Research ArticleAn Efficient Alternating Segment Parallel DifferenceMethod for the Time Fractional Telegraph Equation

Lifei Wu and Xiaozhong Yang

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Correspondence should be addressed to Xiaozhong Yang; yxiaozh@ncepu.edu.cn

Received 25 November 2019; Accepted 16 January 2020; Published 2 March 2020

Academic Editor: Qin Zhou

Copyright © 2020 Lifei Wu and Xiaozhong Yang. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

The fractional telegraph equation is a kind of important evolution equation, which has an important application in signal analysissuch as transmission and propagation of electrical signals. However, it is difficult to obtain the corresponding analytical solution, soit is of great practical value to study the numerical solution. In this paper, the alternating segment pure explicit-implicit (PASE-I)and implicit-explicit (PASI-E) parallel difference schemes are constructed for time fractional telegraph equation. Based on thealternating segment technology, the PASE-I and PASI-E schemes are constructed of the classic explicit scheme and implicitscheme. It can be concluded that the schemes are unconditionally stable and convergent by theoretical analysis. The convergenceorder of the PASE-I and PASI-E methods is second order in spatial direction and 3-α order in temporal direction. Thenumerical results are in agreement with the theoretical analysis, which shows that the PASE-I and PASI-E schemes are superiorto the classical implicit schemes in both accuracy and efficiency. This implies that the parallel difference schemes are efficient forsolving the time fractional telegraph equation.

1. Introduction

Nowadays, the fractional derivative and fractional differentialequation have many applications in various fields, such asboundary layer effects in ducts, colored noise, dielectric polar-ization, electromagnetic waves, fractional kinetics, power-lawphenomenon in fluid and complex network, quantitativefinance, and viscoelastic mechanics [1–3]. The fractional tele-graph equation especially is applied into signal analysis fortransmission, propagation of electrical signals, and so on[4]. Because the analytical solution of the fractional tele-graph equation is difficult to give explicitly or contains spe-cial functions such as the Mittag-Leffler function, which aredifficult to calculate, the development of effectively numericalalgorithms for solving the fractional telegraph equation isimportant [5–7].

Because of the historical dependence and global correlationof fractional calculus, the computational and storage require-ments of fractional differential equations’ numerical simulationare enormous [8–10]. Even with high-performance com-puters, it is difficult to simulate long-term or large-scale com-

putational domains. With the rapid development of multicoreand cluster technology, parallel algorithm has become one ofthe mainstream technologies in improving the computationalefficiency [11]. Zhang et al. constructed a segment implicitscheme by using Saul’yev asymmetric scheme and used thealternative technique to establish a variety of alternatingexplicit-implicit and implicit parallel methods [12]. Themethods have been well applied to numerical solving integerpartial differential equation [13–16]. However, efficient paral-lel numerical methods for integer order differential equationsmay not be effective for fractional order differential equations.It may even produce completely different numerical analysisprocesses. How to extend the existing parallel differencemethod of integer order differential equation to the methodof fractional order differential equation is a great challenge tocomputational mathematics (physics).

In recent years, many scholars have studied the numeri-cal algorithms of the fractional telegraph equation. Forexample, Ford et al. used the finite difference method tonumerically solve the fractional telegraph equation [17]. Saa-datmandi and Mohabbati proposed a numerical solution

HindawiAdvances in Mathematical PhysicsVolume 2020, Article ID 6897815, 11 pageshttps://doi.org/10.1155/2020/6897815

by combining orthogonal Legendre polynomial and Taumethod for the fractional telegraph equation [18]. Niu et al.applied the Chebyshev polynomial to get the numerical solu-tion of the fractional telegraph equation [19]. Zhao and Liapplied the finite difference method and Galerkin finite ele-ment method to solve the time-space fractional telegraphequation numerically [20]. Chen constructed the implicit dif-ference scheme for the Riesz space fractional order telegraphequation [21]. Ren and Liu presented a high-order compactfinite difference method for a class of time fractional Black-Scholes equation [22]. The scheme has the second-ordertemporal accuracy and the fourth-order spatial accuracy.The existing numerical algorithms of fractional differentialequations are mostly serial algorithms, which have low com-putational efficiency.

In recent years, some progress has been made in fastalgorithms for fractional partial differential equations [23,24], most of which are parallel algorithms for algebraic sys-tems based on the view of numerical algebra. For example,Diethelm implemented the second-order Adams-Bashforth-Moulton method of fractional diffusion equations on a paral-lel computer and discussed the accuracy of parallel method[25]. Gong et al. parallelizes the explicit difference schemeof the space fractional reaction-diffusion equation [26]. Gonget al. parallelizes the implicit scheme of the time fractionaldiffusion equation [27] and the core of parallelization is todo parallel computation for the product of matrix and vectorand the addition of vector and vector. Sweilam et al. appliedpreconditioned conjugate gradient method was used to solvediscrete algebraic equations in parallel, based on Crank-Nicolson difference schemes for time fractional parabolicequations [28]. Wang et al. studied the parallel algorithm ofimplicit difference schemes for fractional reaction-diffusionequations [29]. The algorithm is based on the principle ofminimizing communication, allocating computing tasks rea-sonably, and not changing the original serial differenceschemes as much as possible.

