A Compact Difference Scheme for Solving Fractional Neutral ...

Hindawi Journal of Function Spaces Volume 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 1

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

2

Correspondence should be addressed to Yanli Zhou; [email protected] and Xiangyu Ge; xiangyu [email protected] 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. 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.

1. Introduction

𝑤 (0, 𝑡) = 𝛾𝑤 (𝑡) ,

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:

𝑤 (1, 𝑡) = 𝛽𝑤 (𝑡) ,

2 𝜕2 𝑤 (𝑥, 𝑡) 𝜕 𝑤 (𝑥, 𝑝𝑡) 𝜕𝛼 𝑤 = 𝐷( + ) 𝛼 2 𝜕𝑡 𝜕𝑥 𝜕𝑥2

𝑡 ∈ (0, 𝑇] , (1) where 𝐷 > 0 is a constant, 0 < 𝑝 < 1. Let 𝑢(𝑥, 𝑡) = 𝑤(𝑥, 𝑒𝑡 ) for 𝑡 ≥ 𝑡0 + ln(𝑝), where 𝑡0 ≥ 0. Then, 𝑢(𝑥, 𝑡) satisfies the following equation: 𝜕𝛼 𝑢 𝜕2 𝑢 (𝑥, 𝑡) 𝜕2 𝑢 (𝑥, 𝑡 − 𝑠) = 𝐷 ( + ) 𝜕𝑡𝛼 𝜕𝑥2 𝜕𝑥2 + 𝑓𝑤 (𝑢 (𝑥, 𝑡) , 𝑢 (𝑥, 𝑡 − 𝑠) , 𝑥, 𝑒𝑡 ) , (𝑥, 𝑡) ∈ (0, 1) × (𝑡0 , 𝑇] , 𝑢 (𝑥, 𝑡) = 𝜙𝑤 (𝑥, 𝑒𝑡 ) , 𝑢 (0, 𝑡) = 𝛾𝑤 (𝑒𝑡 ) , 𝑢 (1, 𝑡) = 𝛽𝑤 (𝑒𝑡 ) , 𝑡 ∈ (𝑡0 , 𝑇] ,

+ 𝑓𝑤 (𝑤 (𝑥, 𝑡) , 𝑤 (𝑥, 𝑝𝑡) , 𝑥, 𝑡) , (𝑥, 𝑡) ∈ (0, 1) × [0, 𝑇] ,

𝑥 ∈ [0, 1] , 𝑡 ∈ [−𝑠, 𝑡0 ] ,

where 𝑠 = − ln(𝑝).

(2)

2

Journal of Function Spaces

For simplicity, we consider the following fractional neutral parabolic differential equation with delay instead of (1): 𝜕𝛼 𝑢 𝜕2 𝑢 (𝑥, 𝑡) 𝜕2 𝑢 (𝑥, 𝑡 − 𝑠) = 𝐷 ( + ) 𝜕𝑡𝛼 𝜕𝑥2 𝜕𝑥2

𝑘 = 𝛿𝑥 V𝑖+1/2

(3)

+ 𝑓 (𝑢 (𝑥, 𝑡) , 𝑢 (𝑥, 𝑡 − 𝑠) , 𝑥, 𝑡) , (𝑥, 𝑡) ∈ (0, 1) × (0, 𝑇] , 𝑢 (𝑥, 𝑡) = 𝜙 (𝑥, 𝑡) ,

be the grid function space defined on Ωℎ𝜏 . The following notations are used:

𝑥 ∈ [0, 1] , 𝑡 ∈ [−𝑠, 0] ,

(4)

𝑢 (0, 𝑡) = 𝛾 (𝑡) , 𝑢 (1, 𝑡) = 𝛽 (𝑡) ,

(5) 𝑡 ∈ (0, 𝑇] ,

where 𝑠 > 0 is a constant delay term. Time fractional partial derivative (𝜕𝛼 𝑢/𝜕𝑡𝛼 ) (0 < 𝛼 < 1) is defined in the Caputo sense by the following: 𝑡 𝜕𝛼 𝑢 1 −𝛼 𝜕𝑢 (𝑥, 𝜉) = 𝑑𝜉, ∫ (𝑡 − 𝜉) 𝜕𝑡𝛼 Γ (1 − 𝛼) 0 𝜕𝜉

(6)

where Γ( ) 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 𝑂(𝜏2−𝛼 +ℎ4 ). 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

𝛿𝑥2 V𝑖𝑘 =

𝑘 𝑘 − 2V𝑖𝑘 + V𝑖−1 V𝑖+1 , ℎ2

AV𝑖𝑘 =

1 𝑘 𝑘 ), (V + 10V𝑖𝑘 + V𝑖+1 12 𝑖−1

𝛿𝑡𝛼 V𝑘 =

−𝛼

(9)

𝑘−1

𝜏 [𝑑0 V𝑘 − ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) V𝑗 Γ (2 − 𝛼) 𝑗=1 [

− 𝑑𝑘−1 V0 ] , ] where 𝑑𝑗 = (𝑗 + 1)1−𝛼 − 𝑗1−𝛼 , 𝑗 ≥ 0. For the time fractional derivative, we have the following lemma. Lemma 1 (see [13]). Suppose 0 < 𝛼 < 1, 𝑦 ∈ 𝐶2 [0, 𝑡𝑘 ]; it holds that 󵄨󵄨 󵄨󵄨 𝑡𝑘 𝑦󸀠 (𝑠) 𝑑𝑠 1 𝜏−𝛼 [ 󵄨󵄨 𝑑0 𝑦 (𝑡𝑘 ) ∫ 󵄨󵄨 𝛼 − 󵄨󵄨 Γ (1 − 𝛼) 0 (𝑡𝑘 − 𝑠) Γ (2 − 𝛼) 󵄨󵄨 [ 󵄨󵄨 󵄨󵄨 󵄨 − ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 𝑦 (𝑡𝑗 ) − 𝑑𝑘−1 𝑦 (𝑡0 )]󵄨󵄨󵄨 󵄨󵄨 𝑗=1 ]󵄨󵄨 𝑘−1

≤

(10)

1 − 𝛼 22−𝛼 1 [ + − (1 + 2−𝛼 )] Γ (2 − 𝛼) 12 2−𝛼

󵄨 󵄨 ⋅ max 󵄨󵄨󵄨󵄨𝑦󸀠󸀠 (𝑡)󵄨󵄨󵄨󵄨 𝜏2−𝛼 , 0≤𝑡≤𝑡𝑘 and 𝑑𝑗 satisfies the following lemma.

