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

Authors: Wei Gu; Yanli Zhou; Xiangyu Ge
Journal: Journal of Function Spaces
 (2017)

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


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## Body

## 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:(1)∂αw∂tα=D∂2wx,t∂x2+∂2wx,pt∂x2+fwwx,t,wx,pt,x,t,x,t∈0,1×0,T,w0,t=γwt,w1,t=βwt,t∈0,T,where D>0 is a constant, 0<p<1.Letu(x,t)=w(x,et) for t≥t0+ln⁡(p), where t0≥0. Then, u(x,t) satisfies the following equation:(2)∂αu∂tα=D∂2ux,t∂x2+∂2ux,t-s∂x2+fwux,t,ux,t-s,x,et,x,t∈0,1×t0,T,ux,t=ϕwx,et,x∈0,1,t∈-s,t0,u0,t=γwet,u1,t=βwet,t∈t0,T,where s=-ln⁡(p).For simplicity, we consider the following fractional neutral parabolic differential equation with delay instead of (1):(3)∂αu∂tα=D∂2ux,t∂x2+∂2ux,t-s∂x2+fux,t,ux,t-s,x,t,x,t∈0,1×0,T,(4)ux,t=ϕx,t,x∈0,1,t∈-s,0,(5)u0,t=γt,u1,t=βt,t∈0,T,where s>0 is a constant delay term. Time fractional partial derivative ∂αu/∂tα(0<α<1) is defined in the Caputo sense by the following:(6)∂αu∂tα=1Γ1-α∫0tt-ξ-α∂ux,ξ∂ξdξ,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 Section2, 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, assumeu(x,t)∈C6,2([0,1]×(0,T]). Function f(u(x,t),u(x,t-s),x,t) is sufficiently smooth and satisfies (7)fμ+ϵ1,ν+ϵ2,x,t-fμ,ν,x,t≤c0ϵ1+c1ϵ2,where ϵ1,ϵ2 are arbitrary real numbers and c0 and c1 are positive constants.First, letM 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(8)W=vik∣0≤i≤M,-n≤k≤Nbe the grid function space defined on Ωhτ. The following notations are used: (9)δxvi+1/2k=vi+1k-vikh,δx2vik=vi+1k-2vik+vi-1kh2,Avik=112vi-1k+10vik+vi+1k,δtαvk=τ-αΓ2-αd0vk-∑j=1k-1dk-j-1-dk-jvj-dk-1v0,where dj=(j+1)1-α-j1-α,j≥0.For the time fractional derivative, we have the following lemma.Lemma 1 (see [13]).
Suppose0<α<1, y∈C2[0,tk]; it holds that (10)1Γ1-α∫0tky′sdstk-sα-τ-αΓ2-αd0ytk-∑j=1k-1dk-j-1-dk-jytj-dk-1yt0≤1Γ2-α1-α12+22-α2-α-1+2-αmax0≤t≤tky′′tτ2-α,and dj satisfies the following lemma.Lemma 2 (see [14]).
Assume0<α<1; then, it holds that(1)
dj decreases monotonically as j increases, and 0<dj≤1;(2)
d0=1, ∑j=1k-1‍(dk-j-1-dk-j)=d0-dk-1.Lemma 3 (see [5]).
Supposey(x)∈C6[xi-1,xi+1]; then, one has(11)112y′′xi-1+10y′′xi+y′′xi+1-1h2yxi-1-2yxi+yxi+1=h4240y6ιi,where ιi∈(xi-1,xi+1).Considering (3) at the point (xi,tk), we have (12)∂αu∂tαxi,tk=D∂2u∂x2xi,tk+∂2u∂x2xi,tk-s+fuxi,tk,uxi,tk-s,xi,tk,0≤i≤M,0≤k≤N.From Lemma1, we obtain (13)∂αu∂tαxi,tk=δtαUik+rik,where (14)rik=1Γ2-α1-α12+22-α2-α-1+2-αmax0≤t≤tk∂2uxi,tk∂t2τ2-α.From Taylor expansion, we have(15)fuxi,tk,uxi,tk-s,xi,tk=f2Uik-1-Uik-2,Uik-n,xi,tk+τ2∂2u∂t2xi,ηkfμϱik,Uik-n,xi,tk,where ηk∈(tk-2,tk), ϱik 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 (16)AδtαUik=DA∂2u∂x2xi,tk+A∂2u∂x2xi,tk-s+Af2Uik-1-Uik-2,Uik-n,xi,tk+AR0ik,where (17)R0ik=rik+τ2∂2u∂t2xi,ηkfμϱik,Uik-n,xi,tk.From Lemma 3 and Taylor expansion, we have (18)A∂2u∂x2xi,tk=δx2Uik+h4240∂6u∂x6θik,tk,θik∈xi-1,xi+1,A∂2u∂x2xi,tk-s=δx2Uik-n+h4240∂6u∂x6θik-n,tk-n,θik-n∈xi-1,xi+1.Substituting (18) into (16), we have (19)AδtαUik=Dδx2Uik+δx2Uik-n+Af2Uik-1-Uik-2,Uik-n,xi,tk+Rik,where (20)Rik=AR0ik+h4240∂6u∂x6θik,tk+h4240∂6u∂x6θik-n,tk-n.Noticingu(x,t)∈C6,2([0,1]×(0,T]) and (7), we can easily obtain (21)Rik≤CRτ2-α+h4,0≤i≤M,1≤k≤N.Discretizing the initial and boundary conditions of (4) and (5), we obtain (22)Uik=ϕxi,tk,0≤i≤M,-n≤k≤0,(23)U0k=γtk,UMk=βtk,1≤k≤N.Replacing Uik by uik in (19), (22), and (23) and omitting Rik, we can obtain the following compact difference scheme: (24)Aδtαuik=Dδx2uik+δx2uik-n+Af2uik-1-uik-2,uik-n,xi,tk,(25)uik=ϕxi,tk,0≤i≤M,-n≤k≤0,(26)u0k=γtk,uMk=βtk,1≤k≤N.


