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We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order . The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results.
Consider the following stochastic semilinear space–time fractional wave equation driven by fractionally integrated additive noise, with
where D is a bounded domain in with smooth boundary , and and represent the Caputo fractional derivative of order and the Riemann–Liouville fractional integral of order respectively. In addition, is the fractional Laplacian and denotes the space–time noise defined on a complete filtered probability space The initial values and and the nonlinear function (source term) are given functions.
The space–time fractional wave equation, denoted as (1) when devoid of noise, has been extensively explored by researchers due to its wide range of applications in engineering, physics, and biology [1,2,3]. The inclusion of the noise term allows for the characterization of random effects influencing the particle movement within a medium with memory or particles experiencing sticking and trapping phenomena. An example of such noise is the fractionally integrated noise , where the past random effects impact the internal energy . For physical systems, stochastic perturbations arise from many natural sources, which cannot always be ignored. Therefore, it is necessary to include them in the corresponding deterministic model.
It is not possible to find the analytic solution of the space–time fractional Equation (1). Therefore, one needs to introduce and analyze some efficient numerical methods for solving (1). Li et al.  considered the Galerkin finite element method of (1) for the linear case with the additive Gaussian noise, that is, and , and obtain the error estimates. In , the authors studied the Galerkin finite element method for approximating the semilinear stochastic time-tempered fractional wave equations with multiplicative Gaussian noise and additive fractional Gaussian noise, but they only established error estimates for . Extensive theoretical results exist for the stochastic subdiffusion problem with , as seen in works such as [7,8,9,10,11,12], alongside corresponding numerical approximations in works including [13,14,15,16,17]. Regarding the theoretical and numerical findings for the stochastic wave equation, we recommend exploring references such as [18,19,20,21]. For theoretical advancements in fractional-order nonlinear differential equations, recent works such as [22,23,24,25,26,27] and their references provide a comprehensive overview.
In this paper, our focus lies on the application of the Galerkin finite element method to solve (1). Firstly, we establish the existence of a unique solution for (1) using the Banach fixed point theorem. Additionally, we analyze the spatial and temporal regularities of the solution. To approximate the noise, we employ a piecewise constant function in time, resulting in a stochastic regularized equation. This equation is then tackled using the Galerkin finite element method. We provide corresponding error estimates, utilizing the various smoothing properties exhibited by the Mittag–Leffler functions. We extend the error estimates in  from the linear case of (1) with Gaussian additive noise to the semilinear case with the more general integrated additive noise. We also extend the error estimates of  for the stochastic semilinear time fractional wave equation from to .
To establish our error estimates, we employ a similar argument as developed in our recent work , which focused on approximating the stochastic semilinear subdiffusion equation with . We demonstrate that the solution’s spatial and temporal regularities for (1) with surpass those with . Moreover, we observe that the convergence orders of the Galerkin finite element method for (1) with are higher than those with , as expected.
The paper is organized as follows. In Section 2, we provide some preliminaries and notations. In Section 3, we focus on the continuous problem and establish the existence, uniqueness, and regularity results for the problem (1). In Section 4, we discuss the approximation of the noise and obtain an error estimate for the regularized stochastic semilinear fractional superdiffusion problem. In Section 5, we consider the finite element approximation of the regularized problem and derive optimal error estimates. Finally, in Section 6, we present numerical experiments that validate our theoretical findings.
Throughout this paper, we denote C as a generic constant that is independent of the step size and the space step size h, which could be different at different occurrences. Additionally, we always assume is a small positive constant.
2. Notation and Preliminaries
This section provides notations and preliminary results that will be used in subsequent sections. We denote as the space of Lebesgue measurable or square integrable functions on D, with norm and inner product . Additionally, we denote . We assume that with domain is a closed linear self-adjoint positive definite operator with a compact inverse. Moreover, A has the eigenpairs , , subject to homogeneous Dirichlet boundary conditions.
Set or simply for any as a Hilbert space induced by the norm
For we denote by For any function define Let be a separable Hilbert space of all measurable square-integrable random variables with values in such that where denotes the expectation.
where are some real-valued continuous functions rapidly decaying with respect to k. Here, the sequence is mutually independent and identically distributed one-dimensional standard Brownian motions, and the white noise is the formal derivative of the Brownian motion
().(It isometry property) Let be a strongly measurable mapping such that Let denote a real-valued standard Brownian motion. Then, the following isometry equality holds for :
To represent the solution of (1) in the integral form, we utilize the Laplace transform technique to write down the solution representation in terms of the Mittag–Leffler functions. The Mittag–Leffler functions are defined in , and we use them to express the solution in a compact form.
