Zhang et al. Advances in Difference Equations (2018) 2018:215 https://doi.org/10.1186/s13662-018-1665-0

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Error estimates of finite element methods for nonlinear fractional stochastic differential equations Yanpeng Zhang1 , Xiaoyuan Yang1 and Xiaocui Li2* *

Correspondence: [email protected] 2 School of Science, Beijing University of Chemical Technology, Beijing, P.R. China Full list of author information is available at the end of the article

Abstract In this paper, we consider the Galerkin finite element approximations of the initial value problem for the nonlinear fractional stochastic partial differential equations with multiplicative noise. We study a spatial semidiscrete scheme with the standard Galerkin finite element method and a fully discrete scheme based on the Goreno–Mainardi–Moretti–Paradisi (GMMP) scheme. We establish strong convergence error estimates for both semidiscrete and fully discrete schemes. MSC: 60H15; 65M60; 60H35; 65M12 Keywords: Nonlinear fractional stochastic differential equations; Finite element method; Error estimates; Strong convergence; Initial value problem

1 Introduction In the last few years, fractional calculus has attracted lots of attention. The increasing interest in fractional equations is motivated by their applications in various fields of science such as fluid mechanics, heat conduction in materials with memory, physics, chemistry, and engineering [1–5]. As we know, fractional differential equations are highly effective mathematical tools to describe complex behaviors and phenomena of memory processes because of the convolution integral with the power-law memory kernel introduced in the fractional derivatives [6–8]. On the other hand, stochastic perturbations cannot be avoided in physical systems and sometimes even cannot be ignored, so that the corresponding stochastic terms need to be added to the deterministic governing equations. Hence stochastic differential equations with fractional time derivatives have been proposed, which are a more realistic mathematical model of the real-world situations [9], just like the equations (1.1) we are going to discuss in this paper naturally arise from the consideration of the heat equation in a material with thermal memory [10]. In this paper, we consider the following initial value problem for the nonlinear fractional stochastic partial differential equation (SPDE) with multiplicative noise: ⎧ ⎨Dα u(t) + Au(t) = f (u(t)) + g(u(t)) dW (t) , t

⎩u(0) = u0 .

dt

α ∈ (0, 1), t ∈ [0, T],

(1.1)

© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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The random process {u(t)}t∈[0,T] , defined on a filtered probability space (, F , P, {Ft }t≥0 ) with normal filtration {Ft }t≥0 , takes values in a separable Hilbert space H with inner product (·, ·) and norm  · . The initial value u0 is an H-valued and F0 -measurable random variable. The operator A : D(A) ⊂ H → H is not necessarily a bounded, linear, densely defined, and selfadjoint operator with compact inverse. The nonlinear operators f and g are Lipschitz continuous in an appropriate sense. The process W with values in some separable Hilbert space U is a nuclear Q-Wiener process with respect to the filtration. The covariance operator Q is assumed to be selfadjoint and positive semidefinite with finite trace. Here, we denote the Caputo fractional derivative of order α (0 < α < 1) with respect to t by Dαt and define it as [11, 12] Dαt u(t) =

1 (1 – α)



t

(t – s)–α 0

d u(s) ds. ds

It is known that the fractional derivative Dαt recovers the canonical first-order derivative dtd u(t) for the fractional order α = 1, and thus model (1.1) evolves into the standard stochastic partial differential equation (SPDE), whose numerical approximation has been extensively discussed in the literature; see, for example, [13–16]. Stochastic partial differential equations have been applied in many fields such as viscoelasticity, turbulence, electromagnetic theory, heterogeneous flows, and materials [17– 22], so the study of stochastic partial differential equations has recently attracted a lot of attention. In particular, as in [10, 23–26], equations of type (1.1) can be used to model random effects on transport of particles in medium with thermal memory. In [10], a class of SPDEs with time-fractional derivatives was introduced, and the existence and uniqueness of solutions to these equations was proved. The existence of mild solutions for a class of nonlinear fractional stochastic partial differential equations has been discussed in [24]. Foondun and Nane [23] studied asymptotic properties of space–time fractional SPDEs. In [25], the existence and uniqueness of mild solutions for a class of nonlinear fractional Sobolev-type stochastic differential equations under non-Lipschitz conditions was discussed by employing Picard-type approximate sequences. The approximate controllability problem for fractional stochastic differential inclusions with nonlocal conditions and infinite delay has been researched in [26]. Since the random effects on transport of particles in medium with thermal memory can be exactly modeled by fractional stochastic differential systems, it is important and necessary to discuss numerical schemes and error estimation for stochastic fractional equations. However, numerical methods for these kinds of fractional SPDEs are rarely studied, and we only note [27–30]. To the authors’ knowledge, no result has been reported on the error estimation of nonlinear fractional stochastic partial differential equations with multiplicative noise based on the form of mild solutions proposed in [24], so the motivation of this paper is to fill this gap. The main difficulty in the analysis is estimation of nonlinear terms; see Lemmas 3.6 and 3.7. Estimation of a discrete solution operator with limited smoothing properties is also a challenge; see Lemma 4.3. Our main results are as follows. First, in Theorem 3.1, denoting by uh (t) and u(t) the mild solutions to (3.2) and (1.1), we derive a strong convergence error bound for the semidiscrete scheme:   u(t) – uh (t) L

2(,H)

≤ Ch2 .

