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

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.

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. We obtain the error estimates uniformly in space for the proposed finite difference method. The error estimates are based on the bounds of the Green functions and its discrete analogue of (1) as well as the errors between them under some suitable norms. Such Green functions are expressed in terms of the Mittag-Leffler functions involving the parameters 0 < α ≤ 1 and 0 ≤ γ ≤ 1. It is well known that the Mittag-Leffler function E α,β (z), 0 < α ≤ 1, β ∈ R satisfies the following asymptotic properties: Theorem 1.6 in [4], Equation (1.8.28) in [3], with πα 2 < µ < min(π, απ), and which make the numerical analysis of the stochastic subdiffusion Equation (1) much more challenging than the stochastic parabolic equation discussed in [1,5]. To the best of our knowledge, there are no error estimates uniformly in space for the stochastic subdiffusion equations driven by space-time white noise in literature. In this paper, we aim at filling this gap by providing the detailed error estimates based on the error bounds developed in this paper for the Green functions of (1). Let (Ω, F , (F ) t≥0 , P) be a stochastic basis carrying an F t -adapted Brownian sheet W = {W(t, x) : t ≥ 0, x ∈ [0, 1]}. We recall that W is a zero-mean Gaussian random field with covariance [1], p. 3 and [2], E(W(t, x)W(s, y)) = (t ∧ s)(x ∧ y), where E denotes the expectation and t ∧ s := min(s, t) and x ∧ y := min(x, y) for s, t ≥ 0 and x, y ∈ [0, 1].
Under (L), (LG), and the Assumption 1, one may show that the model (1) has a unique solution [2,6] in some suitable spaces.
The model (1) is used to describe the random effects on transport of particles in medium with memory or particles subject to sticking and trapping [6]. The fractional integrated noise reflects the fact that the internal energy depends also on the past random effects. In recent years, the model (1) has been very actively researched [6,[8][9][10][11]. Chen et al. [6] studied the L 2 theory of (1) in both divergence and non-divergence forms. Anh et al. [8] discussed sufficient conditions for a Gaussian solution (in the mean-square sense) and derived temporal, spatial, and spatial-temporal Hölder continuity of the solution. Chen [9] analyzed moments, Hölder continuity and intermittency of the solution for the nonlinear stochastic subdiffusion in one-dimensional case. Liu et al. [11] analyzed the existence and uniqueness of the solution of the model (1) with fairly general quasi-linear elliptic operators.
Let us review some numerical methods for solving (1). Jin et al. [7] considered a fully discrete scheme for approximating (1) with f = 0 and σ(u) = 1 and the space-time noise is the Hilbert space-valued Wiener process with covariance operator Q and the error estimates in the L p , p > 1 norm in space is obtained. Wu et al. [12] introduced the L1 scheme to approximate (1) with f = 0 and σ(u) = 1 and the space-time noise is defined as in Jin et al. [7]. Gunzburger et al. [13] considered the finite element approximation of stochastic partial-differential equations driven by white noise. Li et al. [14] studied the finite element method for stochastic space-time fractional wave equations. Li and Yang [15] considered the finite element method for solving stochastic time fractional partial differential equations. Zou [16] investigated the finite element method for solving stochastic time fractional heat equation.
To the best of our knowledge, we did not find any numerical analysis for solving (1) in the multiplicative (i.e., σ(u) = 1) space-time white noise case in literature. In this paper, we will approximate the derivative ∂ 2 u(t,x) ∂x 2 and the space-time white noise ∂ 2 W(t,x) ∂t∂x with the finite difference methods as in Gyöngy [1] and Anton et al. [5] and obtain a spatial discretization scheme for approximating (1). The convergence rate in the mean-square sense is obtained, uniformly for x ∈ [0, 1].
(i) If f = 0, then there exists a constant C which is independent of t > 0 and the space step size ∆x, such that, with t > 0, (ii) If f = 0, then there exists a constant C which is independent of t > 0 and the space step size ∆x, such that, with t > 0, where and and Remark 2. When α = 1, γ = 0, i.e., the stochastic parabolic equation case, we obtain that, from (8)- (11) in Theorem 1, for the initial data which is consistent with the spatial convergence rate obtained in Theorem 3.1 in [1]. Actually the smoothness assumption of u 0 in this case can be weakened to u 0 ∈ C β ([0, 1]) with β ∈ (0, 1 2 ) and u 0 (0) = u 0 (1) = 0, see Remarks 1 and 9.
Remark 3. We may consider the error estimates with respect to the norm sup k E|u M (t, x k ) − u(t, x k )| 2p for any p ≥ 1 as in Theorem 3.1 in [1]. For simplicity of the notations, we only consider the case with p = 1 in Theorem 1.
(i) If f = 0, then there exists a constant C which is independent of t and the space step size ∆x, such that, for where r 2 and r 3 are defined by (10) and (11), respectively. (ii) If f = 0, then there exists a constant C which is independent of t and the space step size ∆x, such that, for 2 α < 3, that is, 2/3 < α ≤ 1, for all t ∈ [0, T] and M ≥ 1, where r 3 is defined by (11).

