# An Efficient Alternating Segment Parallel Difference Method for the Time Fractional Telegraph Equation

Authors: Lifei Wu; Xiaozhong Yang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

The fractional telegraph equation is a kind of important evolution equation, which has an important application in signal analysis such as transmission and propagation of electrical signals. However, it is difficult to obtain the corresponding analytical solution, so it 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 the alternating segment technology, the PASE-I and PASI-E schemes are constructed of the classic explicit scheme and implicit scheme. It can be concluded that the schemes are unconditionally stable and convergent by theoretical analysis. The convergence order of the PASE-I and PASI-E methods is second order in spatial direction and 3-α order in temporal direction. The numerical results are in agreement with the theoretical analysis, which shows that the PASE-I and PASI-E schemes are superior to the classical implicit schemes in both accuracy and efficiency. This implies that the parallel difference schemes are efficient for solving the time fractional telegraph equation.


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## Body

## 1. Introduction

Nowadays, the fractional derivative and fractional differential equation have many applications in various fields, such as boundary layer effects in ducts, colored noise, dielectric polarization, electromagnetic waves, fractional kinetics, power-law phenomenon in fluid and complex network, quantitative finance, and viscoelastic mechanics [1–3]. The fractional telegraph equation especially is applied into signal analysis for transmission, propagation of electrical signals, and so on [4]. Because the analytical solution of the fractional telegraph equation is difficult to give explicitly or contains special functions such as the Mittag-Leffler function, which are difficult to calculate, the development of effectively numerical algorithms for solving the fractional telegraph equation is important [5–7].Because of the historical dependence and global correlation of fractional calculus, the computational and storage requirements of fractional differential equations’ numerical simulation are enormous [8–10]. Even with high-performance computers, it is difficult to simulate long-term or large-scale computational domains. With the rapid development of multicore and cluster technology, parallel algorithm has become one of the mainstream technologies in improving the computational efficiency [11]. Zhang et al. constructed a segment implicit scheme by using Saul’yev asymmetric scheme and used the alternative technique to establish a variety of alternating explicit-implicit and implicit parallel methods [12]. The methods have been well applied to numerical solving integer partial differential equation [13–16]. However, efficient parallel numerical methods for integer order differential equations may not be effective for fractional order differential equations. It may even produce completely different numerical analysis processes. How to extend the existing parallel difference method of integer order differential equation to the method of fractional order differential equation is a great challenge to computational mathematics (physics).In recent years, many scholars have studied the numerical algorithms of the fractional telegraph equation. For example, Ford et al. used the finite difference method to numerically solve the fractional telegraph equation [17]. Saadatmandi and Mohabbati proposed a numerical solution by combining orthogonal Legendre polynomial and Tau method for the fractional telegraph equation [18]. Niu et al. applied the Chebyshev polynomial to get the numerical solution of the fractional telegraph equation [19]. Zhao and Li applied the finite difference method and Galerkin finite element method to solve the time-space fractional telegraph equation numerically [20]. Chen constructed the implicit difference scheme for the Riesz space fractional order telegraph equation [21]. Ren and Liu presented a high-order compact finite difference method for a class of time fractional Black-Scholes equation [22]. The scheme has the second-order temporal accuracy and the fourth-order spatial accuracy. The existing numerical algorithms of fractional differential equations are mostly serial algorithms, which have low computational efficiency.In recent years, some progress has been made in fast algorithms for fractional partial differential equations [23, 24], most of which are parallel algorithms for algebraic systems based on the view of numerical algebra. For example, Diethelm implemented the second-order Adams-Bashforth-Moulton method of fractional diffusion equations on a parallel computer and discussed the accuracy of parallel method [25]. Gong et al. parallelizes the explicit difference scheme of the space fractional reaction-diffusion equation [26]. Gong et al. parallelizes the implicit scheme of the time fractional diffusion equation [27] and the core of parallelization is to do parallel computation for the product of matrix and vector and the addition of vector and vector. Sweilam et al. applied preconditioned conjugate gradient method was used to solve discrete algebraic equations in parallel, based on Crank-Nicolson difference schemes for time fractional parabolic equations [28]. Wang et al. studied the parallel algorithm of implicit difference schemes for fractional reaction-diffusion equations [29]. The algorithm is based on the principle of minimizing communication, allocating computing tasks reasonably, and not changing the original serial difference schemes as much as possible.In order to obtain more accurate and stable parallel difference schemes, we are prepared to take the parallel path of traditional difference schemes and hope to find another way to parallelize them beyond the difficulty of numerical algebra [30, 31]. In this paper, a kind of alternating segment pure explicit-implicit (PASE-I) and implicit-explicit (PASI-E) parallel difference schemes is obtained. The numerical experiments and theoretical analysis are consistent, showing that the PASE-I and PASI-E schemes are unconditionally stable and convergent. The numerical examples show that the PASE-I and PASI-E difference schemes have obvious parallel computational properties. It shows that the parallel intrinsic difference schemes proposed in this paper is efficient for solving the time fractional telegraph equations.


## 2. PASE-I Scheme for Fractional Telegraph Equation

### 2.1. Time Fractional Telegraph Equation

Consider the following time fractional telegraph equation [3–5]:
(1)∂αux,t∂tα+∂α−1ux,t∂tα−1=K∂2ux,t∂x2+fx,t,0≤x≤L,t≥0,1<α≤2.The initial conditions:ux,0=φ0x,∂ux,0/∂t=φ1x.The boundary conditions:u0,t=0,uL,t=0.