Throughout this paper, assume 𝑢(𝑥, 𝑡) ∈ 𝐶6,2 ([0, 1] × (0, 𝑇]). Function 𝑓(𝑢(𝑥, 𝑡), 𝑢(𝑥, 𝑡 − 𝑠), 𝑥, 𝑡) is sufficiently smooth and satisfies 󵄨󵄨󵄨𝑓 (𝜇 + 𝜖1 , ] + 𝜖2 , 𝑥, 𝑡) − 𝑓 (𝜇, ], 𝑥, 𝑡)󵄨󵄨󵄨 󵄨 󵄨 (7) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ 𝑐0 󵄨󵄨𝜖1 󵄨󵄨 + 𝑐1 󵄨󵄨𝜖2 󵄨󵄨 , where 𝜖1 , 𝜖2 are arbitrary real numbers and 𝑐0 and 𝑐1 are positive constants. First, let 𝑀 and 𝑁 be two positive integers; then, we take ℎ = 1/𝑀, 𝜏 = 𝑠/𝑛 (𝑛 > 0 is a positive integer), 𝑥𝑖 = 𝑖ℎ, 𝑡𝑘 = 𝑘𝜏. Define Ωℎ𝜏 = Ωℎ × Ω𝜏 , where Ωℎ = {𝑥𝑖 | 0 ≤ 𝑖 ≤ 𝑀}, Ω𝜏 = {𝑡𝑘 | −𝑛 ≤ 𝑘 ≤ 𝑁}, 𝑁 = [𝑇/𝜏]. Denote 𝑈𝑖𝑘 = 𝑢(𝑥𝑖 , 𝑡𝑘 ), 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 𝑁 throughout this paper. Let W = {V𝑖𝑘 | 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 𝑁}

𝑘 − V𝑖𝑘 V𝑖+1 , ℎ

(8)

Lemma 2 (see [14]). Assume 0 < 𝛼 < 1; then, it holds that (1) 𝑑𝑗 decreases monotonically as 𝑗 increases, and 0 < 𝑑𝑗 ≤ 1; (2) 𝑑0 = 1, ∑𝑘−1 𝑗=1 (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) = 𝑑0 − 𝑑𝑘−1 . Lemma 3 (see [5]). Suppose 𝑦(𝑥) ∈ 𝐶6 [𝑥𝑖−1 , 𝑥𝑖+1 ]; then, one has 1 [𝑦󸀠󸀠 (𝑥𝑖−1 ) + 10𝑦󸀠󸀠 (𝑥𝑖 ) + 𝑦󸀠󸀠 (𝑥𝑖+1 )] 12 − =

1 [𝑦 (𝑥𝑖−1 ) − 2𝑦 (𝑥𝑖 ) + 𝑦 (𝑥𝑖+1 )] ℎ2

ℎ4 (6) 𝑦 (𝜄𝑖 ) , 240

where 𝜄𝑖 ∈ (𝑥𝑖−1 , 𝑥𝑖+1 ).

(11)


3

Considering (3) at the point (𝑥𝑖 , 𝑡𝑘 ), we have

where

𝜕𝛼 𝑢 𝜕2 𝑢 𝜕2 𝑢 (𝑥 , 𝑡 ) = 𝐷 ( (𝑥 , 𝑡 ) + (𝑥 , 𝑡 − 𝑠)) 𝑖 𝑘 𝑖 𝑘 𝜕𝑡𝛼 𝜕𝑥2 𝜕𝑥2 𝑖 𝑘 + 𝑓 (𝑢 (𝑥𝑖 , 𝑡𝑘 ) , 𝑢 (𝑥𝑖 , 𝑡𝑘 − 𝑠) , 𝑥𝑖 , 𝑡𝑘 ) ,

𝑘 + 𝑅𝑖𝑘 = A𝑅0𝑖

From Lemma 1, we obtain 𝜕𝛼 𝑢 (𝑥 , 𝑡 ) = 𝛿𝑡𝛼 𝑈𝑖𝑘 + 𝑟𝑖𝑘 , 𝜕𝑡𝛼 𝑖 𝑘

(13)

where 𝑟𝑖𝑘 =

1 1 − 𝛼 22−𝛼 [ + − (1 + 2−𝛼 )] Γ (2 − 𝛼) 12 2−𝛼 󵄨󵄨 𝜕2 𝑢 (𝑥 , 𝑡 ) 󵄨󵄨 󵄨 𝑖 𝑘 󵄨󵄨 2−𝛼 󵄨󵄨 𝜏 . ⋅ max 󵄨󵄨󵄨󵄨 2 󵄨󵄨 0≤𝑡≤𝑡𝑘 󵄨 𝜕𝑡 󵄨 󵄨

(14)

Replacing 𝑈𝑖𝑘 by 𝑢𝑖𝑘 in (19), (22), and (23) and omitting 𝑅𝑖𝑘 , we can obtain the following compact difference scheme: (15)

2

where 𝜂𝑘 ∈ (𝑡𝑘−2 , 𝑡𝑘 ), 󰜚𝑖𝑘 in between 𝑢(𝑥𝑖 , 𝑡𝑘 ) and 2𝑈𝑖𝑘−1 −𝑈𝑖𝑘−2 . Substituting (13) and (15) into (12) and applying the operator A on both sides of (12), we obtain 𝜕2 𝑢 𝜕2 𝑢 A𝛿𝑡𝛼 𝑈𝑖𝑘 = 𝐷 (A 2 (𝑥𝑖 , 𝑡𝑘 ) + A 2 (𝑥𝑖 , 𝑡𝑘 − 𝑠)) 𝜕𝑥 𝜕𝑥

(17)

𝜕2 𝑢 ℎ4 𝜕6 𝑢 𝑘 2 𝑘 (𝑥 , 𝑡 ) = 𝛿 𝑈 + (𝜃 , 𝑡 ) , 𝑥 𝑖 𝜕𝑥2 𝑖 𝑘 240 𝜕𝑥6 𝑖 𝑘

𝑢0𝑘 = 𝛾 (𝑡𝑘 ) , 𝑘 = 𝛽 (𝑡𝑘 ) , 𝑢𝑀

(26) 1 ≤ 𝑘 ≤ 𝑁.