In order to obtain more accurate and stable parallel dif-ference schemes, we are prepared to take the parallel pathof traditional difference schemes and hope to find anotherway to parallelize them beyond the difficulty of numericalalgebra [30, 31]. In this paper, a kind of alternating seg-ment pure explicit-implicit (PASE-I) and implicit-explicit(PASI-E) parallel difference schemes is obtained. The numer-ical experiments and theoretical analysis are consistent,showing that the PASE-I and PASI-E schemes are uncon-ditionally stable and convergent. The numerical examplesshow that the PASE-I and PASI-E difference schemeshave obvious parallel computational properties. It showsthat the parallel intrinsic difference schemes proposed inthis paper is efficient for solving the time fractional tele-graph equations.

2. PASE-I Scheme for FractionalTelegraph Equation

2.1. Time Fractional Telegraph Equation. Consider the fol-lowing time fractional telegraph equation [3–5]:

∂αu x, tð Þ∂tα

+ ∂α−1u x, tð Þ∂tα−1

= K∂2u x, tð Þ

∂x2+ f x, tð Þ, 0 ≤ x ≤ L, t ≥ 0, 1 < α ≤ 2:

ð1Þ

The initial conditions: uðx, 0Þ = φ0ðxÞ, ð∂uðx, 0ÞÞ/ð∂tÞ =φ1ðxÞ:

The boundary conditions: uð0, tÞ = 0, uðL, tÞ = 0:

2.2. Construction of PASE-I Scheme. In order to obtain thePASE-I difference scheme for the fractional telegraphequation, the solution region is first meshed: the space andtime steps are taken h = L/M and τ = T/N, where M and Nare natural numbers; xi = ihði = 1, 2,⋯,MÞ,Mh = L; tk = kτðk = 1, 2,⋯,NÞ,Nτ = T; and the grid nodes are ðxi, tkÞ.

The time fractional derivative can be approximated by L1formula [8, 9]:

∂α−1u xi, tk+1ð Þ∂tα−1

= 1Γ 2 − αð Þ〠

k

j=0

u xi, t j+1� �

− u xi, t j� �

τ

ð j+1ð Þτ

jτ

� dξ

tk+1 − ξð Þα−1+O τ3−α� �

≈τ2−α

2 − αð ÞΓ 2 − αð Þ〠k

j=0

u xi, tk+1−j� �

− u xi, tk−j� �

τ

� j + 1ð Þ2−α − j2−α� �

= τ1−α

Γ 3 − αð Þ〠k

j=0cj∇tu xi, tk−j

� �,

ð2Þ

∂αu xi, tk+1ð Þ∂tα

= 1Γ 2 − αð Þ〠

k

j=0

� u xi, tk+1−j� �

− 2u xi, tk−j� �

+ u xi, tk−j−1� �

τ2

�ð j+1ð Þτ

jτ

dξ

tk+1 − ξð Þα−1+O τ3−α� �

≈τ−α

Γ 3 − αð Þ〠k‐1

j=0djδ

2t u xi, tk−j� �

+ 2dk u xi, t1ð Þ − u xi, t0ð Þ − τut xið Þð Þ:ð3Þ

Spatial second derivatives are discretized by central dif-ference method:

∂2u xi, tk+1ð Þ∂x2

= δ2x θu xi, tk+1ð Þ + 1 − θð Þu xi, tkð Þ½ �+O h2� �

:

ð4Þ

Here, δ2xuðxi, tkÞ = ð1/h2Þ½uðxi−1, tkÞ − 2uðxi, tkÞ + uðxi+1,tkÞ�,∇tuðxi, tkÞ = ð1/τÞuðxi, tk+1Þ − uðxi, tkÞ,cj = ðj + 1Þ2−α − j2−α,dj = ðj + 1Þ2−α − j2−α,j = 0, 1, 2,⋯, k − 1:

2 Advances in Mathematical Physics

By substituting formulas (2), (3), (4) into equation (1),the universal difference schemes are obtained as follows:

τ−α

Γ 3 − αð Þ〠k‐1

j=0cjδ

2t u xi, tk−j� �

+ τ1−α

Γ 3 − αð Þ〠k

j=0dj∇tu xi, tk−j

� �

= K

h2δ2x θu xi, tk+1ð Þ + 1 − θð Þu xi, tkð Þ½ � + f xi, tk+1ð Þ:

ð5Þ

Let uki denote the value of uðx, tÞ at the point ðxi, tkÞ,f ki denote the value of f ðx, tÞ at the pointðxi, tkÞ, and a =ðτ−α/ðΓð3 − αÞÞÞ,b = ðτ1−α/ðΓð3 − αÞÞÞ,r = K/h2. The univer-sal difference scheme can be written as follows:

a〠k−1

j=0djδ

2t u

k−ji + 2dk u xi, t1ð Þ − u xi, t0ð Þ − τut xið Þ½ �

+ b〠k−1

j=0cj∇tu

k−ji = rδ2x θuk+1i + 1 − θð Þuki

� �+ f k+1i , k = 0, 1, 2,⋯:

ð6Þ

When θ = 0, (6) is the explicit scheme, its advantage isexplicit parallel computation, but its disadvantage is condi-tional stability. When θ = 1, (6) is the implicit scheme, itsadvantage is unconditional stability. The disadvantage is thatit needs to calculate tridiagonal equations, which is not easyto parallel computation and has low computational efficiency.