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Second, for α ∈ (0, 1), we obtain am L2(,H) -norm error estimate for the fully discrete scheme in Theorem 4.1:   u(tn ) – un  h L

2(;H)

  ≤ C k α + h2 ,

where unh denotes an approximation of the mild solution u(t) at time tn . The parameters h and k, which will be detailed in Sects. 3 and 4, represent the maximal meshsize and time step, respectively. The rest of the paper is organized as follows: In Sect. 2, we introduce some basic notation, present the Laplace transform, and give a representation of the mild solution of equation (1.1) by using basic properties of the Mittag–Leffler function. In Sect. 3, we first give a short review of Galerkin finite element methods and then study the space semidiscrete scheme and derive error estimates for the standard Galerkin finite element method with smooth initial data. Finally, in Sect. 4, using the GMMP scheme, we prove strong error estimates for the fully discrete scheme.

2 Preliminaries In this section, we recall some useful properties on the Mittag–Leffler function, introduce the Laplace transform and present a representation of the mild solution of problem (1.1). Besides, we use the letter C to denote a constant that may vary from one occurrence to another and denote by L(U, H) the space of bounded linear operators from U to H, where U and H are real separable Hilbert spaces with inner product (·, ·) and norms  · U and  · H . 2.1 Mittag–Leffler function The Mittag–Leffler function is defined by Eα,β (z) =

∞ k=0

zk , (kα + β)

z ∈ C,

where (·) is the standard gamma function 

∞

(z) =

t z–1 e–t dt,

R(z) > 0.

0

We give important properties of the Mittag–Leffler function Eα,β (z) essential in our analysis. Lemma 2.1 ([31]) Let 0 < α < 2 and β ∈ R be arbitrary, and let π2α < μ < min(π, απ). Then there exists a constant C = C(α, β, μ) > 0 such that, for μ ≤ | arg(z)| ≤ π , ⎧ C

⎨ 1+|z| β – α ∈ Z– ∪ {0}, 2,

Eα,β (z) ≤ ⎩ C otherwise. 1+|z| Moreover, for λ > 0, α > 0, and t > 0, we have   Dαt Eα,1 –λt α = –λEα,1 –λt α

and

  d Eα,1 –λt α = –λt α–1 Eα,α –λt α . dt

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In our analysis, we also use the Laplace transform. Let π : R+ → H be subexponential, that is, for any ε > 0, the function t → π(t)e–εt belongs to L1 (R+ , H). The Laplace transform of πˆ : C+ → H is denoted by 

+∞

πˆ (z) =

π(t)e–zt dt,

R(z) > 0,

0

where the same notation H represents the complexification of H. Further, we denote by ∗ the Laplace convolution product on [0, t] of two locally integrable subexponential functions π, σ ∈ L1loc (R+ , H), that is,  (π ∗ σ )(t) =

t

π(t – s)σ (s) ds. 0

It is well known that π ∗ σ ∈ L1loc (R+ , H) is subexponential and π ∗ σ = πˆ (z)σˆ (z).

2.2 Solution representation In order to study the representation of the solution of (1.1), we introduce some notation. Let (, F , P) be a probability space. By L2 (, H) we denote the space of H-valued squareintegrable random variables with norm 1  vL2(,H) = Ev2H 2 =



  v(w)2 P(w) H



 12 ,

where E stands for expected value. Let Q ∈ L(U) be a selfadjoint positive semidefinite operator with Tr(Q) < ∞, where Tr(Q) is the trace of Q. Let {(γj , ej )}∞ j=1 be the eigenpairs of Q with orthonormal eigenvectors. The U-valued Q-Wiener process W (t), defined on the probability space (, F , P), has the orthogonal expansion W (t) =

∞

γj1/2 βj (t)ej ,

j=1

where {βj (t)}∞ j=1 are real-valued mutually independent standard Brownian motions. Further, the set L02 = HS(Q1/2 (H), H) expresses the space of all Hilbert–Schmidt operators  0 0 1/2 2 1/2 from Q1/2 (H) to H with norm ψL0 = ( ∞ j=1 ψQ ej  ) , and the subset L2,r ⊂ L2 , r ≥ 0 2 ˙ r with norm ψL0 = is the subspace of all Hilbert–Schmidt operators from Q1/2 (H) to H 2,r t r A 2 ψL0 . It is then possible to define the stochastic integral 0 ψ(s) dW (s) together with 2 Itô’s isometry 2  t  t   2   Eψ(s)L0 ds. E ψ(s) dW (s)  = 2 0

H

(2.1)

0

In a standard way, we present the fractional powers As , s ∈ R, of A as s

A v=

∞ j=1

λsj (v, ϕj )ϕj ,



D A

s 2





 ∞  s 2 s 2   = v ∈ H : A2 v = λj (v, ϕj ) < ∞ , j=1

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∞ where {λj }∞ j=1 and {ϕj }j=1 are respectively the eigenvalues and the orthonormal eigenfunctions of A, that is,

Aϕj = λj ϕj

and

(ϕi , ϕj ) = δi,j

for i, j ≥ 1.