Define the following grid function space on Ωℎ : Vℎ,0 = {𝑢 | 𝑢 = (𝑢0 , 𝑢1 , . . . , 𝑢𝑀) , 𝑢0 = 𝑢𝑀 = 0} .

(27)

𝑀−1

𝜃𝑖𝑘 ∈ (𝑥𝑖−1 , 𝑥𝑖+1 ) , ℎ4 𝜕6 𝑢 𝑘−𝑛 𝜕2 𝑢 (𝜃 , 𝑡𝑘−𝑛 ) , A 2 (𝑥𝑖 , 𝑡𝑘 − 𝑠) = 𝛿𝑥2 𝑈𝑖𝑘−𝑛 + 𝜕𝑥 240 𝜕𝑥6 𝑖

(𝑢, V) = ℎ ∑ 𝑢𝑖 V𝑖 , 𝑖=1

(18)

󵄨 󵄨 ‖𝑢‖ = √(𝑢, 𝑢), ‖𝑢‖∞ = max 󵄨󵄨󵄨𝑢𝑖 󵄨󵄨󵄨 , 1≤𝑖≤𝑀−1 𝑀−1

⟨𝛿𝑥 𝑢, 𝛿𝑥 V⟩ = ℎ ∑ (𝛿𝑥 𝑢𝑖+1/2 ) (𝛿𝑥 V𝑖+1/2 ) ,

𝜃𝑖𝑘−𝑛 ∈ (𝑥𝑖−1 , 𝑥𝑖+1 ) .

𝑖=0

Substituting (18) into (16), we have

󵄨󵄨 󵄨󵄨 󵄨󵄨𝛿𝑥 𝑢󵄨󵄨1 = √⟨𝛿𝑥 𝑢, 𝛿𝑥 𝑢⟩.

A𝛿𝑡𝛼 𝑈𝑖𝑘 = 𝐷 (𝛿𝑥2 𝑈𝑖𝑘 + 𝛿𝑥2 𝑈𝑖𝑘−𝑛 ) −

(25)

If 𝑢, V ∈ Vℎ,0 , we introduce the following inner products and corresponding norms:

From Lemma 3 and Taylor expansion, we have

+

(24)

3. The Solvability, Convergence, and Stability of the Difference Scheme

where

𝑈𝑖𝑘−2 , 𝑈𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 )

𝑢𝑖𝑘 = 𝜙 (𝑥𝑖 , 𝑡𝑘 ) , 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 0,

(16)

𝑘 + A𝑓 (2𝑈𝑖𝑘−1 − 𝑈𝑖𝑘−2 , 𝑈𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ) + A𝑅0𝑖 ,

𝜕2 𝑢 (𝑥 , 𝜂𝑘 ) 𝑓𝜇 (󰜚𝑖𝑘 , 𝑈𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ) . 𝜕𝑡2 𝑖

A𝛿𝑡𝛼 𝑢𝑖𝑘 = 𝐷 (𝛿𝑥2 𝑢𝑖𝑘 + 𝛿𝑥2 𝑢𝑖𝑘−𝑛 ) + A𝑓 (2𝑢𝑖𝑘−1 − 𝑢𝑖𝑘−2 , 𝑢𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ) ,

𝑢 + 𝜏 2 (𝑥𝑖 , 𝜂𝑘 ) 𝑓𝜇 (󰜚𝑖𝑘 , 𝑈𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ) , 𝜕𝑡

A𝑓 (2𝑈𝑖𝑘−1

(23) 1 ≤ 𝑘 ≤ 𝑁.

= 𝑓 (2𝑈𝑖𝑘−1 − 𝑈𝑖𝑘−2 , 𝑈𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 )

A

(22)

𝑈0𝑘 = 𝛾 (𝑡𝑘 ) , 𝑘 = 𝛽 (𝑡𝑘 ) , 𝑈𝑀

𝑓 (𝑢 (𝑥𝑖 , 𝑡𝑘 ) , 𝑢 (𝑥𝑖 , 𝑡𝑘 − 𝑠) , 𝑥𝑖 , 𝑡𝑘 )

𝑘 = 𝑟𝑖𝑘 + 𝜏2 𝑅0𝑖

(20)

Noticing 𝑢(𝑥, 𝑡) ∈ 𝐶6,2 ([0, 1] × (0, 𝑇]) and (7), we can easily obtain 󵄨󵄨󵄨𝑅𝑘 󵄨󵄨󵄨 ≤ 𝐶 (𝜏2−𝛼 + ℎ4 ) , 0 ≤ 𝑖 ≤ 𝑀, 1 ≤ 𝑘 ≤ 𝑁. (21) 󵄨󵄨 𝑖 󵄨󵄨 𝑅 Discretizing the initial and boundary conditions of (4) and (5), we obtain 𝑈𝑖𝑘 = 𝜙 (𝑥𝑖 , 𝑡𝑘 ) , 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 0,

From Taylor expansion, we have

2𝜕

ℎ4 𝜕6 𝑢 𝑘−𝑛 (𝜃 , 𝑡𝑘−𝑛 ) . + 240 𝜕𝑥6 𝑖

(12)

0 ≤ 𝑖 ≤ 𝑀, 0 ≤ 𝑘 ≤ 𝑁.

ℎ4 𝜕6 𝑢 𝑘 (𝜃 , 𝑡 ) 240 𝜕𝑥6 𝑖 𝑘

+

𝑅𝑖𝑘 ,

(19)

It is easy to obtain the following lemma. Lemma 4 (see [12]). ∀𝑢 ∈ Vℎ,0 , one has ‖A𝑢‖2 ≤ ‖𝑢‖2 .

(28)

4


Lemma 5 (see [15]). ∀𝑢 ∈ Vℎ,0 , one has

where 𝜓(𝑥, 𝑡) is the perturbation caused by 𝜙(𝑥, 𝑡). The following difference scheme can be obtained for solving (33):

󵄨󵄨 󵄨󵄨 󵄨𝛿𝑥 𝑢󵄨 ‖𝑢‖∞ ≤ 󵄨 󵄨1 , 2 󵄨󵄨 󵄨󵄨 󵄨𝛿𝑥 𝑢󵄨 ‖𝑢‖ ≤ 󵄨 󵄨1 . √6

A𝛿𝑡𝛼 V𝑖𝑘 = 𝐷 (𝛿𝑥2 V𝑖𝑘 + 𝛿𝑥2 V𝑖𝑘−𝑛 ) + A𝑓 (2V𝑖𝑘−1 − V𝑖𝑘−2 , V𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ) ,

(29)

V𝑖𝑘 = 𝜙 (𝑥𝑖 , 𝑡𝑘 ) + 𝜓𝑖𝑘 ,

The following lemma will be used in the proof of the stability and convergence analysis. 𝑘

Lemma 6 (see [15]). Assume that {𝐹 | 𝑘 ≥ 0} is a nonnegative sequence and satisfies

0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 0,

(34) (35)

V0𝑘 = 𝛾 (𝑡𝑘 ) , 𝑘 = 𝛽 (𝑡𝑘 ) , V𝑀

(36) 1 ≤ 𝑘 ≤ 𝑁.