The matrix form of universal difference scheme is asfollows:

Here, q0 = ad0 + bc0,

q1 = að2d0 − d1Þ + bðc0 − c1Þ, qj = −aðdj−2 − 2dj−1 + djÞ + bðcj−1 − cjÞ,j = 2, 3,⋯, k − 1, qk = a ð−dk−2 + 2dk−1 − dkÞ + bðck−1 − ckÞ,qk+1 = að−dk−1 + dkÞ + bck,bk+1 = ðrðθuk+11 + ð1 − θÞuk1Þ, 0,⋯, 0, rðθuk+1M+1 + ð1 − θÞ ukM+1ÞÞ′,Uk = ½uk1, uk2,⋯, ukM−1�′,U0 = ½φ0ðx1Þ, φ0ðx2Þ,⋯, φ0ðxM−1Þ�′,H = ½φ1ðx1Þ, φ1ðx2Þ,⋯, φ1ðxM−1Þ�′,F = ½ f1, f2, ⋯ ,f M−1�′,A = ðad0 + bc0ÞI + rθG,B = ðað2d0 − d1Þ + bðc0 − c1ÞÞI − rð1 − θÞG,E is the unitmatrix of order ðM − 1Þ,

G =

2 −1−1 2 −1

⋱ ⋱ ⋱

−1 2 −1−1 2

2666666664

3777777775

M−1ð Þ× M−1ð Þ

: ð8Þ

By alternatingly segment applying explicit and implicitschemes, the alternating segment explicit-implicit (PASE-I)scheme for fractional telegraph equation is designed as fol-lows: Let M − 1 = ql, q, l ∈N+, l ≥ 3, q an odd number, andq ≥ 3. The points to be computed in the same even time layerare divided into q segments, which are computed accordingly

by the rule of “explicit segment-implicit segment-explicit seg-ment”. Similarly, the next odd layer is also divided into q seg-ment calculation. The calculation rule is changed to “implicitsegment-explicit segment-implicit segment”. The computa-tional lattice diagram of PASE-I scheme is shown inFigure 1. The blue circle indicates the classical explicit scheme,and the blue square indicates the classical implicit scheme.The PASE-I format will be obtained.

For the PASE-I scheme and i0 ≥ 0, consider the calcula-tion of pointsði0 + i, k + 1Þ, i = 1, 2,⋯, l in implicit segments.The implicit segment format is as follows:

2ad0 + bc0ð ÞI + rG½ �Vk+1

= a 2d0 − d1ð Þ + b c0 − c1ð Þð ÞI½ �Vk

+ q2Vk−1+⋯+qk−1V2 + qkV

1 + qk+1V0

+ 2adkτH1 + bk+1im + F1k+1:

ð9Þ

The explicit segment format is as follows:

2ad0 + bc0ð ÞI½ �Vk+1 = a 2d0 − d1ð Þ + b c0 − c1ð Þð ÞI + rG½ �Vk

+ q2Vk−1+⋯+qk−1V2 + qkV

1

+ qk+1V0 + 2adkτH1 + bk+1ex + F1

k+1:

ð10Þ

. . . . . .. . . . . . . . .

. . . . . . . . . . . . . . .

l l l l l

l l l ll

k+1

k

Figure 1: Computational lattice diagram of PASE-I scheme.

2a + bð ÞI + rθG½ �U1 = 2a + bð ÞI − r 1 − θð ÞG½ �U0 + 2aτH + b1 + F1,

AUk+1 = BUk + q2Uk−1+⋯+qk−2U2 + qk−1U

1 + qkU0 + 2adkτH + bk+1 + Fk+1:

(ð7Þ

3Advances in Mathematical Physics

Here, Vn = ðuni0+1, uni0+2,⋯, uni0+lÞ′,H1 = ½φ1ðxi0+1Þ, φ1 ð

xi0+2Þ,⋯, φ1ðxi0+lÞ�′,F1 = ½ f i0+1, f i0+2,⋯, f i0+l�′,

bk+1im = ðruk+1i0−1, 0,⋯, 0, ruk+1i0+l+1Þ′,

bk+1ex = ðruk1, 0,⋯, 0, ruki0+l+1Þ′:The PASE-I scheme is as follows:

n = 1, 3, 5⋯ , bn+11 = ðrun0 , 0,⋯, 0, runMÞ′,bn+22 = ðrun+20 , 0,⋯, 0, run+2M Þ′,Ql ′ is a zero matrix of order l′,l′ = ððl − 1Þ/l − 2Þ ;G1,G2 is martix of order ðM − 1Þ andthe definition is as follows:

G1 =

Ql−1

Gl+2

Ql−2

Gl+2

⋱

Ql−2

Gl+2

Ql−1

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCA

M−1ð Þ× M−1ð Þ

,

G2 =

~Gl+1

Ql−2

Gl+2

Ql−2

⋱

Gl+2

Ql−2

�Gl+1

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCA

M−1ð Þ× M−1ð Þ

:

ð12Þ

Here,

Gl+2 =

0 0−1 2 −1

⋱ ⋱ ⋱

−1 2 −10 0

0BBBBBBBB@

1CCCCCCCCA

l+2ð Þ× l+2ð Þ

, ð13Þ

~Gl+1 =

2 −1−1 2 −1

⋱ ⋱ ⋱

−1 2 −10 0

0BBBBBBBB@

1CCCCCCCCA

l+1ð Þ× l+1ð Þ

, ð14Þ

�Gl+1 =

0 0−1 2 −1

⋱ ⋱ ⋱

−1 2 −1−1 2

0BBBBBBBB@

1CCCCCCCCA

l+1ð Þ× l+1ð Þ

: ð15Þ

3. Numerical Analysis of PASE-I and PASI-EDifference Schemes

3.1. Existence and Uniqueness of PASE-I Solution

Lemma 1. The matrices ðað2d0 − d1Þ + bðc0 − c1ÞÞI +G1 andðað2d0 − d1Þ + bðc0 − c1ÞÞI +G2 defined by PASE-I formatare nonsingular matrices.