The following Lemma is related to the bounds of the Mittag–Leffler functions.
((Mittag–Leffler function property) ).Let and . Let be defined by (4). Suppose that μ is an arbitrary real number such that Then, there exists a constant such that
Moreover, for it follows that
3. Existence, Uniqueness, and Regularity Results
This section is dedicated to studying the existence, uniqueness, and regularity results of the mild solution of the stochastic semilinear space–time fractional superdiffusion model (1).
There is a positive constant C such that the nonlinear function satisfies
The sequence with its derivative is uniformly bounded by and , respectively, i.e.,
where the series and are convergent.
Let It holds, with
and are the eigenvalues of the Laplacian with
(, Lemma 2.4).An adapted process is called a mild solution to (1) if it satisfies the following integral equation with
(, Lemma 2.5).The solution of the homogenuous problem of (1) satisfies, for ,
(Existence and uniqueness).Let and . Let Assumptions 1–3 hold. Let . Then, there exists a unique mild solution given by (11) to the problem (1) for all
Set , as the set of functions in with the following weighted norm For any fixed this norm is the same as the standard norm on We can therefore define a nonlinear map by
For any the function is a solution of (11) if and only if u is a fixed point of the map . In order to apply the Banach fixed point theorem it suffices to show that for an appropriately chosen is a contraction mapping. We first show that for any . By (15) and the Cauchy–Schwarz inequality we obtain with
Based on the smoothing properties of the solution operators and by Assumption 1, it follows that
By the smoothing property of the operator the isometry property of Brownian motion, we have with that,
Note that and ; we then obtain , which implies that . Now we consider the contraction property. For any given two functions and in it follows from (15) with the smoothing property and boundedness of with and that
With , choose sufficiently large , we obtain
Hence, Therefore, by the Banach contraction mapping theorem, there exists a unique fixed point u of the nonlinear map which is the mild solution given by (15) of the problem (1). □
(Regularity) Let Assume that Assumptions 1–3 hold. Let , with Then, the following regularity results hold for the solution u of (11) with and ,
From the definition of the mild solution (11) and with it follows that, with
here, is the characteristic function for the ith time step length and is some approximations of . The following is the regularized stochastic space–time fractional superdiffusion problem. Let be an approximation of u defined by
For , we note by the Mittage–Leffler function property that
Now we estimate for the different and . We shall show that, with ,
Case 1. We now consider the case . If , then with , it implies that
Since , for and then for ,
and this implies that
Similarly, we may show that for , with ,
Therefore, we obtain, for ,
Case 2. Next, consider the case . If then we obtain,
Similarly, for , it follows that
Therefore, for we obtain,
Thus, we derive the following estimate for . For
Together with these estimates we obtain the following results.
For , it follows that for ,
An application of the Gronwall’s Lemma completes the rest of the proof. □
5. Finite Element Approximation and Error Analysis
Let D be the spatial domain and let be a shape regular and quasi-uniform triangulation of the domain D with spatial discretization parameter , where is the diameter of K. Let , be the piecewise linear finite element space with respect to the triangulation , that is,
Let and be the projection and fractional Ritz projection defined by and . We then have
().(Inverse Estimate in ) For any there exists a constant C independent of h such that
We now consider the error estimates.
Let , , Suppose that Assumptions 1–6 hold. Let and be the solutions of (32) and (57), respectively. Let with . Then, there exists a positive constant C such that for any , with and ,
Introducing as a solution of an intermediate discrete system
We split the error . Again using we split
From Lemma 6 it follows that, with
which means that
To estimate , note that satisfies the following equation
and hence, the representation of solution is written as
Choose and separately, from Lemma 6 with and Lemma 7, it follows that for and that
Now an application of regularity shows
where we used the fact that since and . We now combine (63), (66), and (67) to arrive at an estimate for as, with and ,
Now to estimate , note that satisfies
and therefore we now write in the integral form as
Again, choose . From Lemma 6 with and Lemma 7, it follows for and for that
Combining (68) and (71) it follows for and , and that
An application of the Gronwall’s Lemma completes the rest of the proof. □
The main result of this paper is obtained by combining Theorems 6 and 7.