Denote 𝑘

𝐹𝑘+1 ≤ 𝐴 + 𝐵𝜏∑𝐹𝑙 ,

𝑘 = 0, 1, . . . ;

𝑖=1

then, 𝐹𝑘+1 ≤ 𝐴 exp (𝐵𝑘𝜏) , 𝑘 = 0, 1, 2, . . . ,

𝜔𝑖𝑘 = V𝑖𝑘 − 𝑢𝑖𝑘 ,

(30)

(31)

0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 𝑁.

(37)

Definition 8. Assume that 𝑢𝑖𝑘 satisfies (24)–(26) and V𝑖𝑘 satisfies (34)–(36); then, a numerical scheme for (3)–(5) is stable if one has 󵄩󵄩 𝑘 󵄩󵄩 󵄨 󵄨 󵄩󵄩𝜔 󵄩󵄩 ≤ 𝐶 max 󵄨󵄨󵄨𝜓𝑗 󵄨󵄨󵄨 , (38) 󵄩 󵄩∞ −𝑛≤𝑗≤0 󵄨 󵄨1 where 𝐶 is a bounded constant independent of ℎ and 𝜏.

where 𝐴 and 𝐵 are nonnegative constants. Theorem 7. The difference scheme (24)–(26) has a unique solution. 𝑘 Proof. Denote 𝑢𝑘 = [𝑢1𝑘 , 𝑢2𝑘 , . . . , 𝑢𝑀−1 ]𝑇 ; the difference scheme (24)–(26) is a linear tridiagonal system M𝑢𝑘 = 𝑑, 𝑘 and is where 𝑑 only depends on 𝑢𝑗 (𝑗 ≤ 𝑘 − 1), 𝑢0𝑘 , and 𝑢𝑀 𝑘 independent of 𝑢 .

1 𝐷 10 2𝐷 1 𝐷 M = tridiag ( , − , + − ), 12𝜆 ℎ2 12𝜆 ℎ2 12𝜆 ℎ2

(32)

where 𝜆 = 𝜏𝛼 Γ(2 − 𝛼). We can see that M is a strictly diagonally dominant coefficient matrix. Thus, scheme (24)–(26) has a unique solution.

Theorem 9. Assume 𝑢(𝑥, 𝑡) ∈ 𝐶6,2 ([0, 1] × (−𝑠, 𝑇]) is the solution of (3)–(5); the difference scheme (24)–(26) is stable with respect to the initial perturbation of 𝜓𝑖𝑘 ; that is, ‖𝜔𝑘 ‖∞ ≤ ̂ max−𝑛≤𝑗≤0 |𝜓𝑗 |1 , where 𝐶 ̂ is a positive constant independent 𝐶 of ℎ and 𝜏. Proof. Subtracting (34)–(36) from (24)–(26), respectively, we can obtain the following equations: A𝛿𝑡𝛼 𝜔𝑖𝑘 = 𝐷 (𝛿𝑥2 𝜔𝑖𝑘 + 𝛿𝑥2 𝜔𝑖𝑘−𝑛 ) + A𝑞𝑖𝑘 ,

(39)

𝜔𝑖𝑘 = 𝜓𝑖𝑘 , 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 0,

(40)

𝜔0𝑘 = 0, 𝑘 = 0, 𝜔𝑀

(41) 1 ≤ 𝑘 ≤ 𝑁,

To discuss the stability of the difference scheme (24)–(26), we consider the following problem: A

𝜕𝛼 V 𝜕2 V (𝑥, 𝑡) 𝜕2 V (𝑥, 𝑡 − 𝑠) = 𝐷 ( + ) 𝜕𝑡𝛼 𝜕𝑥2 𝜕𝑥2

where 𝑞𝑖𝑘 = 𝑓(2V𝑖𝑘−1 − V𝑖𝑘−2 , V𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ) − 𝑓(2𝑢𝑖𝑘−1 − 𝑢𝑖𝑘−2 , 𝑢𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ). Multiplying ℎ𝛿𝑡𝛼 𝜔𝑖𝑘 on both sides of (39) and summing up for 𝑖 from 1 to 𝑀 − 1, we obtain 𝑀−1

ℎ ∑ (A𝛿𝑡𝛼 𝜔𝑖𝑘 ) (𝛿𝑡𝛼 𝜔𝑖𝑘 )

+ A𝑓 (V (𝑥, 𝑡) , V (𝑥, 𝑡 − 𝑠) , 𝑥, 𝑡) ,

𝑖=1

(𝑥, 𝑡) ∈ (0, 1) × (0, 𝑇] , V (𝑥, 𝑡) = 𝜙 (𝑥, 𝑡) + 𝜓 (𝑥, 𝑡) , 𝑥 ∈ [0, 1] , 𝑡 ∈ [−𝑠, 0] , V (0, 𝑡) = 𝛾 (𝑡) ,

𝑀−1

(33)

= 𝐷ℎ ∑ (𝛿𝑥2 𝜔𝑖𝑘 + 𝛿𝑥2 𝜔𝑖𝑘−𝑛 ) (𝛿𝑡𝛼 𝜔𝑖𝑘 ) 𝑖=1

𝑀−1

+ ℎ ∑ (A𝑞𝑖𝑘 ) (𝛿𝑡𝛼 𝜔𝑖𝑘 ) ,

𝑢 (1, 𝑡) = 𝛽 (𝑡) ,

𝑖=1

𝑡 ∈ (0, 𝑇] ,

1 ≤ 𝑖 ≤ 𝑀 − 1, 0 ≤ 𝑘 ≤ 𝑁 − 1.