In addition, the sequence {λj }∞ j=1 is an increasing sequence of real numbers, that is, 0 ≤ ˙ s = D(A 2s ) with norm λ1 ≤ λ2 ≤ · · · . Let H 1/2 ∞  s  s 2 λj (v, ϕj ) , vs = A 2 v =

˙ s. v∈H

j=1

¯ by We define the operators E(t) and E(t)

E(t)v =

∞

 Eα,1 –λj t α (v, ϕj )ϕj ,

˙ s, v∈H

j=1

¯ E(t)v =

∞

 t α–1 Eα,α –λj t α (v, ϕj )ϕj ,

˙ s, v∈H

j=1

where α ∈ (0, 1) indicates the order of Caputo fractional derivative. Then, we present the mild solution u(t) of (1.1) [24]: 

t

u(t) = E(t)u0 +

 ¯ – s)f u(s) ds + E(t



0

t

 ¯ – s)g u(s) dW (s). E(t

(2.2)

0

Next, we impose the following conditions on f , g, and u(t), which are the conditions of existence and uniqueness of the mild solution u [24]. Assumption 2.1 For the nonlinear operator f : H → H, there exists a constant C such that   f (x) – f (y) ≤ Cx – y,

  f (x) ≤ Cx.

(2.3)

Assumption 2.2 For the nonlinear operator g : H → L02 , there exists a constant C such that   g(x) – g(y) 0 ≤ Cx – y, L 2

  g(x) 0 ≤ Cx. L 2

(2.4)

Assumption 2.3 The mild solution u : [0, T] ×  → H satisfies 2  sup E u(t)s < +∞,

(2.5)

0≤t≤T

where s ∈ [0, 2]. ¯ Some properties of the operators E(t) and E(t), which are crucial for the semidiscrete error estimates, will be introduced later.

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Lemma 2.2 ([32]) For α ∈ (0, 1), we have the following estimates:   α  p–q  D E(t)v ≤ Ct –α(+ 2 ) vq , t p

t > 0,

where 0 ≤ q ≤ p ≤ 2 for  = 0, and 0 ≤ p ≤ q ≤ 2 and q ≤ p + 2 for  = 1. Lemma 2.3 ([31]) For any t > 0 and 0 ≤ p – q ≤ 4, we have   q–p E(t)v  ≤ Ct –1+α(1+ 2 ) vq . ¯ p

3 Error estimates for spatially semidiscrete approximation In this section, we first review the Galerkin finite element methods and some basic estimates for the finite element projection operators. Then we introduce a representation of the semidiscrete scheme of the mild solution u(t) and some smoothing properties of the operators Eh (t) and E¯ h (t). We close this section with the proof of the semidiscrete error estimates. 3.1 Space discretization Let {Th }h∈(0,1] denote a regular family of triangulations of D , where h is the maximal meshsize, and let Vh denote the space of piecewise linear continuous functions with respect to Th vanishing on ∂ D . Thereby, Vh ⊂ H01 (D) = H˙ 1 = {v ∈ L2 (D), ∇v ∈ L2 (D), v|∂ D = 0}. De˙ 1 → Vh the Ritz projector onto Vh with respect to the inner product note by Rh : H  1 1 a(v, w) = A 2 v, A 2 w ,

˙ 1. v, w ∈ H

Thus we obtain a(Rh v, χ) = a(v, χ),

˙ 1 , χ ∈ Vh . v∈H

Meanwhile, the following error estimate is established: Rh v – v ≤ Chs vs ,

˙ s , 1 ≤ s ≤ 2. v∈H

(3.1)

The semidiscrete problem corresponding to (1.1) is to find a process uh (t) ∈ Vh such that   dW , Dαt uh (t) + Ah uh (t) = Ph f uh (t) + Ph g uh (t) dt

uh (0) = Ph u0 ,

where the mapping Ah : Vh → Vh is a discrete version of the operator A defined by a(ϕ, χ) = (Ah ϕ, χ),

∀ϕ, χ ∈ Vh ,

and Ph is the orthogonal projector Ph : H → Vh ,

(Ph v, χ) = (v, χ),

∀v ∈ H, ∀χ ∈ Vh .

(3.2)

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h N Depending on the eigenvalues and eigenfunctions {λhj }N j=1 and {ϕj }j=1 of the discrete operator Ah , we can introduce a representation of the solution of (3.2). Firstly, we present the ¯ as follows: discrete analogues of operators E(t) and E(t)

Eh (t)vh =

N

  Eα,1 –λhj t α vh , ϕjh ϕjh ,

(3.3)

  t α–1 Eα,α –λhj t α vh , ϕjh ϕjh .

(3.4)

j=1

E¯ h (t)vh =

N j=1

Analogously, the unique solution of the finite element problem (3.2) can be given by 

t

uh (t) = Eh (t)Ph u0 +

 E¯ h (t – s)Ph f uh (s) ds +

0



t

 E¯ h (t – s)Ph g uh (s) dW (s).