Remark 5.
When α = 1, γ = 0, i.e., the stochastic parabolic equation case, we obtain, from Theorem 2, for the sufficiently smooth initial data u 0 , e.g., 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.

Remark 6.
We may consider the error estimates with respect to the norm sup k E|u M (t, x k ) − u(t, x k )| 2p for any p ≥ 1 as in Theorem 3.1 in [1]. For simplicity of the notations of the proof, we only consider the case with p = 1 in Theorem 2.

Remark 7.
We may also consider the case where the nonlinear terms f and g are not Lipschitz continuous as in Gyöngy Section 4 in [1] under the following assumptions: (E). There is a solution u M of (4) for every M ≥ 1.
As we are mainly interested in the error estimates of the stochastic subdiffusion problem (1), for the sake of paper length, we only consider the cases where the nonlinear terms f , σ satisfy the globally Lipschitz conditions and linear growth conditions in this paper.
The paper is organized as follows. In Section 2, we consider the continuous problem (1). We obtain the mild solution of the problem and the spatial regularity of the mild solution. Section 3 is devoted to the spatial discretization of the problem (1). The regularity of the solution of the spatial discretization problem is obtained. In Section 4, we consider the error estimates under the different assumptions for the smoothness of the initial value. Finally, in Appendix A, we consider the bounds and the error estimates of the approximations of the Green functions of (1).
Throughout this paper, we denote by C a generic constant depending on u, u 0 , T, α, γ, but independent of t > 0 and the space step size ∆x, which could be different at different occurrences. Further, > 0 is always a small positive number.

Continuous Problem
In this section, we shall consider the mild solution of (1) and study its spatial regularity. Let {λ j , ϕ j } ∞ j=1 be the eigenpairs of the Laplacian operator It is well known that {ϕ j (x)} ∞ j=1 forms an orthonormal basis in H = L 2 (0, 1). Let E α,β (z), 0 < α ≤ 1, β ∈ R denote the Mittag-Leffler function defined by [4] We have the following differentiation formulas of Mittag-Leffler functions which we shall use frequently in the error estimates of the Green functions in the Appendix A.
2.1. The Mild Solution of (1) In this subsection, we shall give the mild solution of (1).

Proof.
One may prove this lemma by the method of separation of variables. Assume that the solution u(t, x) has the form substituting this form into (1), one may easily obtain the mild solution (16). We omit the details here.
2.2. The Spatial Regularity of the Mild Solution of (1) In this subsection, we shall consider the spatial regularity of the mild solution of (1). To do this, we write the mild solution of (1) into where v(t, x) satisfies the following homogeneous problem with nonzero initial value u 0 , which has the solution and w(t, x) satisfies the following inhomogeneous problem with zero initial value, which has the solution Here, G 1 , G 2 , G 3 are defined by (17), (18), (19), respectively. Let 0 = y 0 < y 1 < · · · < y M−1 < y M = 1 be a partition of [0, 1] and ∆x = 1/M be the space step size. We define the piecewise constant function k M (y), 0 ≤ y ≤ 1 by 2.2.1. The Spatial Regularity of the Homogeneous Problem (20) with the Initial Data In this subsection, we shall consider the spatial regularity of the homogeneous problem (20) with u 0 ∈ C([0, 1]), u 0 (0) = u 0 (1) = 0. We have the following lemma.

Lemma 3.
Let v(t, x) be the solution of the homogeneous problem (20).
Then, there exists a constant C which is independent of t and the space step size ∆x, such that where r 1 is denoted by (9).
Proof. Note that, by Cauchy-Schwarz inequality, where r 1 is defined by (9), which completes the proof of Lemma 3.

The Spatial Regularity of the Homogeneous Problem (20) with the Initial Data
In this subsection, we shall consider the spatial regularity of the homogeneous problem (20) We have the following lemma.