(42)


5 𝑀−1

Then, each term of (42) will be estimated. From the discrete Green formula and inequality −(𝑎+𝑏)2 ≥ −2(𝑎2 +𝑏2 ), we have

𝑘−𝑛 𝑘 ) [𝛿𝑥 𝜔𝑖+1/2 ⋅ ℎ ∑ (𝛿𝑥 𝜔𝑖+1/2 𝑖=0 [

𝑀−1

ℎ ∑ (A𝛿𝑡𝛼 𝜔𝑖𝑘 ) (𝛿𝑡𝛼 𝜔𝑖𝑘 )

𝑘−1

󵄩 󵄩2 ℎ2 𝑀−1 = 󵄩󵄩󵄩󵄩𝛿𝑡𝛼 𝜔𝑘 󵄩󵄩󵄩󵄩 + ℎ ∑ (𝛿𝑥2 (𝛿𝑡𝛼 𝜔𝑖𝑘 )) (𝛿𝑡𝛼 𝜔𝑖𝑘 ) 12 𝑖=1

(43)

≤

󵄩2 ℎ2 󵄩󵄩 𝛼 𝑘 󵄩󵄩2 2 󵄩󵄩 𝛼 𝑘 󵄩󵄩2 󵄩 󵄩󵄩𝛿 𝛿 𝜔 󵄩󵄩 ≥ 󵄩󵄩𝛿𝑡 𝜔 󵄩󵄩 . = 󵄩󵄩󵄩󵄩𝛿𝑡𝛼 𝜔𝑘 󵄩󵄩󵄩󵄩 − 󵄩 12 󵄩 𝑥 𝑡 󵄩 3󵄩

󵄨2 󵄨 󵄨2 󵄨󵄨 󵄨󵄨𝛿𝑥 𝜔𝑘−𝑛 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛿𝑥 𝜔𝑗 󵄨󵄨󵄨 󵄨 󵄨1 󵄨 󵄨 1 + ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 2 𝑗=1

From the discrete Green formula and inequality −𝑎𝑏 ≥ −(𝑎 + 𝑏2 )/2, we have 𝐷ℎ ∑ 𝑖=1

󵄨󵄨 𝑘 󵄨󵄨2 󵄨󵄨󵄨𝛿 𝜔𝑘−𝑛 󵄨󵄨󵄨2 + 󵄨󵄨󵄨𝛿 𝜔0 󵄨󵄨󵄨2 } 󵄨 𝑥 󵄨󵄨1 󵄨󵄨 𝑥 󵄨󵄨1 𝐷 { 󵄨󵄨󵄨𝛿𝑥 𝜔 󵄨󵄨󵄨1 + 𝑑𝑘−1 󵄨 = } 𝜆{ 4 2 } {

(𝛿𝑥2 𝜔𝑖𝑘 ) (𝛿𝑡𝛼 𝜔𝑖𝑘 ) 𝑀−1

𝑘 𝑘 = −𝐷ℎ ∑ (𝛿𝑥 𝜔𝑖+1/2 ) (𝛿𝑡𝛼 𝛿𝑥 𝜔𝑖+1/2 )=− 𝑖=0

𝐷 𝜆

+

𝑀−1

𝑘 𝑘 ⋅ ℎ ∑ (𝛿𝑥 𝜔𝑖+1/2 ) [𝛿𝑥 𝜔𝑖+1/2 𝑖=0 [ 𝑘−1

𝐷 {󵄨󵄨 󵄨2 󵄨󵄨𝛿𝑥 𝜔𝑘 󵄨󵄨󵄨 { 󵄨 󵄨1 𝜆 {

From the Cauchy-Schwarz inequality, Lemma 4, and (7), we have 𝑀−1

ℎ ∑ (A𝑞𝑖𝑘 ) (𝛿𝑡𝛼 𝜔𝑖𝑘 ) 𝑖=1

− ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) ⟨𝛿𝑥 𝜔𝑘 , 𝛿𝑥 𝜔𝑗 ⟩

≤

𝑗=1

0

≤

󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨󵄨 𝑘 󵄨󵄨2 󵄨󵄨𝛿𝑥 𝜔𝑘 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛿𝑥 𝜔0 󵄨󵄨󵄨 } 𝐷 { 󵄨󵄨󵄨𝛿𝑥 𝜔 󵄨󵄨󵄨1 󵄨 󵄨 󵄨 󵄨 1 1 − 𝑑𝑘−1 }=−𝜆 { 2 2 } { 2 󵄨 󵄨󵄨 󵄨2 󵄨 󵄨󵄨𝛿 𝜔𝑗 󵄨󵄨 𝑘−1 󵄨󵄨𝛿𝑥 𝜔0 󵄨󵄨󵄨 } 󵄨󵄨 𝑥 󵄨󵄨1 󵄨 󵄨1 , − ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) − 𝑑𝑘−1 } 2 2 𝑗=1 }

≤

𝑀−1 2 1 󵄩󵄩 𝛼 𝑘 󵄩󵄩2 󵄩󵄩𝛿𝑡 𝜔 󵄩󵄩 + 𝜀𝑐02 ℎ ∑ (2𝜔𝑖𝑘−1 − 𝜔𝑖𝑘−2 ) 󵄩 󵄩 2𝜀 𝑖=1 𝑀−1

+ 𝜀𝑐12 ℎ ∑ (𝜔𝑖𝑘−𝑛 )

𝑀−1

2

𝑖=1

𝑖=1

𝑖=0

1 󵄩󵄩 𝛼 𝑘 󵄩󵄩2 󵄩󵄩𝛿 𝜔 󵄩󵄩 2𝜀 󵄩 𝑡 󵄩 𝜀 𝑀−1 󵄨 󵄨 󵄨 󵄨 2 + ℎ ∑ (𝑐0 󵄨󵄨󵄨󵄨2𝜔𝑖𝑘−1 − 𝜔𝑖𝑘−2 󵄨󵄨󵄨󵄨 + 𝑐1 󵄨󵄨󵄨󵄨𝜔𝑖𝑘−𝑛 󵄨󵄨󵄨󵄨) 2 𝑖=1