(3.5)

0

Then, similarly to Lemmas 2.2 and 2.3, we show some vital properties of Eh (t) and E¯ h (t) in the following: Lemma 3.1 ([32]) Let Eh (t) be defined by (3.3), and let χ ∈ Vh . Then, for α ∈ (0, 1) and p, q ∈ [–1, 1], we have   α  p–q  D Eh (t)χ  ≤ Ct –α(+ 2 ) χq , t p where q ≤ p and 0 ≤ p – q ≤ 2 for  = 0, and p ≤ q ≤ p + 2 for  = 1. Lemma 3.2 ([32]) Let E¯ h be defined by (3.4), and let χ ∈ Vh . Then, for all t > 0, ⎧ ⎨Ct –1+α(1+ q–p 2 ) χq ,   p – 2 ≤ q ≤ p, E¯ h (t)χ  ≤ p ⎩Ct –1+α χq , p < q, where p, q ∈ [–1, 1]. Based on this lemma, we have the following conclusion. Lemma 3.3 Let E¯ h be defined by (3.4), and let v ∈ H, Ph v = vh . For all t > 0, we have   E¯ h (t)Ph v ≤ Ct α–1 v. Proof By Lemma 3.2 with p = q = 0 we get   E¯ h (t)vh  ≤ Ct α–1 vh . Since vh = Ph v, we get   E¯ h (t)vh  ≤ Ct α–1 Ph v ≤ Ct α–1 v, which completes the proof. Moreover, we need the following estimate of uh (t).



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Lemma 3.4 For any t ∈ [0, T] and α ∈ ( 12 , 1), let uh (t) be the mild solution of (3.2). Then there exists a constant C > 0 such that  2 sup uh (t)L

0≤t≤T

≤ Cu0 2L2(;H) .

2(;H)

Proof For any t ∈ [0, T], from (3.5) by Lemma 3.1 with  = p = q = 0, Lemma 3.3, Assumptions 2.1 and 2.2, and Itô’s isometry we obtain  t  2   2   2      ¯ E uh (t) ≤ 4E Eh (t)Ph u0 + 4E Eh (t – s)Ph f uh (s) ds  0

 t 2     ¯ + 4E Eh (t – s)Ph g uh (s) dW (s)  0  t   2  2 ≤ 4EEh (t)Ph u0  + 4 EE¯ h (t – s)Ph f uh (s)  ds 0



t

+4

  2 EE¯ h (t – s)Ph g uh (s) L0 ds 2

0



t

≤ CEu0 2 + C

 2 (t – s)2α–2 Euh (s) ds

0



t

+C

 2 (t – s)2α–2 Euh (s) ds.

0

Thus, applying the integral version of Gronwall’s lemma, we deduce that

 t   2 sup Euh (t) ≤ CEu0 2 exp C (t – s)2α–2 ds ≤ CEu0 2 . 0≤t≤T

0



3.2 Semidiscrete finite element approximation In this subsection, we first present and prove some lemmas, which are crucial for the derivation of the semidiscrete error estimate for the nonlinear fractional stochastic differential equation. Then we give a detailed proof of the semidiscrete error estimate. Lemma 3.5 ([28]) Let 0 ≤ ν ≤ μ ≤ 2 and Fh (t) = E(t) – Eh (t)Ph . Then, for α ∈ (0, 1), there exists a constant C such that   Fh (t)x ≤ Chμ t –α μ–ν 2 x , ν

˙ ν. x∈H

¯ – E¯ h (t)Ph . Then, for t ∈ [0, T], there exists a Lemma 3.6 Let 1 < q ≤ 2 and F¯ h (t) = E(t) constant C such that    

T 0

2   2q ≤ Ch F¯ h (t)h(t) dt  

0

 h(t)2 dt, q–2

T

Proof By the definition of F¯ h (t) we split  0

t

F¯ h (t – s)h(t) ds =



t

t 0

F¯ h (t – s)h(t) ds into two additional terms:

¯ – s)h(t) ds – E(t

0

= v(t) – vh (t)

˙ q–2 . h(t) ∈ H



t 0

E¯ h (t – s)Ph h(t) ds

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  = v(t) – Ph v(t) + Ph v(t) – vh (t) = η(t) + ξ (t), where v(t) and vh (t) are the solutions of the following equations: ⎧ ⎨Dα v(t) + Av(t) = h(t), t

(3.6)

⎩v(0) = 0, ⎧ ⎨Dα v (t) + A v (t) = P h(t), h h h t h ⎩vh (0) = Ph v(0) = 0.

(3.7)

To bound ξ , we note that by our definitions  Dαt ξ (t) + Ah ξ (t) = Ah Rh v(t) – Ph v(t) ,

ξ (0) = 0.

By the Laplace transforms of both sides of this equation, we recover za ξˆ (z) + Ah ξˆ (z) = Ah (Rh – Ph )ˆv(z). Therefore –1  ξˆ (z) = zα I + Ah Ah (Rh – Ph )ˆv(z). Since the operator Ah generates an analytic contraction semigroup, there exists a constant C, depending only on φ and α, such that  α   z I + Ah –1  ≤ Cz–α ,

∀z ∈ φ ,

where φ = {z ∈ C : | arg z| ≤ φ}. By the identity 

zα I + Ah

–1

 –1 Ah = I – zα zα I + Ah

we get     α   z I + Ah –1 Ah  ≤ 1 + zα zα I + Ah –1  ≤ 1 + C ≤ C. Using the inverse Laplace transform and inequality (3.1), we obtain     ξ (t) ≤ C (Rh – Ph )v(t)       ≤ C (Rh – I)v(t) + C (I – Ph )v(t) ≤ Chq v(t)q . Then by Theorem 2.1 of [31] we get  0

