Operational Matrix Method for Variable-Order Fractional Stochastic Systems

Introduction: Variable-order fractional stochastic integro-differential equations (VO-FSIDEs) have attracted increasing attention due to their capability to model complex dynamical systems involving memory effects, hereditary properties, and stochastic perturbations. Such equations arise naturally in a wide range of applications, including anomalous diffusion, viscoelastic materials, population dynamics, and financial mathematics, where uncertainty and time-dependent fractional behavior play a significant role.

Method: In this study, an efficient numerical framework based on the operational matrix method is presented for solving VO-FSIDE . The unknown stochastic solution is approximated using Chelyshkov, Chebyshev, and Euler polynomial expansions. Operational matrices corresponding to the variable-order Caputo fractional differential operator are constructed for each polynomial basis. By employing Newton–Cotes nodes as collocation points, the proposed scheme transforms the original variable-order fractional stochastic integro-differential equations into a system of algebraic equations, which can be solved efficiently using standard numerical techniques such as the Gauss elimination method and Newton’s iterative method.

Results: A comparative numerical study is conducted to evaluate the performance of the proposed polynomial-based methods in terms of accuracy, convergence rate, and computational efficiency. Several numerical examples, including problems with known exact solutions, are presented to validate the proposed methodology. The numerical results demonstrate excellent agreement with the exact solutions and reveal that the Chelyshkov polynomial-based operational matrix method achieves higher accuracy with fewer basis functions compared to the Chebyshev and Euler polynomial approaches.

Conclusions: The proposed operational matrix framework provides a reliable, accurate, and computationally efficient tool for solving VO-FSIDEs. The comparative results highlight the effectiveness of the Chelyshkov polynomial approximation for capturing variable-order fractional dynamics, making the proposed approach well suited for practical applications in science and engineering.