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

Define the following grid function space onΩh: (27)Vh,0=u∣u=u0,u1,…,uM,u0=uM=0.If u,v∈Vh,0, we introduce the following inner products and corresponding norms: (28)u,v=h∑i=1M-1uivi,u=u,u,u∞=max1≤i≤M-1ui,δxu,δxv=h∑i=0M-1δxui+1/2δxvi+1/2,δxu1=δxu,δxu.It is easy to obtain the following lemma.Lemma 4 (see [12]).
∀
u
∈
V
h
,
0, one has Au2≤u2.Lemma 5 (see [15]).
∀
u
∈
V
h
,
0, one has (29)u∞≤δxu12,u≤δxu16.The following lemma will be used in the proof of the stability and convergence analysis.Lemma 6 (see [15]).
Assume that{Fk∣k≥0} is a nonnegative sequence and satisfies (30)Fk+1≤A+Bτ∑i=1kFl,k=0,1,…;then, (31)Fk+1≤Aexp⁡Bkτ,k=0,1,2,…,where A and B are nonnegative constants.Theorem 7.
The difference scheme (24)–(26) has a unique solution.Proof.
Denoteuk=[u1k,u2k,…,uM-1k]T; the difference scheme (24)–(26) is a linear tridiagonal system Muk=d, where d only depends on uj(j≤k-1), u0k, and uMk and is independent of uk. (32)M=tridiag112λ-Dh2,1012λ+2Dh2,112λ-Dh2,where λ=ταΓ(2-α). We can see that M 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: (33)A∂αv∂tα=D∂2vx,t∂x2+∂2vx,t-s∂x2+Afvx,t,vx,t-s,x,t,x,t∈0,1×0,T,vx,t=ϕx,t+ψx,t,x∈0,1,t∈-s,0,v0,t=γt,u1,t=βt,t∈0,T,where ψ(x,t) is the perturbation caused by ϕ(x,t). The following difference scheme can be obtained for solving (33):(34)Aδtαvik=Dδx2vik+δx2vik-n+Af2vik-1-vik-2,vik-n,xi,tk,(35)vik=ϕxi,tk+ψik,0≤i≤M,-n≤k≤0,(36)v0k=γtk,vMk=βtk,1≤k≤N.Denote(37)ωik=vik-uik,0≤i≤M,-n≤k≤N.Definition 8.
Assume thatuik satisfies (24)–(26) and vik satisfies (34)–(36); then, a numerical scheme for (3)–(5) is stable if one has (38)ωk∞≤C-max-n≤j≤0ψj1,where C- is a bounded constant independent of h and τ.Theorem 9.
Assumeu(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∞≤C^max-n≤j≤0|ψj|1, where C^ is a positive constant independent of h and τ.Proof.
Subtracting (34)–(36) from (24)–(26), respectively, we can obtain the following equations: (39)Aδtαωik=Dδx2ωik+δx2ωik-n+Aqik,(40)ωik=ψik,0≤i≤M,-n≤k≤0,(41)ω0k=0,ωMk=0,1≤k≤N,where qik=f(2vik-1-vik-2,vik-n,xi,tk)-f(2uik-1-uik-2,uik-n,xi,tk).
