Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020
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Wei Gu, "Investigations for a Type of Variable Coefficient Fractional Subdiffusion Equation with Multidelay", Journal of Function Spaces, vol. 2020, Article ID 6964159, 8 pages, 2020. https://doi.org/10.1155/2020/6964159
Investigations for a Type of Variable Coefficient Fractional Subdiffusion Equation with Multidelay
Abstract
A difference scheme is constructed for a type of variable coefficient time fractional subdiffusion equation with multidelay. Stability and convergence results of the scheme are obtained, and theoretical results are proved by two numerical tests.
1. Introduction
This paper will consider the following variable coefficient time fractional subdiffusion equation with arbitrary multidelay where is the variable diffusion coefficient, are delay terms, and ; time fractional partial derivative is defined as where is the gamma function.
For simplicity, we consider the following equation with arbitrary two different delays: where are two delay terms and satisfy , , are integers, and , .
Applications of delay differential equations (DDEs) can be found in many fields, such as electrotechnics, population dynamics, biology, and economics [1â€“5]. However, one can obtain the analytical solutions of DDEs in few cases. The researchers turn to numerical methods for solving DDEs in general cases [6â€“9]. Most studies about DDEs consider one delay; in fact, more than one delays should be considered in some systems. Tian and Kuang considered Hmethod and GPstability for DDEs with several delay terms [10]; the applications of such DDEs can be found in [11â€“14].
Fractional delay differential equations (FDDEs) are widely used in automatic control, population dynamics, finance, etc. [15â€“17]. Numerical solutions for FDDEs can be found in [18â€“25], where [18â€“20] considered numerical solutions for FDDEs, and [21â€“25] considered numerical solutions for fractional delay partial differential equations (FDPDEs). However, the researchers seldom considered the numerical methods for FDDEs with multidelays; some studies considered such FDDEs without theoretical analysis.
In this paper, there is an effective difference scheme for solving systems (3)â€“(5) with multidelay. Some stability and convergence results of the scheme are obtained by mathematical proof, and the theoretical results are proved by two numerical tests.
We organize the paper as the follows. Section 2 constructs a numerical scheme to solve (3)â€“(5). Section 3 provides the stability and convergence results by proof. Section 4 presents two numerical tests. Section 5 gives a brief discussion.
2. The Construction of the Second Finite Difference Method
The following assumptions are assumed to be true in this paper. (H1)Assume to be sufficiently smooth function which fulfills (H2)Suppose that is sufficiently smooth and satisfies [26, 27]where are arbitrary real numbers and , and are three constants.
Taking and to be two integers, we have space and time stepsize and , respectively, and discrete points and . Assume , , , and . Define the following grid function space on :
We have where
We introduce the following two Lemmas,
Lemma 1 [28]. Suppose , ; it holds that
Lemma 2 [29]. Assume ; then, it holds that (i) decrease monotonically with increases, and (ii),
Considering (3) at the point , we have
From Lemma 1, we obtain where
From Taylorâ€™s expansion, we have
Subtracting (13) and (14) and dividing the result by , we obtain where , , , and is a real number between and .
Substituting (11), (15), and (16) into (10), we obtain where
From Assumptions (H1) and (H2), we can easily obtain
We have the discretization of (4) and (5):
Replacing by in (17), (20), and (21), and omitting , we have the following numerical scheme:
3. The Solvability, Convergence, and Stability of the Difference Scheme
Assume the following to be defined on
If , the following inner products and corresponding norms are introduced
We can obtain Lemma 3 from Assumption (H1):
Lemma 3. For , we have .
Lemma 4 [30]. For , one can obtain
We also introduce Lemma 5 to be utilized in the following proof.
Lemma 5 [30]. Assuming the sequence satisfies then where and are two constants.
Theorem 6. Under the assumptions of (H1) and (H2), the solution of the difference schemes (22)â€“(24) is unique.
Proof. Difference schemes (22)â€“(24) can be reformed as follows:
When , we have
When , we have where , . From (30) and (31), we can see that the schemes (22)â€“(24) are a linear tridiagonal system with strictly diagonally dominant coefficient matrix. Thus, schemes (22)â€“(31) have a unique solution.
Denote , subtracting (22)â€“(24) from (17), (20), and (21), respectively, the following error equations can be obtained: where
Theorem 7. Denote to be the solution of systems (3)â€“(5), and (22)â€“(24) has numerical solution . Then, we have where is a positive constant independent of and .
Proof. Multiplying (32) with , one can have
Then each term of (36) will be estimated.
By the discrete Green formula, we have
From the CauchySchwarz inequality, we have
From the CauchySchwarz inequality and Assumption (H2), we have
Inserting (37)â€“(40) into (36), we obtain
Multiplying (41) by , and letting , we have
From Lemmas 3 and 4, and noticing (19), we obtain denoting
Noticing , one can obtain from . By (43), one can have
From Lemmas 2 and 5, we have where . From Lemma 4, we have
We complete the proof.
To discuss the stability of the difference schemes (22)â€“(24), we consider the following problem: where is the perturbation caused by . The following difference scheme solving for (48)â€“(50) can be obtained:
Denote where is a bounded constant independent of and .
Definition 8. Assume satisfy (22)â€“(24) and satisfy (51)â€“(53), a numerical scheme for (3)â€“(5) is stable if we have
Similar to the proof of Theorem 7, the following stability result can be obtained.
Theorem 9. Under the same condition as Theorem 7, numerical schemes (22)â€“(24) satisfy .
4. Numerical Test
Now, we present the following numerical tests to testify the schemes (22)â€“(24). Introducing the following notations, when considering for , should be fixed and small enough. While considering for , should be fixed and small enough.
Example 10. Let .
Equation (57) has exact solution , where
Table 1 presents the maximum errors in the temporal directions, where , and . From the results, we can draw a conclusion that the convergence order in the temporal directions is . Table 2 presents the maximum errors in the spatial directions, where , and . From the results, we can draw a conclusion that the convergence order in the spatial directions is 2.


Figure 1 gives the error planes of Example 10, where , and . Figure 1 tells us that bigger can bring bigger error.
Example 11. Let
Equation (59) has exact solution , where
Tables 3 and 4 show the computational results for a different of Example 11. When , the maximum errors and convergence orders are provided in Table 3. It can be shown that the order accuracy in temporal direction is verified, while when , from Table 4, we can see that the 2 order accuracy in spatial direction is verified.