T

 ξ (t)2 dt ≤ Ch2q

 0

T

 v(t)2 dt ≤ Ch2q q

 0

T

 h(t)2 dt. q–2

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According to inequality (3.1) and Theorem 2.1 of [31], the estimate of η yields 

T

 η(t)2 dt ≤ C

0



 (Rh – I)v(t)2 dt ≤ Ch2q

T

0



T

 v(t)2 dt q

0



T

≤ Ch2q 0

 h(t)2 dt. q–2

Since 

  

T  t

0

0

2  t 2     F¯ h (t)h ds dt = C(T) F¯ h (t – s)h ds    0



T

 η(t) + ξ (t)2 dt

= 0



T

 η(t)2 dt + 2

≤2 0



≤ Ch2q 0



T

 ξ (t)2 dt

0 T

 h(t)2 dt, q–2

we get the conclusion    

T 0

2   ¯Fh (t)h dt  ≤ Ch2q 

T

 h(t)2 dt. q–2

0



˜ ∈H ¯ – E¯ h (t)Ph . Then, for t ∈ [0, T] and h(s) ˙ q, Lemma 3.7 Let 1 < q ≤ 2 and F¯ h (t) = E(t) there exists a constant C such that 2  t  t   2  2q   ˜ ¯ (t – s)2α–2 E˜h(s)L0 ds. E Fh (t – s)h(s) dW (s) ≤ Ch 2,q 0

0

Proof Just like in the proof of Lemma 3.6, we split terms: 

t

˜ dW (s) = F¯ h (t – s)h(s)

0



t

t

˜ dW (s) – ¯ – s)h(s) E(t

0

0

F¯ h (t – s)h(t) ds into two additional



t

˜ dW (s) E¯ h (t – s)Ph h(s)

0

  = v˜ (t) – v˜ h (t) = v˜ (t) – Ph v˜ (t) + Ph v˜ (t) – v˜ h (t) = (t) + ϑ(t),

where v˜ (t) and v˜ h (t) are the solutions of the following equations: ⎧ ⎨Dα v˜ (t) + A˜v(t) = h(t) ˜ dW (t) , t

⎩v˜ (0) = 0,

dt

(3.8)

⎧ ⎨Dα v˜ (t) + A v˜ (t) = P h(t) ˜ dW (t) , h h h t h dt ⎩v˜ h (0) = Ph v˜ (0) = 0.

(3.9)

To bound ϑ, we note that by our definitions  Dαt ϑ(t) + Ah ϑ(t) = Ah Rh v˜ (t) – Ph v˜ (t) ,

ϑ(0) = 0.

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As in the proof of Lemma 3.6, taking the Laplace transform and inverse Laplace transform on both sides of this equation, we eventually get     ϑ(t) ≤ C (Rh – Ph )˜v(t)       ≤ C (Rh – I)˜v(t) + C (I – Ph )˜v(t) ≤ Chq v˜ (t)q . Thus by Lemma 2.3 with p = q ∈ (1, 2] and Itô’s isometry we derive  t 2  2   2  2q  2q     ˜ ¯ E ϑ(t) ≤ Ch E v˜ (t) q = Ch E E(t – s)h(s) dW (s)  0

q

 t 2   q ˜ dW (s) ¯ – s)h(s) 2 E(t = Ch2q E A   0  t  q  ˜ 2 0 ds ¯ – s)h(s) = Ch2q EA 2 E(t L 2

0

 ≤ Ch2q

∞ 2   t 1   ˜ 2 e  ds h(s)Q (t – s)2α–2 E j   0 j=1



t

= Ch2q 0

q

  q ˜ 2 0 ds. (t – s)2α–2 EA 2 h(s) L 2

According to inequality (3.1) and Lemma 2.3, the estimate of  yields  2  2  2 E(t) ≤ C (Rh – I)˜v(t) ≤ Ch2q Ev˜ (t)q  t   q ˜ 2 0 ds. ≤ Ch2q (t – s)2α–2 EA 2 h(s) L 2

0

Thereby, 2  t  t     ˜ dW (s) ≤ Ch2q (t – s)2α–2 E˜h(s)2 0 ds. ¯ h (t – s)h(s) F E   L 0

2,q

0



Now, we will give the semidiscrete error estimate in space for the stochastic fractional differential equation (1.1). Theorem 3.1 Let u(t) and uh (t) be the solutions of (1.1) and (3.2), respectively. Then, for ˙ s ), s ∈ [0, 2], we have t ≥ 0, α ∈ ( 12 , 1), and u0 ∈ L2 (, H   u(t) – uh (t) L

2(,H)

≤ Ch2 .