Proof. From the definition of G1 and að2d0 − d1Þ + bðc0 −c1Þ > 0, we can know that ðað2d0 − d1Þ + bðc0 − c1ÞÞI +G1 isa strictly diagonally dominant matrix, and the principal diag-onal element is a positive real number. So ðað2d0 − d1Þ +bðc0 − c1ÞÞI + G1 is a nonsingular matrix, and its inversematrix also exists. Thus, there is the following theorem.

Theorem 2. The PASE-I difference scheme (11) for the timefractional telegraph equation is uniquely solvable.

3.2. Stability Analysis of PASE-I Scheme

Lemma 3. If the matrix C is a nonnegative real matrix, for anyparameter ρ ≥ 0,0 ≤ σ1 ≤ σ2, there is an estimate

σ1I − ρCð Þ σ2I + ρCð Þ−1�� ��2≤ 1: ð16Þ

Proof. kðσ1I − ρCÞðσ2I + ρCÞ−1k22 = maxφ∈Rn ,φ≠0

fððσ1I − ρCÞðσ2I+ ρCÞ−1φ, ðσ1I − ρCÞðσ2I + ρCÞ−1φÞ/ðφ, φÞg:

Make a transformation ψ = ðσ2I + ρCÞ−1φ, then

σ1I − ρCð Þ σ2I + ρCð Þ−1�� ��22

= maxφ∈Rn ,φ≠0

σ1I − ρCð Þψ, σ1I − ρCð Þψð Þσ2I + ρCð Þψ, σ2I + ρCð Þψð Þ

= maxψ∈Rn ,ψ≠0

σ12 ψ, ψð Þ − 2σ1ρ Cψ, ψð Þ + ρ2 Cψ, Cψð Þ

σ22 ψ, ψð Þ + 2σ2ρ Cψ, ψð Þ + ρ2 Cψ, Cψð Þ :

ð17Þ

q0I + rG1ð ÞUn+1 = q1I − rG2ð ÞUn + q2Un−1 ⋯ +qn−1U2 + qnU

1 + qn+1U0 + 2adkτH + bn+11 + Fk+1,

q0I + rG2ð ÞUn+2 = q1I − rG1ð ÞUn+1 + q2Un ⋯ +qnU2 + qn+1U

1 + qn+2U0 + 2adk+1τH + bn+22 + Fn+2:

(ð11Þ

4 Advances in Mathematical Physics

From 0 ≤ σ1 ≤ σ2, we have kðσ1I − ρCÞðσ2I + ρCÞ−1k2≤ 1:

Let ~uni be the approximate solution of the differencescheme, and uni be the exact solution of difference scheme.

Let εni = ~uni − uni ,En = ðεn1 , εn3 ,⋯,εnM−1Þ,1 ≤ n ≤N + 1,εn0 = 0, εnM = 0, so we have

where n = 1, 3,⋯,w1 = ad0 + bc0,w2 = að2d0 − d1Þ + bðc0 − c1Þ:

When n ≥ 3, we have

En+2 = q0I +G2ð Þ−1 q1I −G1ð Þ q0I +G1ð Þ−1 q1I − G2ð ÞEn

+ q0I + G2ð Þ−1 q1I −G1ð Þ q0I +G1ð Þ−1� q2E

n−1+⋯+qn−1E2 + qnE1� �

+ q0I + G2ð Þ−1 q2En+⋯+qnE2 + qn+1E

1� �:

ð19Þ

Take norms on both sides of formula (19):

En+2�� �� = q0I + G2ð Þ−1 q1I −G1ð Þ q0I +G1ð Þ−1 q1I −G2ð Þ�� ��� Enk k + q0I + G2ð Þ−1 q1I −G1ð Þ q0I +G1ð Þ−1�� ��� q2E

n−1+⋯+qn−1E2 + qnE1� ��� �� + q0I +G2ð Þ−1�� ��

� q2En+⋯+qnE2 + qn+1E

1� ��� ��:ð20Þ

The growth matrix of PASE-I scheme is T =ðq0I +G2Þ−1ðq1I −G1Þðq0I + G1Þ−1ðq1I −G2Þ. According tothe definition of the matrix, Gl,G1, and G2 have the sameeigenvalue. Let ~T = ðq0I +G2ÞTðq0I +G2Þ−1, suppose theeigenvalue of G1 or G2 in the matrix is r. From theLemma 3, we have

Tk k = ~T��� ��� = q1I − rG1ð Þ q0I + rG1ð Þ−1��� q1I − rG2ð Þ q0I + rG2ð Þ−1��

=max q1 − rq0 + r

2�����

�����( )

≤ 1:

ð21Þ

If q1 > r, let q1 = μrðμ > 1Þ, because jðq1 − rÞ/ðq0 + rÞj ≤ 1,we have μ ≤ 2.