Lemma 4.
Let v(t, x) be the solution of the homogeneous problem (20).
Then, there exists a constant C which is independent of t and the space step size ∆x, such that where r 2 is defined by (10).
where G 2 is defined by (18). Thus, by using Cauchy-Schwarz inequality, with y k ≤ y ≤ y k+1 , k = 0, 1, . . . , M − 1, By Lemma A4 and using the error estimates of the liner interpolation function, we obtain , which completes the proof of Lemma 4. (22) In this subsection, we shall consider the spatial regularity of the inhomogeneous problem (22). We have the following lemma.

The Spatial Regularity of the Inhomogeneous Problem
Lemma 5. Assume (L), (LG) and Assumption 1 hold. Let w(t, x) be the solution of the inhomogeneous problem (22). Then there exists a constant C which is independent of t and the space step size ∆x, such that where r 2 and r 3 are defined by (10), (11), respectively.

Proof.
Denote h(s, z) = f (u(s, z)) or σ(u(s, z)). One may easily prove (we omit the proof here due to the length of the paper) that, under the assumptions (L) and (LG), Denote where G2 and G3 are defined by (18) and (19), respectively. We will show that where r 2 and r 3 are defined by (10) and (11), respectively. We only prove (29) here since the proof of (28) is similar. By Burkholder's inequality [1], p. 9 and the boundedness of h in (27), we have where r 3 is defined by (11).
The proof of Lemma 5 is complete.

Spatial Discretization
In this section, we shall consider the spatial discretization of (1). (4) Let {λ M j , ϕ M j } M−1 j=1 be the eigenpairs of the following discrete Laplacian matrix A defined by

The Mild Solution of the Spatial Discretization Problem
It is well known that (2.4) in page 4 in [1] and ϕ M j , j = 1, 2, . . . , M − 1 forms an orthonormal basis in R M−1 .
Proof. We write (4) into the following matrix form: where The solution of (37) can then be written into the following integration form: Replacing ϕ j (x k ) by the piecewise linear interpolation function ϕ M j (x) and writing the summation terms ∑ M l=1 into the integral forms in (38), we then obtain the following piecewise linear interpolation function of u M (t, x k ), k = 0, 1, 2, . . . , M, which shows (32), where k M (y) and ∂ 2 W M (s,y) ∂s∂y are defined by (24) and (36), respectively. The proof of Lemma 6 is now complete.

Spatial Regularity of the Spatial Discretization Problem
In this subsection, we shall consider the regularity of the mild solution (32) of the spatial discretization problem. To do this, we write the solution where v M (t, x) is the solution of the corresponding homogeneous problem defined by and w M (t, x) is the solution of the corresponding inhomogeneous problem defined by 3.2.1. Spatial Regularity of the Homogeneous Spatial Discretization Problem with the Initial Data In this subsection, we shall consider the spatial regularity of the homogeneous spatial discretization problem with the initial data u 0 ∈ C([0, 1]), u 0 (0) = u 0 (1) = 0. We have the following lemma.
Then there exists a constant C which is independent of t and the space step size ∆x, such that where r 1 is defined by (9).
Proof. Note that, by Cauchy-Schwarz inequality, which completes the proof of Lemma 7.
Then, there exists a constant C which is independent of t and the space step size ∆x, such that where r 2 is defined by (10).
For the first term of the last equality in (42), we have, with k = 0, 1, 2, . . . , M, and noting that ϕ M j (y) is the piecewise linear interpolation function of ϕ j (y) on y j , j = 0, 1, . . . , M, Therefore, ∑ M−1 j=1 1 0 ϕ M j (y)ϕ j (k M (z))u 0 (k M (z)) dz is the piecewise linear interpolation function of u 0 (y k ), k = 0, 1, 2, . . . , M and we denote Further, we assume that the following equality holds at the moment: which we shall prove later. We then get where G M 2 is defined by (34). By using the Cauchy-Schwarz inequality, we have By Lemma A4, and using the error estimates of the liner interpolation function and mean-value theorem, we obtain It remains to prove (43). In fact, we have, noting that ϕ j (y 0 ) = 0, where we use the fact u 0 (y 0 ) = u 0 (y M ) = 0 in the last equality in (45). Hence (43) holds. The proof of Lemma 8 is now complete.

Spatial Regularity of the Inhomogeneous Spatial Discretization Problem
In this subsection, we shall consider the spatial regularity of the inhomogeneous spatial discretization problem. Following the same lines of the proof of Lemma 5, we may prove the following.