𝐷ℎ ∑ (𝛿𝑥2 𝜔𝑖𝑘−𝑛 ) (𝛿𝑡𝛼 𝜔𝑖𝑘 ) 𝑘−𝑛 𝑘 = −𝐷ℎ ∑ (𝛿𝑥 𝜔𝑖+1/2 ) (𝛿𝑡𝛼 𝛿𝑥 𝜔𝑖+1/2 )=−

1 󵄩󵄩 𝛼 𝑘 󵄩󵄩2 󵄩󵄩𝛿 𝜔 󵄩󵄩 2𝜀 󵄩 𝑡 󵄩 𝜀 𝑀−1 󵄨 󵄨 󵄨 󵄨 2 + ℎ ∑ (A [𝑐0 󵄨󵄨󵄨󵄨2𝜔𝑖𝑘−1 − 𝜔𝑖𝑘−2 󵄨󵄨󵄨󵄨 + 𝑐1 󵄨󵄨󵄨󵄨𝜔𝑖𝑘−𝑛 󵄨󵄨󵄨󵄨]) 2 𝑖=1

} 𝐷 {󵄨 󵄨2 − 𝑑𝑘−1 ⟨𝛿𝑥 𝜔 , 𝛿𝑥 𝜔 ⟩} ≤ − {󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘 󵄨󵄨󵄨󵄨1 𝜆 } { 󵄨󵄨󵄨𝛿 𝜔𝑘 󵄨󵄨󵄨2 + 󵄨󵄨󵄨𝛿 𝜔𝑗 󵄨󵄨󵄨2 𝑘−1 󵄨 𝑥 󵄨󵄨1 󵄨󵄨 𝑥 󵄨󵄨1 − ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 󵄨 2 𝑗=1

𝑀−1

2

󵄨2 󵄨󵄨 󵄨󵄨𝛿𝑥 𝜔𝑗 󵄨󵄨󵄨 󵄨1 󵄨 + ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 2 𝑗=1 𝑘−1

(44)

𝑘−1

𝑘

󵄨2 󵄨 3 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−𝑛 󵄨󵄨󵄨󵄨1

󵄨󵄨 󵄨2 󵄨󵄨𝛿𝑥 𝜔0 󵄨󵄨󵄨 } 󵄨 󵄨1 . + 𝑑𝑘−1 2 } }

𝑗

0 ] − ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 𝛿𝑥 𝜔𝑖+1/2 − 𝑑𝑘−1 𝛿𝑥 𝜔𝑖+1/2 𝑗=1 ]

=−

𝐷 { 1 󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨󵄨𝛿 𝜔𝑘 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛿 𝜔𝑘−𝑛 󵄨󵄨󵄨 󵄨1 𝜆 { 4 󵄨 𝑥 󵄨1 󵄨 𝑥 { 𝑘−1

2

𝑀−1

𝑗

0 ] − ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 𝛿𝑥 𝜔𝑖+1/2 − 𝑑𝑘−1 𝛿𝑥 𝜔𝑖+1/2 𝑗=1 ]

𝑖=1

≤ 𝐷 𝜆

1 󵄩󵄩 𝛼 𝑘 󵄩󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩𝛿 𝜔 󵄩󵄩 + 8𝜀𝑐02 (󵄩󵄩󵄩𝜔𝑘−1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜔𝑘−2 󵄩󵄩󵄩 ) 󵄩 󵄩 󵄩 󵄩 2𝜀 󵄩 𝑡 󵄩 󵄩 󵄩2 + 𝜀𝑐12 󵄩󵄩󵄩󵄩𝜔𝑘−𝑛 󵄩󵄩󵄩󵄩 .

(45)

6


Substituting (43)–(45) into (42) and taking 𝜀 = 3/4 in (45), we obtain

From Lemma 5, we have 󵄩󵄩 𝑘 󵄩󵄩 ̂ max 󵄨󵄨󵄨󵄨𝜓𝑗 󵄨󵄨󵄨󵄨 . 󵄩󵄩𝜔 󵄩󵄩 ≤ 𝐶 󵄩 󵄩∞ −𝑛≤𝑗≤0 󵄨 󵄨1

𝐷 󵄨󵄨 󵄨2 3𝐷 󵄨󵄨 󵄨2 󵄨󵄨𝛿𝑥 𝜔𝑘 󵄨󵄨󵄨 ≤ 󵄨󵄨𝛿𝑥 𝜔𝑘−𝑛 󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨1 1 4𝜆 2𝜆 +

𝐷 𝑘−1 󵄨2 󵄨 − 𝑑𝑘−𝑗 ) 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑗 󵄨󵄨󵄨󵄨1 ∑ (𝑑 𝜆 𝑗=1 𝑘−𝑗−1

The proof is completed.

𝐷 󵄨 󵄨2 𝑑 󵄨󵄨󵄨𝛿 𝜔0 󵄨󵄨󵄨 𝜆 𝑘−1 󵄨 𝑥 󵄨1 󵄩2 󵄩 󵄩2 󵄩 + 6𝑐02 (󵄩󵄩󵄩󵄩𝜔𝑘−1 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝜔𝑘−2 󵄩󵄩󵄩󵄩 )

Denote 𝑒𝑖𝑘 = 𝑈𝑖𝑘 − 𝑢𝑖𝑘 , 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 𝑁; by subtracting (24)–(26) from (19), (22), and (23), respectively, the following error equations can be obtained:

+

3 󵄩 󵄩2 + 𝑐12 󵄩󵄩󵄩󵄩𝜔𝑘−𝑛 󵄩󵄩󵄩󵄩 4 ≤

(46)

𝐷 󵄨2 󵄨 − 𝑑𝑘−𝑗 ) 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑗 󵄨󵄨󵄨󵄨1 ∑ (𝑑 𝜆 𝑗=1 𝑘−𝑗−1

𝑒0𝑘 = 0,

1 ≤ 𝑘 ≤ 𝑁, where 𝑝𝑖𝑘 = 𝑓(2𝑈𝑖𝑘−1 − 𝑈𝑖𝑘−2 , 𝑈𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ) − 𝑓(2𝑢𝑖𝑘−1 − 𝑢𝑖𝑘−2 , 𝑢𝑖𝑘−𝑛 , 𝑥𝑖 , 𝑡𝑘 ). Similar to the proof of Theorem 9, the following convergence result can be obtained.

