Fractional Runge-Kutta schemes for stochastic fractional equations Conflict of Interest

Abstract

Stochastic fractional differential equations provide a flexible mathematical framework for processes in which hereditary memory and random fluctuations act simultaneously. Such models arise in anomalous diffusion, viscoelasticity, finance, biology and control, but their numerical treatment remains difficult because the Caputo derivative is nonlocal while the stochastic term is driven by nonsmooth Brownian paths. This paper develops a high-order fractional Runge-Kutta scheme for nonlinear stochastic fractional differential equations of Caputo type. The proposed stochastic fractional Runge-Kutta method combines a two-stage fractional drift approximation with an Itô-type stochastic increment and a memory quadrature correction. Under standard global Lipschitz and linear growth assumptions, we establish existence-compatible moment bounds, derive the local truncation estimate, prove mean-square stability for a scalar test equation, and obtain strong convergence in the mean-square norm. The stability result gives an explicit step-size criterion involving the fractional order, drift coefficient and diffusion intensity. Numerical experiments for linear and nonlinear test problems compare the proposed method with fractional Euler and fractional Milstein-type schemes. The results indicate improved error reduction and stable behaviour for fractional orders below one. Tables, convergence plots, stability diagrams and computational-cost graphs are supplied to support the analysis. The paper is written as a publication-ready applied mathematics article with theorem-proof structure, numerical validation and reproducible algorithmic details.