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

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 O(τ2−α + h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.


Introduction
In the past few years, more and more scholars have been attracted to the research of delay partial differential equations (DPDEs) [1][2][3].However, most DPDEs have no exact solutions.Constructing efficient numerical methods for DPDEs is of great importance [4][5][6][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  > 0 is a constant, 0 <  < 1.
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 ( 2− +ℎ 4 ).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.
For the time fractional derivative, we have the following lemma.

The Solvability, Convergence, and Stability of the Difference Scheme
Define the following grid function space on Ω ℎ : If , V ∈ V ℎ,0 , we introduce the following inner products and corresponding norms: It is easy to obtain the following lemma.
Lemma 5 (see [15]).∀ ∈ V ℎ,0 , one has The following lemma will be used in the proof of the stability and convergence analysis.

Numerical Test
In this section, a numerical test is used to validate the performance of scheme ( 24 For Rate  , we require ℎ to be fixed and small enough, while for Rate ℎ ,  should be fixed and small enough. the exact solution of (56) is (, ) =    2 (1 − ) 2  2+ , and ⋅ ( 2+ + ( − 0.1) 2+ ) . (57) From Table 1, we can see the maximum errors between the numerical solution and the exact solution in the temporal directions for  = 0.3, 0.5, 0.8, respectively, where the spatial step is fixed to be ℎ = 1/400.The results show that the temporal convergence order matches well the theoretical convergence order of 2 − .
Table 2 shows the maximum errors in the spatial directions for  = 0.2 when the temporal step is fixed at  = 1/2000.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  = 0.2, 0.4, 0.6, 0.8, respectively.From this figure, we can see that the error becomes larger when a larger  is taken.

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 ( 2− + ℎ 4 ).A numerical experiment is presented to support the theoretical results and validate the efficiency of the difference scheme.

Table 2 :
Maximum errors and convergence order in spatial direction with  = 1/2000 and  = 0.2 for Example 1.