Spatial Discretization for Stochastic Semi-Linear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative Space-Time White Noise

Junmei Wang,James Hoult,Yubin Yan

doi:10.3390/math9161917

Abstract

Spatial discretization of the stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative space-time white noise is considered. The nonlinear terms f and σ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the fractionally integrated multiplicative space-time white noise are discretized by using the finite difference methods. Based on the approximations of the Green functions expressed by the Mittag–Leffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under some suitable smoothness assumptions of the initial value.

Highlights

We will consider the spatial discretization of the following stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative spacetime white noise, with 0 < α ≤ 1, 0 ≤ γ ≤ 1 [1,2]
The main aim of this paper is to extend the spatial discretization schemes discussed in Gyöngy [1] and Anton et al [5] for the stochastic quasi-linear parabolic partial differential equations driven by multiplicative space-time white noise to the stochastic subdiffusion equations driven by integrated multiplicative space-time white noise
There are many works for the numerical methods for solving the stochastic parabolic equations driven by additive or multiplicative noises, i.e., the case with α = 1, γ = 0 in (1), see, e.g., in [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] and the references therein. Most of these references are concerned with an interpretation of stochastic partial differential equations in Hilbert spaces and error estimates are provided in the L2((0, 1)) norm

Summary

Introduction

We will consider the spatial discretization of the following stochastic semi-linear subdiffusion equations driven by fractionally integrated multiplicative spacetime white noise, with 0 < α ≤ 1, 0 ≤ γ ≤ 1 [1,2],. There are many works for the numerical methods for solving the stochastic parabolic equations driven by additive or multiplicative noises, i.e., the case with α = 1, γ = 0 in (1), see, e.g., in [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] and the references therein Most of these references are concerned with an interpretation of stochastic partial differential equations in Hilbert spaces and error estimates are provided in the L2((0, 1)) norm (or similar norms). When α = 1, γ = 0, i.e., the stochastic parabolic equation case, we obtain, from Theorem 2, for the sufficiently smooth initial data u0, e.g., u0 ∈ C3([0, 1]), u0(0) = u0(1) = 0, sup E|uM(t, xk) − u(t, xk)|2 ≤ C∆x, k which is consistent with the spatial convergence rate obtained in Theorem 3.1 in [1] for the stochastic parabolic equation driven by space-time white noise.

Continuous Problem

Spatial Regularity of the Spatial Discretization Problem

Spatial Regularity of the Inhomogeneous Spatial Discretization Problem

Error Estimates

Proof of Theorem 2

Proof of Theorem 1