The following is proven by mathematical induction kEn

k ≤ kE1k.When n = 1,

E2�� �� = q0I + rG1ð Þ−1 q1I − rG2ð ÞE1�� �� ≤ E1�� ��: ð22Þ

When n = 2, case I max fq1, rg = q1, and r < q1 ≤ 2r:

E3�� �� ≤ q0I + rG2ð Þ−1 q1I − rG1ð Þ q0I + rG1ð Þ−1 q1I − rG2ð Þ�� ��� E1�� �� + q0I + rG2ð Þ−1�� �� q2E

1�� ��≤max q1 − r

q0 + r

��������2

( )E1�� �� +max q2

q0 + r

��������

� E1�� ��

≤max r − q2q0 + r

��������

� E1�� �� ≤ E1�� ��:

ð23Þ

When max fq1, rg = r,

E3�� �� ≤ q0 I + rG2ð Þ−1 q1I − rG1ð Þ q0I + rG1ð Þ−1 q1I − rG2ð Þ�� ��� E1�� �� + q0I + rG2ð Þ−1�� �� q2E

1�� ��≤max q1 − r

q0 + r

��������

� E1�� �� +max q2

q0 + r

��������

� E1�� ��

≤max r + q2j jr + q0

� E1�� �� ≤ E1�� ��:

ð24Þ

Suppose kEnk ≤ kE1kwhen n ≤ k + 1:When n = k + 2, weconsider the following two situations.

Case 1. max fq1, rg ≤ q1, and r < q1 ≤ 2r.

Ek+2��� ��� = q0I + r�G2

� �−1 q1I − r�G1� ����

� q0I + r�G1� �−1 q1I − r�G2

� ����� Ek��� ��� + q0I + r�G2

� �−1 q1I − r�G1� �

q0I + r�G1� �−1��� ���

� q2Ek−1+⋯+qk−1E2 + qkE

1� ���� ���

+ q0I + r�G2� �−1��� ��� q2E

k+⋯+qkE2 + qk+1E1

� ���� ���≤

rq0 + r

2E1�� �� + r q0 − q1j j

q0 + rð Þ2 E1�� �� + q0 − q1j jq0 + r

E1�� ��≤

rq0 + r

E1�� �� + q0 − q1j jq0 + r

E1�� �� ≤ E1�� ��:ð25Þ

a0I +G1ð ÞEn+1 = w1I − G2ð ÞEn +w2En−1 ⋯ +wn−1E

2 + anE1,

a0I +G2ð ÞEn+2 = w1I − G1ð ÞEn+1 +w2En ⋯ +wnE

2 + an+1E1:

(ð18Þ

5Advances in Mathematical Physics

Case 2. max fq1, rg ≤ r.

Ek+2��� ��� = q0I + �G2

� �−1 q1I − �G1� �

q0I + �G1� �−1 q1I − �G2

� ���� ���� Ek��� ��� + q0I + �G2

� �−1 q1I − �G1� �

q0I + �G1� �−1��� ���

� q2Ek−1+⋯+qk−1E2 + qkE

1� ���� ��� + q0I + �G2

� �−1��� ���� q2E

k+⋯+qkE2 + qk+1E1

� ���� ���≤

ra0 + r

2E1�� �� + r q1 − q0ð Þ

a0 + rð Þ2 E1�� �� + q1 − q0a0 + r

E1�� ��= rq0 + r

rq0 + r

+ q1 − q0q0 + r

E1�� �� + q1 − q0

q0 + rE1�� ��

≤r

q0 + rE1�� �� + q1 − q0

q0 + rE1�� �� ≤ E1�� ��:

ð26Þ

To sum up, we have the following theorem.

Theorem 4. The PASE-I difference scheme (11) for the timefractional telegraph equation is unconditionally stable.

3.3. Convergence of PASE-I Scheme. Firstly, the accuracy ofexplicit and implicit schemes is analyzed, respectively, andTaylor expansion will be carried out at ðxi, tn+1Þ. The trunca-

tion errors are recorded as T1ðτ, hÞ and T2ðτ, hÞ. It is knownthat C

0Dαt uðxi, tn+1Þ has second-order accuracy. We have

T1 τ, hð Þ =Dαt u +Dα−1

t u − K

uxx + τuxxt −

τ2

2 uxxtt

−h2

12 uxxxx +τh2

12 uxxxxt

!+O τ3−α + h2� �

,

T2 τ, hð Þ =Dαt u +Dα−1

t u − K

uxx − τuxxt −

τ2

2 uxxtt

−h2

12 uxxxx −τh2

12 uxxxxt

!+O τ3−α + h2� �

:

ð27Þ

For the PASE-I scheme, the explicit and implicit schemesare alternately used for each grid point in spatial direction.For T1ðτ, hÞ and T2ðτ, hÞ, the coefficients of uxxt and uxxxxtare equal and symbolically opposite. When the explicit andimplicit schemes are alternately used, the errors of the twoterms will be offset. Therefore, the accuracy of the PASE-Ischeme is second order in spatial direction and 3 − α orderin time direction.

Let uðxi, tnÞ, ði = 1, 2,⋯,M ; n = 1, 2,⋯,NÞ be the exactsolution of equation (1) at grid point ðxi, tnÞ: Let eni = uðxi,tnÞ − uni ,en = ðen1 , en3 ,⋯, enM−1ÞT . eni = uðxi, tnÞ − uni is intro-duced into the PASE-I scheme,

Obviously, e1 = 0, Rn =O ≤ ðτ3−α + h2Þ, there exists a pos-itive constant C, such that ταRn = ταCðτ3−α + h2Þ.