Let , . Suppose that Assumptions 1–6 hold. Let u and be the solutions of (1) and (57), respectively. Let with Then, there exists a positive constant C such that, for any with and
In particular, when the noise is the trace class noise, i.e.,
In this case we have , , , where , , , we obtain with
which are consistent with the results for the stochastic heat equation.
6. Numerical Simulations
In this section, we will explore three numerical examples of the stochastic semilinear fractional wave equation. For simplicity, we will focus on the Laplacian operator, that is, in Equation (1). Our goal is to approximate Equation (1) with various functions and examine the experimentally determined orders of convergence in time. We consider both the cases of trace class and white noises. Specifically, we choose the following functions: , , and . By comparing the experimentally determined orders of convergence with the theoretical findings in Theorem 8, we observe consistent results, as expected. All the numerical simulations in this paper are performed on an Acer Aspire 5 Laptop.
To complete this, let us first introduce the numerical method for solving (8). Consider, with ,
where and are given smooth functions. Here, with ,
where are the Brownian motions. Here, denotes the eigenfunctions of the operator with . Further, let be the eigenpairs of the covariance operator Q of the stochastic process , that is,
We shall consider two cases in our numerical simulations.
Case 1: The white noise case, e.g., with , which implies that
where denotes the trace of the operator Q.
Case 2: The trace class case, e.g., with , which implies that
The numerical methods for solving stochastic time fractional partial differential equations are similar to the numerical methods for solving deterministic time fractional partial differential equations. The only difference is that we have the extra term g in the stochastic case and we need to consider how to approximate g.
Let . Then (75)–(77) can be written as the following:
Since the initial values in (79)–(81), it is easier to consider the numerical analysis for the time discretization scheme of (79)–(81).
Let be a partition of the time interval and the time step size. Let be a partition of the space interval and h the space step size.
Let be the piecewise linear finite element space defined by
The finite element method of (79)–(81) is to find such that, ,
where denotes the projection operator.
Let be the approximation of . We define the following time discretization scheme: find , with , such that, ,
where the weights are generated using the Lubich’s convolution quadrature formula, with ,
Let be the linear finite element basis functions defined by, with ,
To find the solution , we assume that
for some coefficients . Choose in (84), we have, with ,
where we assume the initial values and have the following expressions:
To solve (86) using MATLAB, we need to write (86) into the matrix form, which we demonstrate below.
After some simple calculations, we may obtain the following mass and stiffness metrics
respectively. Then, (86) can be written as the following matrix form, ,
Hence can be calculated using the following formula
We now consider how to calculate . The kth component in can be approximated by using the following formula:
where, with ,
See the MATLAB code in Appendix A.1 for calculating kth element of in .
We next consider how to calculate , which is more complicated than . Approximating the Riemann–Liouville fractional integral by the Lubich first-order convolution quadrature formula and truncating the noise term to terms, we obtain the lth component of by, with ,
where are generated by the Lubich first-order method, with ,
To solve (91), we first need to generate Brownian motions , . This can be performed by using MathWorks MATLAB function fbm1d.m , which gives the value of the fractional Brownian motion with the Hurst parameter at any fixed time T. Let and and let denote the reference time step size. Let be the time partition of . We generate the fractional Brownian motions with the Hurst number by using the MATLAB code in Appendix A.2. When , fbm1d.m generates the standard Brownian motions.
Since we do not know the exact solution of the system, we shall use the reference time step size and the space step size to calculate the reference solution . The spacial discretization is based on the linear finite element method.
We then choose and consider the different time step size to obtain the approximate solutions at .
Let us discuss how to calculate the lth component of in MATLAB. Denote
The lth component of the vector satisfies
Finally, we shall consider how to calculate the projections and of and , respectively. Here, we only consider the case . The calculation of is similar. Assume that
By the definition of , we obtain
Hence, can be calculated by
Consider the following stochastic time fractional PDE (Partial Differential Equation), with ,
where and the initial value and is defined by (78).
