractional Stochastic Integro-Differential Equations Driven by Multi-Dimensional Fractional Brownian Motion

Document Type : Original Paper

Authors

Zahra Zarei

Afshin Babaei

Somayeh Nemati

Hassan Ranjbar

‎Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran‎.

10.22055/jamm.2026.50122.2383

Abstract

‎In this paper‎, ‎a numerical method based on cubic hat functions is presented for solving a class of fractional stochastic integro-differential equations driven by multidimensional fractional Brownian motion with Hurst parameter‎ ‎$ H \in \left(\frac{1}{2}‎ , ‎1\right) $‎.‎The fractional derivative is considered in the Caputo sense‎. ‎First‎, ‎an approximation of the Caputo derivative of the unknown function with unknown coefficients is constructed based on the basis functions‎. ‎Then‎, ‎using this approximation and the properties of the Caputo derivative and the Riemann-Liouville integral‎, ‎approximations for the solution and other functions appearing in the equation are introduced‎. ‎By substituting these approximations into the equation and using suitable nodal points along with a backward approximation for simulating the fractional Gaussian noise‎, ‎the original equation is transformed into a system of algebraic equations‎. ‎Due to the properties of the cubic hat functions and their operational matrix of integration‎, ‎the resulting system has low computational complexity‎. ‎Solving this system yields an approximation of the solution to the original problem‎. ‎Furthermore‎, ‎the error analysis of the proposed method is investigated‎. ‎Finally‎, ‎several numerical examples are provided to confirm the efficiency of the proposed method‎.

Keywords

‎Fractional stochastic integro-differential equations‎

‎Multi-Dimensional fractional Brownian motion‎

‎Cubic hat functions‎

‎Operational matrix

Subjects