Proof For t ∈ [0, T], by (1.1) and (3.2) we have   u(t) – uh (t) L

2(,H)

  ≤  E(t) – Eh (t)Ph u0 

L2(,H)

 t        ¯ ¯ E(t – s)f u(s) – Eh (t – s)Ph f uh (s) ds +  0

L2(,H)

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 t        ¯ ¯ E(t – s)g u(s) – Eh (t – s)Ph g uh (s) dW (s) +  0

L2(,H)

= I + II + III. For I, by Lemma 3.5 with ν = μ = 1 + r (r ∈ (0, 1]) we have I ≤ Ch1+r u0 L2(;H˙ 1+r ) . We dominate II by two additional terms:   t     ¯ – s)f u(s) – E¯ h (t – s)Ph f uh (s) ds E(t II =    0

L2(,H)

 t        ¯ ≤  Eh (t – s)Ph f u(s) – f uh (s) ds  0

L2(,H)

 t     ¯ h (t – s)f u(s) ds + F   0

L2(,H)

= I1 + I2 . We estimate each term separately. First, note that by Lemma 3.3 and Assumption 2.1 we have 

t

I1 ≤ 0

    E¯ h (t – s)Ph f u(s) – f uh (s)  L

2(,H)



t

≤C

2(,H)

0



t

≤C

    (t – s)α–1 f u(s) – f uh (s) L   (t – s)α–1 u(s) – uh (s)L

2(,H)

0

ds

ds

ds.

The term I2 is reckoned by applying Lemma 3.6, Assumptions 2.1 and 2.3. Then we get  t    2 ¯ h (t – s)f u(s) ds I22 = E F   0



≤ Ch4

t

  2 Ef u(s)  ds

0

 ≤ Ch4

t

 2 sup Eu(s) ds

0 0≤s≤T

≤ Ch . 4

A combination of the estimates I1 and I2 gives  II ≤ Ch + C 2

4

0

t

 2 (t – s)2(α–1) u(s) – uh (s)L

2(,H)

ds.

In a similar way as for II, we dominate III by two additional terms:   t      ¯ ¯ III =  E(t – s)g u(s) – Eh (t – s)Ph g uh (s) dW (s)  0

L2(,H)

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 t        ¯ ≤  Eh (t – s)Ph g u(s) – g uh (s) dW (s)  0

L2(,H)

 t     ¯ h (t – s)g u(s) dW (s) F +   0

L2(,H)

= I3 + I4 . As in an estimate for I1 , we can get an estimate for I3 by using Lemma 3.3 together with Assumption 2.2 and Itô’s isometry:  t 2      ¯ h (t – s)Ph g u(s) – g uh (s) dW (s) E I32 = E   0



t

= 

0

=

    2 EE¯ h (t – s)Ph g u(s) – g uh (s) L0 ds 2

∞  t     1 2 E¯ h (t – s)Ph g u(s) – g uh (s) Q 2 ej  ds E

0

j=1



t

≤C

   2 (t – s)2α–2 Eg u(s) – g uh (s) L0 ds 2

0



t

≤C

2  (t – s)2α–2 u(s) – uh (s)L

2(,H)

0

ds.

For the estimate of term I4 , we apply Lemma 3.7, Assumptions 2.2 and 2.3, and Itô’s isometry:  t 2    ¯ h (t – s)g u(s) dW (s) I42 = E F   0



≤ Ch4

t

0

 ≤ Ch4

 2 (t – s)2α–2 Eu(s)2 ds 

t

t

(t – s)2α–2 ds

 2 sup Eu(s)2 ds

0 0≤s≤T

0

≤ Ch . 4

In total, we have by I3 and I4 that  III 2 ≤ Ch4 + C

t

 2 (t – s)2α–2 u(s) – u(t)L

2(,H)

0

ds.

Let ϕ(t) = u(s) – uh (s)2L2(,H) . Since I 2 ≤ Ch2+2r u0 L2(;H˙ 1+r ) = Ch4 u0 L2(;H˙ 2 ) ,  t  2 II 2 ≤ Ch4 + C (t – s)2(α–1) u(s) – uh (s)L

2(,H)

0

 III ≤ Ch + C 2

4

0

t

 2 (t – s)2α–2 u(s) – u(t)L

2(,H)

ds,

ds,

Zhang et al. Advances in Difference Equations (2018) 2018:215

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according to the integral version of Gronwall’s lemma, we get ϕ(t) ≤ Ch4 . Then we have   u(s) – uh (s) L

2(,H)

≤ Ch2 .



4 Error estimates for fully discrete approximation In this section, we first introduce the GMMP scheme. Then we give a fully discrete scheme and the corresponding fully discrete error estimate, together with some lemmas, which are significant in the proof of the fully discrete error estimate. 4.1 The GMMP scheme We denote the time mesh points by tn = nk, n = 0, 1, . . . , N , with a fixed time step k > 0, such that 0 ≤ tn ≤ T and k = NT . Now let us present the GMMP scheme derived by Gorenflo, Mainardi, Moretti, and Paradisi [33]. The Caputo fractional derivative (when 0 < α < 1) can be approximated by n  1 α wm u(tn–m ) – u(0) α k m=0  n  1 α w u(tn–m ) – φn u(0) , = α k m=0 m

Dαt u(tn ) ≈

(4.1)

where wαm = φn =

(m – α) , (–α)(m + 1)

n m=0

wαm =

(n + 1 – α) , (1 – α)(n + 1)

(4.2) n ≥ 0.

(4.3)

Moreover, wαm and φn have the following properties. Lemma 4.1 ([34, 35]) For α > 0, n = 1, 2, . . . , we have:   α α (1) wα0 = 1, wαn < 0, |wαn+1 | ≤ |wαn |, and 0 < – nm=1 wαm < – ∞ m=1 wm = w0 ; α (2) φn – φn–1 = wn < 0, that is, φn < φn–1 < φn–2 < · · · < φ0 = 1.