Let and transform the system (93)–(95) of u into the system of v. We shall consider the approximation of v at . We choose the space step size and the time step size to obtain the reference solution vref. To observe the time convergence orders, we consider the different time step sizes with to obtain the approximate solution V. We choose simulations to calculate the following L2 error at with the different time step sizes
By Theorem 8, the convergence order should be
In Table 1, we consider the case of trace class noise, where for . We observe that the experimentally determined time convergence orders are consistent with our theoretical convergence orders, as indicated in the numbers in the brackets. We have included the CPU time in seconds for running 20 simulations in each experiment. The CPU times exhibit similarity across the other tables; hence, we have decided not to include them in subsequent tables.
In Table 2, we consider the case of white noise, where for . We observe that the experimentally determined time convergence orders are slightly lower than the orders in the trace class noise case, as we expected.
In Figure 1, we plot the experimentally determined orders of convergence with and as shown in Table 1 for the trace class noise. The expected convergence order is . To get the plot, we choose the different time step sizes and calculate the corresponding errors . By error estimate we have with the convergence order which implies that
The reference line in Figure 1 is determined by four points and the blue line in Figure 1 is determined by the four points . If these two lines are parallel, then we may conclude that the experimentally determined convergence order is almost p. We use the similar approach to obtain other figures below.
In Figure 2, we plot the experimentally determined orders of convergence with and as shown in Table 2 for the white noise. We observe that the convergence order is almost in the figure, where the reference line represents the order .
Consider the following stochastic time fractional PDE, with ,
where and the initial values and is defined by (78).
We use the same notations as in Example 1. In Table 3, we consider the case of trace class noise, where for . We observe that the experimentally determined time convergence orders are consistent with our theoretical convergence orders, as indicated in the numbers in the brackets.
In Table 4, we consider the case of white noise, where for . We observe that the experimentally determined time convergence orders are slightly lower than the orders in the trace class noise case, as expected.
In Figure 3, we plot the experimentally determined orders of convergence with and for the trace class noise as shown in Table 3. The expected convergence order is . The reference line in the figure represents the order , which is consistent with our observations.
In Figure 4, we plot the experimentally determined orders of convergence with and as shown in Table 4 for the white noise. We observe that the convergence order is almost in the figure, where the reference line represents the order .
Consider the following stochastic time fractional PDE, with ,
where and the initial values and are defined by (78).
We use the same notations as in Example 1. In Table 5, we consider the trace class noise, i.e., , and observe that the experimentally determined time convergence orders are consistent with our theoretical convergence orders. The numbers in the brackets denote the theoretical convergence orders.
In Table 6, we consider the white noise, i.e., , and observe that the experimentally determined time convergence orders are slightly less than the orders in the trace class noise case, as expected.
In Figure 5, we plot the experimentally determined orders of convergence with and in Table 5 for the trace class noise. The expected convergence order is . We indeed observe this in the figure where the reference line is for the order .
In Figure 6, we plot the experimentally determined orders of convergence with and in Table 6 for the white noise. We observe that the convergence order is almost in the figure where the reference line is for the order .
In this paper, we explore a numerical approach to approximate the stochastic semilinear space–time fractional wave equation. We establish the existence of a unique solution for this equation by using the Banach fixed point theorem, assuming that the nonlinear function satisfies the global Lipschitz condition. To obtain the stochastic regularized problem, we approximate the noise using a piecewise constant function in time. The finite element method is then employed to approximate the stochastic regularized problem. Furthermore, we propose a natural extension of this work, which involves considering more general nonlinear functions, such as the Allen–Cahn equation, as well as exploring different types of noise, including fractional noise with a Hurst parameter . Additionally, it might be very interesting to investigate more advanced fractal–fractional derivatives  in our stochastic space–time fractional wave equation.
We have the equal contributions to this work. B.A.E. considered the theoretical analysis, performed the numerical simulation, and wrote the original version of the work. Y.Y. introduced and guided this research topic. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
The author is grateful to the reviewers for their valuable and constructive comments, which improved the quality of the manuscript significantly.
Conflicts of Interest
The authors declare no conflict of interest.
In this Appendix, we include some MATLAB codes used in Section 6.
Appendix A.1. Calculate the kth Element of (f(u(tn)),φk) in Fn in (90)
% find (fu, phi)
% f(u)= sin(u)
F0=sin(U0); F_1=sin(U_1); F1=sin(U1);
Appendix A.2. Generate the Fractional Brownian Motions (t0), (t1),⋯ (tN), m = 1, 2, ⋯, M − 1 with the Hurst Number H ∈ [1/2,1]
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