When n = 1,

e2 = I +G1ð Þ−1 I −G2ð Þe1 + I +G1ð Þ−1R2

= I +G1ð Þ−1R:ð29Þ

Combination Lemma 3 has

e2�� ��

∞ = I +G1ð Þ−1R1�� ��∞ ≤ Rk k∞ ≤ C τ3−α + h2

� �= q−11 C τ3−α + h2

� �:

ð30Þ

When n = 2,

a0I + G2ð Þe3 = q1I − G1ð Þe2 + q2e1 + R3,

e3�� �� = q0I +G2ð Þ−1 q1I − G1ð Þe2 − q2e

1 + R3� �≤ q0I + G2ð Þ−1�� �� q1I −G1k k − q2 − q3½ �q−13 R:

ð31Þ

Case 3. max fw1, rg ≤w1,

e3�� �� ≤ q0I +G2ð Þ−1�� �� q1I −G1k k − q2 − q3½ �q−13 R

≤1

q0 + rq1 + q1 − q0ð Þð Þq−13 R

� �≤ q−13 R

≤ q−13 C τ3−α + h2� �

:

ð32Þ

Case 4. max fw1, rg ≤ r,

e3�� �� = q0I + G2ð Þ−1 q1I −G1ð Þe2 − q2e

1 + R3� �≤ q0I +G2ð Þ−1�� �� w1I −G1k k − q2 − q3½ �q−13 R

≤1

q0 + rr − q1 − q0 − q1ð Þð Þq−13 R

� �≤2q1 − q0 + r

q0 + rq−13 R ≤ q−13 R ≤ q−13 C τ3−α + h2

� �:

ð33Þ

Suppose the inequality kenk ≤ ke1k holds, when n ≤ k + 1.When n = k + 2, reference to the proof process of stability,have

a0I +G1ð Þen+1 = q1I − G2ð Þen + q2en−1 ⋯ +qn−1e2 + qne

1 + ταRn+1,a0I +G2ð Þen+2 = q1I − G1ð Þen+1 + q2e

n ⋯ +qne2 + qn+1e1 + ταRn+2:

(ð28Þ

6 Advances in Mathematical Physics

ek+2��� ��� = q0I +G2ð Þ−1 q1I − G1ð Þ q0I +G1ð Þ−1 q1I −G2ð Þ�� ��

� ek��� ��� + q0I +G2ð Þ−1 q1I −G1ð Þ q0I +G1ð Þ−1�� ��� q2e

k−1+⋯+qk−1e2 + qke1

� ���� ���+ q0I +G2ð Þ−1�� �� q2e

k+⋯+qke2 + qk+1e1

� ���� ���+ q0I +Gð Þ−1�� ��Rk+2 ≤

rq0 + r

2R + r 1 −w1ð Þ

q0 + rð Þ2 R

+ 1q0 + r

q2+⋯+qk + qk+1ð Þq−1k+1R

≤r2

q0 + rð Þ2 + r q0 − q1j jq0 + rð Þ2 + q0 − q1j j

q0 + r

" #a−1k+1R

≤r

q0 + rr

q0 + r+ q0 − q1j j

q0 + r

+ q0 − q1j j

q0 + r

� �q−1k+1R

≤r

q0 + r+ q0 − q1j j

q0 + r

� �q−1k+1R ≤ q−1k+1R

≤ q−1k+1C τ3−α + h2� �

:

ð34Þ

Because qk = −aðdk−1 − dkÞ + bck, then there is a posi-tive finite constant C1, makes the inequality ken+1k ≤ q−1n+1C1ðτ3−α + h2Þ hold. We have the following conclusions:

u xi, tnð Þ − unik k∞ ≤ C1 τ3−α + h2� �

: ð35Þ

In summary, the theorem is obtained.

Theorem 5. The PASE-I difference scheme (11) for the timefractional telegraph equation is convergent, and kuðxi, tnÞ −uni k∞ ≤ Cðτ3−α + h2Þ, C is a positive number.

3.4. PASI-E Parallel Difference Scheme. The PASI-E schemeof the time fractional telegraph equation can be obtained bychanging the computational order of explicit segment andimplicit segment. In the even time layer, the PASI-E schemeis obtained by using the rule “implicit segment-explicitsegment-explicit segment” and the rule “explicit segment-implicit segment-explicit segment” in the odd time layer.The PASI-E scheme for solving the time fractional telegraphequation is as follows:

Here, n = 1, 3, 5⋯ , the definition of G1,G2,H, Fn, bn1 isdefined as before.

Since the difference between PASE-I format (11) andPASI-E (36) lies only in the order of using explicit format

and implicit format, the computational complexity of thetwo formats should be equal in theory.

By the same proof process, the following theorem can beobtained.

