Numerical approximation of semilinear stochastic subdiffusion driven by fractionally integrated fBm via the Mittag-Leffler Euler scheme

Abstract

This paper introduces a numerical method for solving the stochastic semilinear subdiffusion equation with fractionally integrated additive fractional Brownian motion (fBm). The temporal discretization is carried out using the Mittag-Leffler Euler method, while the spatial variable is approximated by the spectral Galerkin method. We prove that the resulting fully discrete scheme attains the optimal strong convergence rate O ( τ min { α , H + α + γ − 1 } ) for all H ∈ (0, 1), thereby overcoming the 1/2-order restriction reported in [1]. Here, τ denotes the time step size, H ∈ (0, 1) is the Hurst parameter, α ∈ (0, 1) is the order of the Caputo fractional derivative, and γ ∈ [0, 1] is the order of the Riemann-Liouville fractional integral. Numerical experiments are presented to demonstrate the accuracy of the method and to confirm the sharpness of the theoretical convergence results.