𝑘−𝑛 󵄨2

󵄨󵄨 , 󵄨󵄨1

where Lemma 5 has been used. Multiplying (46) by 4𝜆/𝐷, we have

Theorem 10. Assume 𝑢(𝑥, 𝑡) ∈ 𝐶6,2 ([0, 1] × (−𝑠, 𝑇]) is the solution of (3)–(5) and {𝑢𝑖𝑘 | 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 𝑁} is the solution of (24)–(26). Then, one has

𝑘−1 󵄨 󵄨 󵄨2 󵄨2 󵄨2 󵄨󵄨 󵄨󵄨𝛿𝑥 𝜔𝑘 󵄨󵄨󵄨 ≤ 4 ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 󵄨󵄨󵄨𝛿𝑥 𝜔𝑗 󵄨󵄨󵄨 + 4 󵄨󵄨󵄨𝛿𝑥 𝜔0 󵄨󵄨󵄨 󵄨 󵄨 󵄨1 󵄨1 󵄨1 󵄨 𝑗=1

+

4𝜆 2 󵄨󵄨 󵄨2 󵄨 󵄨2 𝑐0 (󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−1 󵄨󵄨󵄨󵄨1 + 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−2 󵄨󵄨󵄨󵄨1 ) 𝐷

+ (6 +

(52)

𝑘 = 0, 𝑒𝑀

𝐷 󵄨 󵄨2 𝑑 󵄨󵄨󵄨𝛿 𝜔0 󵄨󵄨󵄨 𝜆 𝑘−1 󵄨 𝑥 󵄨1 󵄨2 󵄨 󵄨2 󵄨 + 𝑐02 (󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−1 󵄨󵄨󵄨󵄨1 + 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−2 󵄨󵄨󵄨󵄨1 ) +

3𝐷 1 2 󵄨󵄨 + 𝑐 ) 󵄨󵄨𝛿 𝜔 2𝜆 8 1 󵄨 𝑥

A𝛿𝑡𝛼 𝑒𝑖𝑘 = 𝐷 (𝛿𝑥2 𝑒𝑖𝑘 + 𝛿𝑥2 𝑒𝑖𝑘−𝑛 ) + A𝑝𝑖𝑘 + 𝑅𝑖𝑘 , 𝑒𝑖𝑘 = 0, 0 ≤ 𝑖 ≤ 𝑀, −𝑛 ≤ 𝑘 ≤ 0,

𝑘−1

+(

(51)

(47)

󵄩󵄩 𝑘 󵄩󵄩 󵄩󵄩𝑒 󵄩󵄩 ≤ 𝐶 (𝜏2−𝛼 + ℎ4 ) , 󵄩 󵄩∞

1 ≤ 𝑘 ≤ 𝑁,

(53)

where 𝐶 is a positive constant independent of ℎ and 𝜏.

𝜆 2 󵄨󵄨 󵄨2 𝑐 ) 󵄨󵄨𝛿 𝜔𝑘−𝑛 󵄨󵄨󵄨󵄨1 . 2𝐷 1 󵄨 𝑥

4. Numerical Test

Denote 𝐶𝑘 = max {

4Γ (2 − 𝛼) 2 Γ (2 − 𝛼) 2 𝑐0 , 𝑐1 + 6} > 0. 𝐷 2𝐷

(48)

Noticing 𝜆 = 𝜏𝛼 Γ(2 − 𝛼) for 0 < 𝜏 < 1, we have 𝜏𝛼 < 1. Then, from (47), we obtain 𝑘−1

󵄨󵄨 󵄨2 󵄨2 󵄨2 󵄨 󵄨 󵄨󵄨𝛿𝑥 𝜔𝑘 󵄨󵄨󵄨 ≤ 4 ∑ (𝑑𝑘−𝑗−1 − 𝑑𝑘−𝑗 ) 󵄨󵄨󵄨𝛿𝑥 𝜔𝑗 󵄨󵄨󵄨 + 4 󵄨󵄨󵄨𝛿𝑥 𝜔0 󵄨󵄨󵄨 󵄨 󵄨1 󵄨1 󵄨1 󵄨 󵄨 𝑗=1

(49)

󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 + 𝐶𝑘 (󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−1 󵄨󵄨󵄨󵄨1 + 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−2 󵄨󵄨󵄨󵄨1 + 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑘−𝑛 󵄨󵄨󵄨󵄨1 ) . From Lemmas 2 and 6, we have 󵄨󵄨 󵄨2 󵄨󵄨𝛿𝑥 𝜔𝑘 󵄨󵄨󵄨 󵄨 󵄨1 󵄨 󵄨2 ≤ (3𝐶𝑘 + 4) exp (3𝐶𝑘 + 4 (1 − 𝑑𝑘−1 )) max 󵄨󵄨󵄨󵄨𝛿𝑥 𝜔𝑗 󵄨󵄨󵄨󵄨1 (50) −𝑛≤𝑗≤0 󵄨 󵄨2 ≤ (3𝐶𝑘 + 4) exp (3𝐶𝑘 + 4) max 󵄨󵄨󵄨󵄨𝛿𝑥 𝜓𝑗 󵄨󵄨󵄨󵄨1 . −𝑛≤𝑗≤0

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 󵄩 󵄩 𝑒 (ℎ, 𝜏) = max 󵄩󵄩󵄩󵄩𝑈𝑘 − 𝑢𝑘 󵄩󵄩󵄩󵄩∞ ; 1≤𝑘≤𝑁

(54)

the convergence order in time and space is defined, respectively, as Rate𝜏 =

log (𝑒 (ℎ, 𝜏1 ) /𝑒 (ℎ, 𝜏2 )) , log (𝜏1 /𝜏2 )

log (𝑒 (ℎ1 , 𝜏) /𝑒 (ℎ2 , 𝜏)) . Rateℎ = log (ℎ1 /ℎ2 )

(55)

For Rate𝜏 , we require ℎ to be fixed and small enough, while for Rateℎ , 𝜏 should be fixed and small enough.


7  = 0.2

 = 0.4 ×10−6

×10−7 5

2.5

4

2

3

1.5

2

1

1

0.5

0 1 t

1

0.5

0.5 0

0

0 1 t

x

1

0.5

0.5 0

0

 = 0.6

x

 = 0.8

×10−5 1

×10−5 4

0.8

3

0.6

2

0.4 0.2

1

0 1

0 1

t

1

0.5

0.5 0

0

t

x

1

0.5

0.5 0

0

x

Figure 1: Error planes for 𝜏 = ℎ = 1/100 for different 𝛼 in Example 1.

− 𝑒𝑥 (2 − 8𝑥 + 𝑥2 + 6𝑥3 + 𝑥4 )

Example 1. Consider the following problem:

⋅ (𝑡2+𝛼 + (𝑡 − 0.1)2+𝛼 ) .