0.50.500

PASI-E schemePASE-I scheme

Exact solution Implicit scheme

tx

0.750.750.250.25

110

0.5

u (x

,t)

0

0.5

u (x

,t)

0

0.5

u (x

,t)

0

0.5

u (x

,t)

0.50x

0.750.25

1

0.50x

0.750.25

10.5

0x

0.750.25

1

0.50

t

0.750.25

1

0.50

t

0.750.25

10.5

0t

0.750.25

1

0.50.500

PASI-E schemePASE-I schemetx

0.750.750.250.25

10

0.5

u (x

,t)

0

0.5

u(x

,t)

0.50x

0.750.25

0.50.750 25 0.50.75

0 25

1

0.50

t

0.750.20 5

1

0.5 0.750 25

10.5 0.75

0 25

Figure 2: Exact and numerical solutions of surfaces (M = 50, N = 100).

q0I +G2ð ÞVn+1 = q1I −G1ð ÞVn + q2Vn−1 ⋯ +qn−1V2 + qnV

1 + 2adkτH + bn+11 + Fk+1,q0I +G1ð ÞVn+2 = q1I −G2ð ÞVn+1 + q2V

n ⋯ +qnV2 + qn+1V1 + 2adk+1τH + bn+21 + Fn+2:

(ð36Þ

7Advances in Mathematical Physics

Theorem 6. The PASI-E difference scheme (36) of the timefractional telegraph equation is unconditionally stable andconvergent. Moreover, kuðxi, tnÞ − uni k∞ ≤ Cðτ3−α + h2Þ, C isa positive number.

4. Numerical Experiments

The numerical experiment is based on Intel Core i5-2400CPU@3.10GHz and is carried out under the environmentof MATLAB R2014a. We will verify the theoretical analysisby numerical experiments.

Example 1. Consider the following time fractional telegraphequation, take α = 1:8 [6, 7].

∂1:8u x, tð Þ∂t1:8

+ ∂0:8u x, tð Þ∂t0:8

= ∂2u x, tð Þ∂x2

+ f x, tð Þ, 0 ≤ x ≤ 1, 0 ≤ t ≤ 1:ð37Þ

Here, f ðx, tÞ = xð1 − xÞðð2/ðΓð1:2ÞÞÞt0:2 + ð2/ðΓð2:2ÞÞÞt1:2Þ + 2ð1 + t2Þ:

The initial condition: uðx, 0Þ = xð1 − xÞ, utðx, 0Þ = 0, 0 <x < 1:

The boundary condition: uð0, tÞ = 0, uð1, tÞ = 0.Then, the equation has exact solutions: uðx, tÞ = ð1 + t2Þ

xð1 − xÞ.

When M = 50, N = 100, the analytical solution surfaceand numerical solution surface of the three differenceschemes of the fractional telegraph equation are shown inFigure 2. From Figure 2, we can see that the PASE-I andPASI-E difference schemes and implicit schemes are smoothand can approximate analytical solutions very well. The errorsurfaces of the three difference schemes are shown inFigure 3. From Figure 3, we can see that the error of implicitdifference schemes is less than 2.5e-3, and the error limits ofPASE-I and PASI-E schemes are less than 3e-4. The accuracy

of PASE-I or PASI-E scheme is better than the implicit differ-ence scheme.

Because the accuracy of PASE-I and PASI-E are similar,we take PASE-I as an example to investigate the variationof relative error (RE) of PASE-I in time direction. The rela-tive error is defined as follows. The analytical solution isregarded as the control solution, and the solution of PASE-Ischeme is regarded as the perturbation solution. The formulafor relative error is

RE ið Þ = 〠M

j=1

uij − �uij��� ���

uij: ð38Þ

The space step h is 1/50 and the time step is 1/100,respectively. The RE of PASE-I is shown in Figure 4. FromFigure 4, we can know that the RE of PASE-I scheme is lessthan 0.16. The RE decreases with the advance of time step,

0 10 20 30 40 50 60 70 80 90 100Time step

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16RE

Figure 4: The curve of RE changes with time.

0.50.5

00 t

x

0.750.75

−0.5

0

0

1

2

3

Error surface of implicit Error surface of PASE-I Error surface of PASE-E

0.5

1

1.5

2

2.5

0.250.25

11

0.5

0

x

0.75

0.25

1

0.5

0

x

0.75

0.25

1

0.5

0 t

0.75

0.25

1

0.5

0 t

0.75

0.25

1

u (x

,t)

u (x

,t)

0

1

2

3

u (x

,t)

×10–4×10–4

×10–3

0.50.5

tt

0.7575

0

1

2

3

0.250.25

10.5

00

x

0.75

0.25

1

0.5

00

x

0.75

0.25

1

0.5tt

0.75

0.25

1

0.5tt

0.7555

0.25u

(x,t)

0

1

2

3

u (x

,t)

Figure 3: The error surface of numerical solutions.

8 Advances in Mathematical Physics

which indicates that the PASE-I scheme of time fractionaltelegraph equation is computationally stable.

Next, time convergence order (order1) and space conver-gence order (order2) are defined by the maximum moduluserror. The theoretical analysis is validated by the followingnumerical experiments. With fixed space step h, the order1is defined as follows [8, 9]:

order1 = log2E∞ τ3−α + h2� �

E∞ τ/2ð Þ3−α + h2� �

!≈ 3 − α: ð39Þ

Let τ3−α = h2, the order2 is defined as follows:

order2 = log2E∞ τ1

3−α + h2� �

E∞ τ23−α + h/2ð Þ2� �

!= log2

E∞ h2� �

E∞ h/2ð Þ2� � !