Multiplyinghδtαωik on both sides of (39) and summing up for i from 1 to M-1, we obtain (42)h∑i=1M-1Aδtαωikδtαωik=Dh∑i=1M-1δx2ωik+δx2ωik-nδtαωik+h∑i=1M-1Aqikδtαωik,1≤i≤M-1,0≤k≤N-1.Then, each term of (42) will be estimated. From the discrete Green formula and inequality -(a+b)2≥-2(a2+b2), we have (43)h∑i=1M-1Aδtαωikδtαωik=δtαωk2+h212h∑i=1M-1δx2δtαωikδtαωik=δtαωk2-h212δxδtαωk2≥23δtαωk2.From the discrete Green formula and inequality -ab≥-(a2+b2)/2, we have (44)Dh∑i=1M-1δx2ωikδtαωik=-Dh∑i=0M-1δxωi+1/2kδtαδxωi+1/2k=-Dλh∑i=0M-1δxωi+1/2kδxωi+1/2k-∑j=1k-1dk-j-1-dk-jδxωi+1/2j-dk-1δxωi+1/20=-Dλδxωk12-∑j=1k-1dk-j-1-dk-jδxωk,δxωj-dk-1δxωk,δxω0≤-Dλδxωk12-∑j=1k-1dk-j-1-dk-jδxωk12+δxωj122-dk-1δxωk12+δxω0122=-Dλδxωk122-∑j=1k-1dk-j-1-dk-jδxωj122-dk-1δxω0122,Dh∑i=1M-1δx2ωik-nδtαωik=-Dh∑i=0M-1δxωi+1/2k-nδtαδxωi+1/2k=-Dλh∑i=0M-1δxωi+1/2k-nδxωi+1/2k-∑j=1k-1dk-j-1-dk-jδxωi+1/2j-dk-1δxωi+1/20≤Dλ14δxωk12+δxωk-n12+∑j=1k-1dk-j-1-dk-jδxωk-n12+δxωj122+dk-1δxωk-n12+δxω0122=Dλδxωk124+3δxωk-n122+∑j=1k-1dk-j-1-dk-jδxωj122+dk-1δxω0122.
From the Cauchy-Schwarz inequality, Lemma4, and (7), we have (45)h∑i=1M-1Aqikδtαωik≤12εδtαωk2+ε2h∑i=1M-1Ac02ωik-1-ωik-2+c1ωik-n2≤12εδtαωk2+ε2h∑i=1M-1c02ωik-1-ωik-2+c1ωik-n2≤12εδtαωk2+εc02h∑i=1M-12ωik-1-ωik-22+εc12h∑i=1M-1ωik-n2≤12εδtαωk2+8εc02ωk-12+ωk-22+εc12ωk-n2.Substituting (43)–(45) into (42) and taking ε=3/4 in (45), we obtain (46)D4λδxωk12≤3D2λδxωk-n12+Dλ∑j=1k-1dk-j-1-dk-jδxωj12+Dλdk-1δxω012+6c02ωk-12+ωk-22+34c12ωk-n2≤Dλ∑j=1k-1dk-j-1-dk-jδxωj12+Dλdk-1δxω012+c02δxωk-112+δxωk-212+3D2λ+18c12δxωk-n12,where Lemma 5 has been used.
Multiplying (46) by 4λ/D, we have (47)δxωk12≤4∑j=1k-1dk-j-1-dk-jδxωj12+4δxω012+4λDc02δxωk-112+δxωk-212+6+λ2Dc12δxωk-n12.Denote (48)Ck=max⁡4Γ2-αDc02,Γ2-α2Dc12+6>0.Noticing λ=ταΓ(2-α) for 0<τ<1, we have τα<1. Then, from (47), we obtain (49)δxωk12≤4∑j=1k-1dk-j-1-dk-jδxωj12+4δxω012+Ckδxωk-112+δxωk-212+δxωk-n12.From Lemmas 2 and 6, we have (50)δxωk12≤3Ck+4exp⁡3Ck+41-dk-1max-n≤j≤0δxωj12≤3Ck+4exp⁡3Ck+4max-n≤j≤0δxψj12.From Lemma 5, we have (51)ωk∞≤C^max-n≤j≤0ψj1.The proof is completed.Denoteeik=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: (52)Aδtαeik=Dδx2eik+δx2eik-n+Apik+Rik,eik=0,0≤i≤M,-n≤k≤0,e0k=0,eMk=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 Theorem9, the following convergence result can be obtained.Theorem 10.
Assumeu(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 (53)ek∞≤Cτ2-α+h4,1≤k≤N,where C is a positive constant independent of h and τ.


