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

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.

Spatial Discretization for Stochastic Semilinear Superdiffusion Driven by Fractionally Integrated Multiplicative Space–Time White Noise

“…Green Function G 1 (t, x, y) and Its Discretized Analogue G M 1 (t, x, y) In this subsection, we shall give the estimates of the Green function G 1 (t, x, y) and its discretized analogue G M 1 (t, x, y), defined in Lemmas 4 and 6, respectively. The proofs are similar to the proofs of ( [22], Lemmas A1-A3). We omit the proofs here.…”

“…We only consider the estimate related to the nonlinear term F(u). The estimate related to the multiplicative noise term g(u) can be obtained by using a similar method as that in the proof of ( [22], Lemma 5).…”

“…We observe that the spatial convergence order decreases as α traverses the range from 1 to 2 due to the white noise in the spatial direction. Throughout this paper, we shall consider the time discretization, and we expect that as α traverses the range from 1 to 2, the time convergence orders will increase as in the subdiffusion case with α ∈ (0, 1) [22].…”

“…The methodology employed in this paper shares similarities with the work of Gyöngy [1], which considered the spatial discretization of (1) with α = 1, and the work of Wang et al [22], which examined the spatial discretization of (1) for the subdiffusion problem with α ∈ (0, 1). However, the error estimates in our paper are more complicated than [1,22] due to the presence of an additional initial value u (0), which requires the studies of the estimates related to the initial value…”