Journal of Function Spaces
Volume 2017 (2017), Article ID 3679526, 8 pages
https://doi.org/10.1155/2017/3679526

Research Article

A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay

Wei Gu,¹ Yanli Zhou,² and Xiangyu Ge¹

¹School of Statistics & Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
²School of Finance, Zhongnan University of Economics and Law, Wuhan 430073, China

Correspondence should be addressed to Yanli Zhou; ylzhou8507@gmail.com and Xiangyu Ge; xiangyu_ge@163.com

Received 5 July 2017; Accepted 17 September 2017; Published 18 October 2017

Academic Editor: Xinguang Zhang

Copyright © 2017 Wei Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. Let

be 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 has

where .

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.

Table 1: Maximum errors and convergence order in temporal direction with and for Example 1.

1/100

1/200

1.744

1.519

1.203

1/300

1.739

1.515

1.202

1/400

1.737

1.513

1.202

Table 2: Maximum errors and convergence order in spatial direction with and for Example 1.

1/8

1/12

3.988

1/16

4.011

1/20

3.985

Figure 1: Error planes for for different in Example 1.

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).

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