≈ 2:

ð40Þ

From Table 1, the convergence order of PASE-I andPASI-E schemes in time direction is 3 − α order, and theaccuracy of implicit schemes is equal to that of PASE-Iand PASI-E schemes. From Table 2, we can see the spatialconvergence orders of PASE-I and PASI-E schemes areboth of second order, which is consistent with the theoreticalanalysis.

Example 2. Consider the following time fractional telegraphequation [1, 5].

∂1:8u x, tð Þ∂t1:8

+ ∂0:8u x, tð Þ∂t0:8

= ∂2u x, tð Þ∂x2

, 0 ≤ x ≤ 1, 0 ≤ t ≤ 1:

ð41Þ

The initial conditions: uðx, 0Þ = xð1 − xÞ, utðx, 0Þ = 0,0 < x < 1:

The boundary conditions: uð0, tÞ = 0, uð1, tÞ = 0.When the time division is 500 and the space mesh points

is 50, the numerical solution surfaces of the three schemes areshown in Figure 5. From Figure 5, it can be seen that thenumerical solution surfaces of the PASE-I and PASI-Eschemes are the same and smooth as those of the classicalimplicit scheme.

We define the speedup as Sp = T1/Tp (T1 is the CPU timeof implicit, Tp is the CPU time of parallel scheme) and theefficiency as Ep = Sp/p (p is the number of processors in par-allel processor) [30, 31]. We use four cores for this numericalexperiment. When the number of spatial grid points is 100,200, 400, 800, 1600, and 3200, respectively, the CPU timerequired for the three schemes is shown in Table 3. FromTable 3, it can be seen that the CPU time of serial differenceincreases exponentially, and the parallel difference schemesincrease relatively slowly, with the increase of the numberof spatial grid points. Compared with serial differenceschemes, the computational efficiency of the PASE-I andPASI-E parallel schemes are greatly improved with therefinement of the spatial mesh. When the number of spatialgrids is small (100), the computing time of parallel differencescheme is almost the same as that of the serial implicitscheme, because the communication between modules con-sumes a lot of CPU time. With the increase of computationaldomain, the parallel computing characteristics of PASE-I andPASI-E schemes will become more prominent. When thenumber of grid points is 1600, the efficiency of parallel dif-ference schemes is optimal in this example. The linearacceleration ratio can be achieved when the number ofspace grid points is more than 800. Compared with theserial difference scheme, the computation time of PASE-I

Table 1: Time order (order1) of difference schemes (h = 1/200).

τ Implicit E∞Implicitorder1

PASE-I E∞PASE-Iorder1

PASI-E E∞PASI-Eorder1

1/200 2.987928e-5 2.792870e-5 2.794564e-5

1/400 1.520501e-5 0.974597 1.197192e-5 1.222094 1.197806e-5 1.222228

1/800 7.720232e-6 0.977831 5.127047e-6 1.223454 5.129525e-6 1.223497

1/1600 3.911829e-6 0.980800 2.190633e-6 1.226780 2.191679e-6 1.226788

1/3200 1.978420e-6 0.983494 9.332082e-7 1.231076 9.336585e-7 1.231069

Table 2: Space order (order2) of difference schemes.

h Implicit E∞Implicitorder2

PASE-I E∞PASE-Iorder2

PASI-E E∞PASI-Eorder2

1/10 1.500088e-4 6.358423e-4 6.304834e-4

1/20 4.820269e-5 1.637862 1.597008e-4 1.993297 1.596684e-4 1.981378

1/40 1.576648e-5 1.612252 3.994980e-5 1.999111 3.998717e-5 1.997470

1/80 5.148237e-6 1.614710 1.001469e-5 1.996069 1.002560e-5 1.995848

1/160 1.676390e-6 1.618720 2.529035e-6 1.985459 2.531798e-6 1.985453

1/320 5.420003e-7 1.628992 6.398285e-7 1.982830 6.405186e-7 1.982850

9Advances in Mathematical Physics

and PASI-E scheme can save about 80% when the numberof grid points is great 800.

5. Conclusion

In this paper, a class of alternating segment pure explicit-implicit (PASE-I) and implicit-explicit (PASI-E) parallel dif-ference methods are constructed for the time fractionaltelegraph equations. From theoretical analysis, it can be con-cluded that the parallel difference methods are uncondition-ally convergent and stable, with second-order convergence inspatial direction and 3 − α order in temporal direction. Thenumerical experiments show that the parallel differenceschemes are more efficient than the serial difference schemewith the increase of space mesh generation. The numericalexperiments are in agreement with the theoretical analysis,which show that the numerical algorithm is efficient and fea-sible for solving the time fractional telegraph equation.

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

The research was partly supported by the Subproject ofMajor Science and Technology Program of China(2017ZX07101001-1) and the Fundamental Research Fundsfor the Central Universities (2018MS168).

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0.5

00 t

0.75

0.25

1

0.5

0 t

0.75

0.25

1

0.5

0 t

0.75

0.25

10.5

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PASI-E scheme PASE-I schemeImplicit scheme

0.5t

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0.5t

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100 29.26 18.72 18.74 1.56 0.39

200 52.45 23.63 23.72 2.21 0.55

400 95.11 33.69 33.50 2.82 0.70

800 226.02 48.19 48.91 4.68 1.17

1600 433.85 81.76 83.42 5.30 1.32

3200 895.42 170.69 170.18 5.24 1.31

10 Advances in Mathematical Physics

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11Advances in Mathematical Physics