Time discretization schemes for stochastic subdiffusion and fractional wave equations with integrated additive noise

Abstract

In this paper, we introduce a time discretization scheme for solving the stochastic subdiffusion equation based on the two-fold integral-differential and two step backward differentiation formula (ID2-BDF2). We prove that this scheme attains a convergence rate of O ( τ α + γ − 1 / 2 ) for 1 / 2 < α + γ < 2 with α ∈ (0, 1) and γ ∈ [0, 1]. Our approach regularizes the additive noise through a two-fold integral-differential (ID2) calculus and discretizes the equation using BDF2 convolution quadrature, achieving superlinear convergence in solving the stochastic subdiffusion. Furthermore, we extend the scheme to solve the stochastic fractional wave equation, proving that the scheme achieves a convergence rate of O ( τ min { 2 , α + γ − 1 / 2 } ) for α ∈ (1, 2) and γ ∈ [0, 1]. Numerical examples are presented to validate the theoretical results for the linear problem. The numerical observations further indicate that the same convergence rates also apply to stochastic semilinear time-fractional equations.