4.2 Error estimates By using the GMMP scheme (4.1) we indicate the approximation of u(tn ) by un ≈ u(tn ). Then the fully discrete scheme for equation (1.1) can be defined by  n    n 1 tn  1 α n–m 0 n dW (s). w u – φn uh + Ah uh = Ph f uh + Ph g un–1 h k α m=0 m h k tn–1

(4.4)

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Furthermore, we define R(λ, X) = (λI – X)–1 , λ > 0, and E˜ kh = R(k –α , –Ah ) = (k –α I + Ah )–1 . Then scheme (4.4) can be rewritten as n

unh = k –α φn E˜ kh u0h – k –α E˜ kh +

1 k



 wαm un–m + E˜ kh Ph f unh h

m=1

 dW (s). E˜ kh Ph g un–1 h

tn tn–1

(4.5)

Besides, the semidiscretized version of mild solution (3.5) at time tn should be shown: 

tn

uh (tn ) = Eh (tn )Ph u0 + 

 E¯ h (tn – s)Ph f uh (s) ds

0 tn

+

 E¯ h (tn – s)Ph g uh (s) dW (s).

(4.6)

0

Now let us introduce and prove some lemmas, which will play an important role later on. Lemma 4.2 ([30]) For any k > 0 and h ∈ (0, 1), there exists a constant C > 0 such that E˜ kh v ≤ Ck α v,

E˜ kh Ph v ≤ Ck α v,

∀v ∈ H.

Lemma 4.3 For any t > 0 and p, q ∈ [–1, 1] such that 0 ≤ p – q < 2, we have   (2+q–p)α Eh (t)vh – vh  ≤ Ct 2 vh q+2 , p

∀vh ∈ Vh .

Proof The definition of Eh (t)vh in (3.3) and Lemma 2.1 yield   Eh (t)vh – vh 2 p =

N  h p   2  2 λj 1 – Eα,1 –λhj t α vh , ϕjh j=1

= t (q–p)α

N  h α p–q   2  h q  2 λj t 1 – Eα,1 –λhj t α λj vh , ϕjh j=1

= t (q–p)α

 t 2 N  h α p–q   q  2 λj t λhj sα–1 Eα,α –λhj sα ds λhj vh , ϕjh 0

j=1

≤ Ct (q–p)α

N 

λhj t α

p–q



= Ct

N 

p–q λhj t α



j=1

≤ Ct (q–p)α

N  j=1

λhj t α

λhj sα–1

0

j=1

(q–p)α

t

p–q

t

0

t

2

(λhj sα )(

p–q h α ( 2 )

1 + (λhj sα )2

λhj sα–1 p–q h α ( 2 )

(λj s )

2 ds

 h q  2 λj vh , ϕjh 2

p–q 2 )

λhj sα–1 (λj s )

0



1 ds 1 + (λhj sα )2

ds

 h q  2 λj vh , ϕjh

 h q  2 λj vh , ϕjh

Zhang et al. Advances in Difference Equations (2018) 2018:215

= Ct

(q–p)α

 N  α p–q t j=1

t

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sα–1 p–q

0 (sα ) 2

2 ds

 h q+2  2 λj vh , ϕjh

N  α p–q 2α–(p–q)α  h q+2  2 t λj vh , ϕjh ·t

≤ Ct (q–p)α

j=1

= Ct (2+q–p)α vh q+2 .



Lemma 4.4 ([30]) For any λ > 0 and μ ∈ R, there exists a constant C such that     μR(λ, Ah ) – I Ph v ≤ Cλ–1 v. Based on the previous discussion, we are ready to prove the error estimates for the fully discrete approximation. Theorem 4.1 Let unh and u(tn ) be solutions of (4.4) and (1.1), respectively, for t ≥ 0, α ∈ ˙ s ), s ∈ [0, 2]. Then there exists a constant C > 0 such that ( 12 , 1), and u0 ∈ L2 (, H   u(tn ) – un 2 h L

2(;H)

  ≤ C k 2α + h4 .

Proof By the triangle inequality we have   u(tn ) – un  h L

2(;H)

  ≤ u(tn ) – uh (tn )L

2(;H)

  = ρ n L

  + θ n L

2(;H)

  + uh (tn ) – unh L

2(;H)

2(;H)

.

Since we have estimated the error of ρ n L2(;H) in Theorem 3.1, we only need to estimate θ n L2(;H) . Using equations (4.6) and (4.5), we obtain  n θ  L

2(;H)

  ≤ Eh (tn )Ph u0 – k –α φn E˜ kh Ph u0 L 2(;H)   n     + –k –α E˜ kh wαm un–m  h   m=1

  + 

tn

0

L2(;H)

   ¯Eh (tn – s)Ph f uh (s) ds 

   + –E˜ kh Ph f unh    + 

0

L2(;H)

L2(;H)

tn

   ¯Eh (tn – s)Ph g uh (s) dW (s) 

   1 + –  k

tn

tn–1

    dW (s) E˜ kh Ph g un–1 h 

L2(;H)

L2(;H)

= I1 + I2 + I3 + I4 + I5 + I6 . For I1 , by the triangle inequality, we separate I12 into two additional terms: 2  I12 = EEh (tn )Ph u0 – k –α φn E˜ kh Ph u0 