Error Estimates
In this section, we will consider the error estimates of u M (t, x) for approximating u(t, x) under the suitable smoothness assumptions of the initial value u 0 . We need the following Grönwall Lemma Lemma 3.4 in [1]. Lemma 10. Let z : R + → R + be a Borel function satisfying for all t ∈ [0, T] the inequality with some constants a ≥ 0, K > 0 and σ > −1. Then, there exists a constant C = C(σ, K, T) such that z(t) ≤ aC for all t ∈ [0, T].

Proof of Theorem 2
In this subsection, we shall prove Theorem 2 where the initial data We first consider the case (i), that is, f = 0. We divide the proof into two steps.
Step 1. We consider the approximation of the homogeneous problem of (1). Recall that the solution of the homogeneous problem of (1) has the form, by (21), The approximate solution of the homogeneous problem of (1) has the form, by (39), By (26) where I h u 0 (x) is the piecewise linear interpolation function of u 0 (x k ), k = 0, 1, 2, . . . , M. Therefore, one gets For I 1 , using the error estimates of the linear interpolation function, one gets For I 2 , we obtain, by Cauchy-Schwartz inequality, Thus, by Lemma A9, , where r 2 is defined by (10).
Step 2. We now consider the approximation of the inhomogeneous problem of (1). Recall that, by (23), the solution of the inhomogeneous problem of (1) has the form, as f = 0, and, by (40), the approximate solution of the inhomogeneous problem of (1) has the form Thus, we have For I 1 , we have By Burkholder inequality [1], p. 9, Proof of Proposition 3.5, one gets By the Assumptions (L) and (LG), we have, using the boundedness of the solution u M , Therefore, by Lemmas A4 and A6, where r 3 is defined by (11). For I 2 , we have By (36), we have, for y k ≤ȳ ≤ y k+1 , k = 0, 1, 2, . . . , M − 1, Thus, we get Note that, for y k ≤ y,ȳ ≤ y k+1 , k = 0, 1, 2, . . . , M − 1, which implies that Thus, we get By the spatial regularity Lemmas 8 and 9 and the error estimate (47) for E|v M (s, y) − v(s, y)| 2 , we obtain E|w M (s, y) − w(s, y)| 2 ds.
By using Grönwall Lemma 10, we get, if 2(α + γ − 1) − α 2 > −1, i.e., 2(1−2γ) We now consider the case (ii), that is, f = 0. In this case, the approximation to the solution of the homogeneous problem of (1) is the same as the case (i). For the inhomogeneous problem of (1), the solution has the form The approximate solution of the inhomogeneous problem of (1) has the form Following the same arguments as in Step 2 above, we may get, E|w M (s, y) − w(s, y)| 2 ds, By using Grönwall Lemma 10, we get, if 2(α − 1) − α 2 > −1, i.e., 2 α < 3, Together this with (47) shows (ii). The proof of Theorem 2 is now complete.
We first consider the case (i), that is, f = 0. We divide the proof into two steps.
Step 1. We consider the approximation of the homogeneous problem of (1). By Cauchy-Schwarz inequality, we have Therefore, by mean-value theorem, Further, applying Lemmas A1 and A3, one obtains where r 1 is defined by (9).
Step 2. We now consider the approximation of the inhomogeneous problem of (1). Following the proof of (50), we get where r 3 is defined by (11).
By Grönwall Lemma 10, we get where r 1 and r 3 are defined by (9) and (11), respectively.
By Grönwall Lemma 10, we get Together with these estimates, we obtain where r 1 and r 3 are defined by (9) and (11), respectively. We now consider the case (ii), that is, f = 0. In this case, the approximation of the solution for the homogeneous problem of (1) is the same as in the case (i). For the inhomogeneous problem of (1), we have, following the same arguments as in Step 2, where J 3 (t) and J 4 (t) are defined as in (55) as v(s, y) and v M (s, y) are the same as in the case (i).

Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A.1. Green Function G 1 (t, x, y) and Its Approximation G M 1 (t, x, y) In this subsection, we will consider the bounds of G 1 (t, x, y), G M 1 (t, x, y), as well as the error bounds of G 1 (t, x, y) − G M 1 (t, x, y) in some suitable norms.
For (A4), we get, by (A3), Together these estimates complete the proof of Lemma A1.
Together these estimates complete the proof of Lemma A3.
Appendix A.2. Green Function G 3 (t, x, y) and Its Approximation G M 3 (t, x, y) In this subsection, we will consider the bounds of G 3 (t, x, y) and its approximation G M 3 (t, x, y) and the error bounds of G 3 (t, x, y) − G M 3 (t, x, y) in some suitable norms.