## 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 (54)eh,τ=max1≤k≤NUk-uk∞;the convergence order in time and space is defined, respectively, as (55)Rateτ=log⁡eh,τ1/eh,τ2log⁡τ1/τ2,Rateh=log⁡eh1,τ/eh2,τlog⁡h1/h2.For Rateτ, we require h to be fixed and small enough, while for Rateh, τ should be fixed and small enough.Example 1.
Consider the following problem:(56)∂αu∂tα=∂2ux,t∂x2+∂2ux,t-0.1∂x2-ux,t2+ux,t-0.1+Gx,t,x,t∈0,1×0,1,ux,t=exx21-x2t2+α,x∈0,1,t∈-0.1,0,u0,t=0,u1,t=0,t∈0,1;the exact solution of (56) is u(x,t)=exx2(1-x)2t2+α, and (57)Gx,t=e2xx41-x4t4+2α-exx21-x2t-0.12+α+Γ3+α2t2exx21-x2-ex2-8x+x2+6x3+x4t2+α+t-0.12+α.
From Table1, 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.
Table2 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.
Figure1 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.Table 1
Maximum errors and convergence order in temporal direction withh=1/400 and α=0.3,0.5,0.8 for Example 1.
τ
α
=
0.3
α
=
0.5
α
=
0.8
e
(
h
,
τ
)
R
a
t
e
τ
e
(
h
,
τ
)
R
a
t
e
τ
e
(
h
,
τ
)
R
a
t
e
τ
1/100
1.083
e
-
006
∗
4.502
e
-
006
∗
3.800
e
-
005
∗
1/200
3.234
e
-
007
1.744
1.571
e
-
006
1.519
1.651
e
-
005
1.203
1/300
1.598
e
-
007
1.739
8.501
e
-
007
1.515
1.014
e
-
005
1.202
1/400
69.694
e
-
008
1.737
5.501
e
-
007
1.513
7.176
e
-
006
1.202Table 2
Maximum errors and convergence order in spatial direction withτ=1/2000 and α=0.2 for Example 1.
h
e
(
h
,
τ
)
R
a
t
e
h
1/8
7.949
e
-
005
∗
1/12
1.578
e
-
005
3.988
1/16
4.976
e
-
006
4.011
1/20
2.045
e
-
006
3.985Figure 1
Error planes forτ=h=1/100 for different α 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 isO(τ2-α+h4). A numerical experiment is presented to support the theoretical results and validate the efficiency of the difference scheme.



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