Zhang et al. Advances in Difference Equations (2018) 2018:215

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   2 = E Eh (tn )Ph u0 – Ph u0 + Ph u0 – k –α φn E˜ kh Ph u0    2 2 ≤ 2E Eh (tn )Ph u0 – Ph u0  + 2E Ph u0 – k –α φn E˜ kh Ph u0  = I11 + I12 . For I11 , by Lemma 4.3 with p = q = 0 we get  2 I11 = 2E Eh (tn )Ph u0 – Ph u0  ≤ Ctn2α EPh Au0 2 ≤ Ck 2α Eu0 22 . For I12 , setting μ = k –α φn and using Lemma 4.4, we have I12 = 2EμE˜ kh Ph u0 – Ph u0 2 2    = 2E μR k –α , Ah – I Ph u0  ≤ Ck 2α Eu0 2 . By Lemma 4.1 we have

n

α α m=1 |wm | < w0

= 1. Together with Lemmas 4.2 and 3.4, we obtain

 2 n     wαm un–m I22 = Ek –α E˜ kh  h   m=1

 2 n    –α ˜ α  n–m  = Ek Ekh wm uh – uh (tn–m ) + uh (tn–m )    m=1

≤C

n

n 2   2 Eθ n–m  + C EE˜ kh uh (tn–m ) )

m=1

≤C

n

m=1

2   Eθ n–m  + Ck 2α Eu0 2 .

m=1

The term I3 is estimated by applying Lemma 3.2, Assumption 2.1, and Lemma 3.4: for 0 < tn ≤ T = Nk, we get I32

 tn    2  ¯ = E Eh (tn – s)Ph f uh (s) ds  0  tn   2 ≤ EE¯ h (tn – s)Ph f uh (s)  ds 0



≤C

tn

 2 (tn – s)2α–2 Euh (s) ds

0

≤ Ck 2α Eu0 2 . By Lemma 4.2, Lemma 3.4, and Assumption 2.1 we get the following estimate for I4 :      2  2 I42 ≤ 2EE˜ kh Ph f unh – f uh (tn )  + 2EE˜ kh Ph f uh (tn )       2 2 ≤ Ck 2α Ef unh – f uh (tn )  + Ck 2α Ef uh (tn ) 

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 2  2 ≤ Ck 2α Eunh – uh (tn ) + Ck 2α Euh (tn )  2 ≤ Ck 2α Eθ n  + Ck 2α Eu0 2 . For I5 , by Lemma 3.2, Assumption 2.2, Lemma 3.4, and Itô’s isometry, we obtain   I52 = E 

tn

0



  2 EE¯ h (tn – s)Ph g uh (s) L0 ds

tn

= 0

2   E¯ h (tn – s)Ph g uh (s) dW (s) 

2



≤C

tn

 2 (tn – s)2α–2 Euh (s) ds



Eu0 2 .

0

≤ Ck

2α

For I6 , by Lemma 4.2, Lemma 3.4, Assumption 2.2, and Itô’s isometry we have 2     1 tn   ˜ kh Ph g un–1 dW (s) I62 = E – E h   k tn–1   tn 2 1      ˜ kh Ph g un–1 – g u ≤ 2E (t ) dW (s) E h n–1 h k  tn–1   tn 2 1    ˜ + 2E Ekh Ph g uh (tn–1 ) dW (s)  k tn–1     2 2 tn  – g uh (tn–1 ) L0 ds = EE˜ kh Ph g un–1 h 2 k tn–1  tn   2 2 + EE˜ kh Ph g uh (tn–1 ) L0 ds 2 k tn–1    2 ≤ Ck 2α Eθ n–1  + Ck 2α Eu0 2 . Therefore, coming back to θ n L2(;H) , combining I1 , I2 , I3 , I4 , I5 , and I6 and applying a discrete version of Gronwall’s lemma, we have  n 2 θ  L

2(;H)

≤ Ck 2α .

By the triangle inequality we obtain   u(tn ) – un 2 h L

2(;H)

 2 ≤ θ n L

which completes the proof.

2(;H)

 2 + ρ n L

2(;H)

 ≤ C k 2α + h4 , 

5 Conclusions and discussions In this paper, we have studied semidiscrete and fully discrete schemes for nonlinear timefractional SPDEs. The semidiscrete scheme employs a standard Galerkin finite element method, and the time direction of the fully discrete scheme is based on the GMMP scheme. The strong convergence error estimates for the semidiscrete and fully discrete schemes in the L2 -norm are demonstrated. However, there are several possible extensions of the

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work. First, we only consider the initial value condition in our given problem; the complex boundary condition in our future study will be discussed. Second, numerical investigations on time-space fractional SPDEs are an interesting direction for our future research.

Acknowledgements The authors would like to express their sincere gratitude to the anonymous reviewers for their careful reading of the manuscript and their comments, which led to a considerable improvement of the original manuscript. Funding This research is supported by the National Natural Science Foundation of China under Grant 61671002 and the Fundamental Research Funds for the Central Universities under grant ZY1821. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors participated in drafting and checking the manuscript and approved the final manuscript. Author details 1 LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, P.R. China. 2 School of Science, Beijing University of Chemical Technology, Beijing, P.R. China.

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