𝜕𝛼 𝑢 𝜕2 𝑢 (𝑥, 𝑡) 𝜕2 𝑢 (𝑥, 𝑡 − 0.1) = ( + ) − 𝑢 (𝑥, 𝑡)2 𝜕𝑡𝛼 𝜕𝑥2 𝜕𝑥2

(57)

+ 𝑢 (𝑥, 𝑡 − 0.1) + 𝐺 (𝑥, 𝑡) , (𝑥, 𝑡) ∈ (0, 1) × (0, 1] , 𝑢 (𝑥, 𝑡) = 𝑒𝑥 𝑥2 (1 − 𝑥)2 𝑡2+𝛼 ,

(56) 𝑥 ∈ [0, 1] , 𝑡 ∈ [−0.1, 0] ,

𝑢 (0, 𝑡) = 0, 𝑢 (1, 𝑡) = 0, 𝑡 ∈ (0, 1] ;

the exact solution of (56) is 𝑢(𝑥, 𝑡) = 𝑒𝑥 𝑥2 (1 − 𝑥)2 𝑡2+𝛼 , and 𝐺 (𝑥, 𝑡) = 𝑒2𝑥 𝑥4 (1 − 𝑥)4 𝑡4+2𝛼 − 𝑒𝑥 𝑥2 (1 − 𝑥)2 Γ (3 + 𝛼) 2 𝑥 2 ⋅ (𝑡 − 0.1)2+𝛼 + 𝑡 𝑒 𝑥 (1 − 𝑥)2 2

From Table 1, we can see the maximum errors between the numerical solution and the exact solution in the temporal directions for 𝛼 = 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 − 𝛼. Table 2 shows the maximum errors in the spatial directions for 𝛼 = 0.2 when the temporal step is fixed at 𝜏 = 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 𝛼 = 0.2, 0.4, 0.6, 0.8, respectively. From this figure, we can see that the error becomes larger when a larger 𝛼 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

8

Journal of Function Spaces Table 1: Maximum errors and convergence order in temporal direction with ℎ = 1/400 and 𝛼 = 0.3, 0.5, 0.8 for Example 1.

𝜏 1/100 1/200 1/300 1/400

𝛼 = 0.3 𝑒(ℎ, 𝜏) 1.083𝑒 − 006 3.234𝑒 − 007 1.598𝑒 − 007 69.694𝑒 − 008

Rate𝜏 ∗ 1.744 1.739 1.737

𝛼 = 0.5 𝑒(ℎ, 𝜏) 4.502𝑒 − 006 1.571𝑒 − 006 8.501𝑒 − 007 5.501𝑒 − 007

Table 2: Maximum errors and convergence order in spatial direction with 𝜏 = 1/2000 and 𝛼 = 0.2 for Example 1. ℎ 1/8 1/12 1/16 1/20

𝑒(ℎ, 𝜏) 7.949𝑒 − 005 1.578𝑒 − 005 4.976𝑒 − 006 2.045𝑒 − 006

Rateℎ ∗ 3.988 4.011 3.985

global convergence of the scheme in the maximum norm are proved. The convergence order of the considered scheme is 𝑂(𝜏2−𝛼 + ℎ4 ). A numerical experiment is presented to support the theoretical results and validate the efficiency of the difference scheme.

[5]

[6]

[7]

[8]

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).

References [1] A. V. Rezounenko and J. Wu, “A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors,” Journal of Computational and Applied Mathematics, vol. 190, no. 1-2, pp. 99–113, 2006. [2] B. Zubik-Kowal, “Solutions for the cell cycle in cell lines derived from human tumors,” Computational and Mathematical Methods in Medicine. An Interdisciplinary Journal of Mathematical, Theoretical and Clinical Aspects of Medicine, vol. 7, no. 4, pp. 215– 228, 2006. [3] D. Li, C. Zhang, and W. Wang, “Long time behavior of nonFickian delay reaction-diffusion equations,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1401–1415, 2012. [4] Z.-X. Sun and Z. B. Zhang, “A linearized compact difference scheme for a class of nonlinear delay partial differential equations,” Applied Mathematical Modelling: Simulation and

[9]

[10]

[11]

[12]

[13]

[14]

[15]

Rate𝜏 ∗ 1.519 1.515 1.513

𝛼 = 0.8 𝑒(ℎ, 𝜏) 3.800𝑒 − 005 1.651𝑒 − 005 1.014𝑒 − 005 7.176𝑒 − 006

Rate𝜏 ∗ 1.203 1.202 1.202

Computation for Engineering and Environmental Systems, vol. 37, no. 3, pp. 742–752, 2013. Q. Zhang and C. Zhang, “A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equations,” Applied Mathematics Letters, vol. 26, no. 2, pp. 306–312, 2013. W. Gu, “A compact difference scheme for a class of variable coefficient quasilinear parabolic equations with delay,” Abstract and Applied Analysis, Article ID 810352, Art. ID 810352, 8 pages, 2014. D. Li, C. Zhang, and J. Wen, “A note on compact finite difference method for reaction-diffusion equations with delay,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 5-6, pp. 1749–1754, 2015. C. R. Jin, Z. H. Yu, and R. N. Qu, “An implicit difference scheme for solving a neutral delay parabolic differential equation,” Journal of Shandong University. Natural Science. Shandong Daxue Xuebao. Lixue Ban, vol. 46, no. 8, pp. 13–16, 2011. R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, pp. 1–77, 2000. S. Chen, F. Liu, I. Turner, and V. Anh, “An implicit numerical method for the two-dimensional fractional percolation equation,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4322–4331, 2013. F. A. Rihan, “Computational methods for delay parabolic and time-fractional partial differential equations,” Numerical Methods for Partial Differential Equations, vol. 26, no. 6, pp. 1556–1571, 2010. Q. Zhang, M. Ran, and D. Xu, “Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay,” Applicable Analysis: An International Journal, vol. 96, no. 11, pp. 1867–1884, 2017. Z. Sun and X. Wu, “A fully discrete difference scheme for a diffusion-wave system,” Appl. Numer. Math, vol. 56, pp. 193–209, 2006. S. Chen, F. Liu, P. Zhuang, and V. Anh, “Finite difference approximations for the fractional Fokker-Planck equation,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 33, no. 1, pp. 256–273, 2009. Z. Z. Sun, The numerical methods for partial equations, Science Press, Beijing, China, 2005.

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