Abstract
A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be . A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.
1. Introduction
In the past few years, more and more scholars have been attracted to the research of delay partial differential equations (DPDEs) [1–3]. However, most DPDEs have no exact solutions. Constructing efficient numerical methods for DPDEs is of great importance [4–7]. For details on numerically solving neutral delay parabolic differential equations (NDPDEs), the reader is referred to [5, 8]. Recently, fractional delay partial differential equations have been of great interest due to their application in automatic control, population dynamics, economics, and so forth [9, 10]. For details on numerical solutions to fractional delay partial differential equations, we refer the reader to [11, 12]. The work in [11] considers the numerical method without theoretical analysis, and the work in [12] considers the numerical method for a type of semilinear fractional partial differential equation with time delay.
In this paper, we consider the following fractional neutral parabolic differential equation with proportional delay: where is a constant, .
Let for , where . Then, satisfies the following equation:where .
For simplicity, we consider the following fractional neutral parabolic differential equation with delay instead of (1):where is a constant delay term. Time fractional partial derivative is defined in the Caputo sense by the following:where is the Gamma function.
In this paper, a linearized compact finite difference scheme is constructed for solving (3)–(5). By the energy method, the unconditional stability of the scheme is then proved, and the convergence order of the scheme is proved to be . A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.
The rest of the paper is organized as follows. In Section 2, a compact difference scheme is constructed to solve (3)–(5). Section 3 considers the solvability, convergence, and stability of the provided difference scheme. In Section 4, a numerical test is presented to illustrate the validity of the theoretical results. Section 5 gives a brief conclusion of this paper.
2. The Construction of the Compact Difference Scheme
Throughout this paper, assume . Function is sufficiently smooth and satisfies where are arbitrary real numbers and and are positive constants.
First, let and be two positive integers; then, we take , ( is a positive integer), , . Define , where , , . Denote throughout this paper. Letbe the grid function space defined on . The following notations are used: where
For the time fractional derivative, we have the following lemma.
Lemma 1 (see [13]). Suppose , ; it holds that and satisfies the following lemma.
Lemma 2 (see [14]). Assume ; then, it holds that(1) decreases monotonically as increases, and ;(2), .
Lemma 3 (see [5]). Suppose ; then, one haswhere .
Considering (3) at the point , we have
From Lemma 1, we obtain where
From Taylor expansion, we have where , in between and .
Substituting (13) and (15) into (12) and applying the operator on both sides of (12), we obtain where From Lemma 3 and Taylor expansion, we have Substituting (18) into (16), we have where
Noticing and (7), we can easily obtain
Discretizing the initial and boundary conditions of (4) and (5), we obtain Replacing by in (19), (22), and (23) and omitting , we can obtain the following compact difference scheme:
3. The Solvability, Convergence, and Stability of the Difference Scheme
Define the following grid function space on : If , we introduce the following inner products and corresponding norms:
It is easy to obtain the following lemma.
Lemma 4 (see [12]). , one has .
Lemma 5 (see [15]). , one has
The following lemma will be used in the proof of the stability and convergence analysis.
Lemma 6 (see [15]). Assume that is a nonnegative sequence and satisfies then, where and are nonnegative constants.
Theorem 7. The difference scheme (24)–(26) has a unique solution.
Proof. Denote ; the difference scheme (24)–(26) is a linear tridiagonal system , where only depends on , , and and is independent of . where . We can see that is a strictly diagonally dominant coefficient matrix. Thus, scheme (24)–(26) has a unique solution.
To discuss the stability of the difference scheme (24)–(26), we consider the following problem: where is the perturbation caused by . The following difference scheme can be obtained for solving (33):
Denote
Definition 8. Assume that satisfies (24)–(26) and satisfies (34)–(36); then, a numerical scheme for (3)–(5) is stable if one has where is a bounded constant independent of and .
Theorem 9. Assume is the solution of (3)–(5); the difference scheme (24)–(26) is stable with respect to the initial perturbation of ; that is, , where is a positive constant independent of and .
Proof. Subtracting (34)–(36) from (24)–(26), respectively, we can obtain the following equations: where
Multiplying on both sides of (39) and summing up for from to , we obtain Then, each term of (42) will be estimated. From the discrete Green formula and inequality , we have From the discrete Green formula and inequality , we have From the Cauchy-Schwarz inequality, Lemma 4, and (7), we have Substituting (43)–(45) into (42) and taking in (45), we obtain where Lemma 5 has been used.
Multiplying (46) by , we have Denote Noticing for , we have . Then, from (47), we obtain From Lemmas 2 and 6, we have From Lemma 5, we have The proof is completed.
Denote ; by subtracting (24)–(26) from (19), (22), and (23), respectively, the following error equations can be obtained: where
Similar to the proof of Theorem 9, the following convergence result can be obtained.
Theorem 10. Assume is the solution of (3)–(5) and is the solution of (24)–(26). Then, one has where is a positive constant independent of and .
4. Numerical Test
In this section, a numerical test is used to validate the performance of scheme (24)–(26). Denote the maximum error at all grid points as the convergence order in time and space is defined, respectively, as For , we require to be fixed and small enough, while for , should be fixed and small enough.
Example 1. Consider the following problem: the exact solution of (56) is , and From Table 1, we can see the maximum errors between the numerical solution and the exact solution in the temporal directions for , respectively, where the spatial step is fixed to be . The results show that the temporal convergence order matches well the theoretical convergence order of .
Table 2 shows the maximum errors in the spatial directions for when the temporal step is fixed at . From the results, we can see that the spatial convergence order is 4, which coincides with the theoretical result.
Figure 1 gives the error plane for , respectively. From this figure, we can see that the error becomes larger when a larger is taken.
5. Conclusion
This paper presents a compact finite difference scheme for solving the fractional neutral parabolic differential equation with proportional delay. The unconditional stability and the global convergence of the scheme in the maximum norm are proved. The convergence order of the considered scheme is . A numerical experiment is presented to support the theoretical results and validate the efficiency of the difference scheme.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research work is supported by the National Natural Science Foundation of China (11401591) and the Humanities and Social Science Foundation of the Ministry of Education of China (17YJC630236). The first author acknowledges the Project of the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (2013693).