### 2.2. Construction of PASE-I Scheme

In order to obtain the PASE-I difference scheme for the fractional telegraph equation, the solution region is first meshed: the space and time steps are takenh=L/M and τ=T/N, where M and N are natural numbers; xi=ihi=1,2,⋯,M,Mh=L; tk=kτk=1,2,⋯,N,Nτ=T; and the grid nodes are xi,tk.The time fractional derivative can be approximated by L1 formula [8, 9]:
(2)∂α−1uxi,tk+1∂tα−1=1Γ2−α∑j=0kuxi,tj+1−uxi,tjτ∫jτj+1τdξtk+1−ξα−1+Oτ3−α≈τ2−α2−αΓ2−α∑j=0kuxi,tk+1−j−uxi,tk−jτj+12−α−j2−α=τ1−αΓ3−α∑j=0kcj∇tuxi,tk−j,(3)∂αuxi,tk+1∂tα=1Γ2−α∑j=0kuxi,tk+1−j−2uxi,tk−j+uxi,tk−j−1τ2∫jτj+1τdξtk+1−ξα−1+Oτ3−α≈τ−αΓ3−α∑j=0k‐1djδt2uxi,tk−j+2dkuxi,t1−uxi,t0−τutxi.Spatial second derivatives are discretized by central difference method:(4)∂2uxi,tk+1∂x2=δx2θuxi,tk+1+1−θuxi,tk+Oh2.Here,δx2uxi,tk=1/h2uxi−1,tk−2uxi,tk+uxi+1,tk,∇tuxi,tk=1/τuxi,tk+1−uxi,tk,cj=j+12−α−j2−α,dj=j+12−α−j2−α,j=0,1,2,⋯,k−1.By substituting formulas (2), (3), (4) into equation (1), the universal difference schemes are obtained as follows:
(5)τ−αΓ3−α∑j=0k‐1cjδt2uxi,tk−j+τ1−αΓ3−α∑j=0kdj∇tuxi,tk−j=Kh2δx2θuxi,tk+1+1−θuxi,tk+fxi,tk+1.Letuik denote the value of ux,t at the point xi,tk, fik denote the value of fx,t at the pointxi,tk, and a=τ−α/Γ3−α,b=τ1−α/Γ3−α,r=K/h2. The universal difference scheme can be written as follows:
(6)a∑j=0k−1djδt2uik−j+2dkuxi,t1−uxi,t0−τutxi+b∑j=0k−1cj∇tuik−j=rδx2θuik+1+1−θuik+fik+1,k=0,1,2,⋯.Whenθ=0, (6) is the explicit scheme, its advantage is explicit parallel computation, but its disadvantage is conditional stability. When θ=1, (6) is the implicit scheme, its advantage is unconditional stability. The disadvantage is that it needs to calculate tridiagonal equations, which is not easy to parallel computation and has low computational efficiency.The matrix form of universal difference scheme is as follows:(7)2a+bI+rθGU1=2a+bI−r1−θGU0+2aτH+b1+F1,AUk+1=BUk+q2Uk−1+⋯+qk−2U2+qk−1U1+qkU0+2adkτH+bk+1+Fk+1.Here,q0=ad0+bc0,q1=a2d0−d1+bc0−c1,qj=−adj−2−2dj−1+dj+bcj−1−cj,j=2,3,⋯,k−1, qk=a−dk−2+2dk−1−dk+bck−1−ck,qk+1=a−dk−1+dk+bck,bk+1=rθu1k+1+1−θu1k,0,⋯,0,rθuM+1k+1+1−θuM+1k′,Uk=u1k,u2k,⋯,uM−1k′,U0=φ0x1,φ0x2,⋯,φ0xM−1′,H=φ1x1,φ1x2,⋯,φ1xM−1′,F=f1,f2,⋯,fM−1′,A=ad0+bc0I+rθG,B=a2d0−d1+bc0−c1I−r1−θG,E is the unit matrix of order M−1,
(8)G=2−1−12−1⋱⋱⋱−12−1−12M−1×M−1.By alternatingly segment applying explicit and implicit schemes, the alternating segment explicit-implicit (PASE-I) scheme for fractional telegraph equation is designed as follows: LetM−1=ql, q,l∈N+, l≥3, q an odd number, and q≥3. The points to be computed in the same even time layer are divided into q segments, which are computed accordingly by the rule of “explicit segment-implicit segment-explicit segment”. Similarly, the next odd layer is also divided into q segment calculation. The calculation rule is changed to “implicit segment-explicit segment-implicit segment”. The computational lattice diagram of PASE-I scheme is shown in Figure 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.Figure 1
Computational lattice diagram of PASE-I scheme.For the PASE-I scheme andi0≥0, consider the calculation of pointsi0+i,k+1, i=1,2,⋯,l in implicit segments. The implicit segment format is as follows:
(9)2ad0+bc0I+rGVk+1=a2d0−d1+bc0−c1IVk+q2Vk−1+⋯+qk−1V2+qkV1+qk+1V0+2adkτH1+bimk+1+F1k+1.The explicit segment format is as follows:(10)2ad0+bc0IVk+1=a2d0−d1+bc0−c1I+rGVk+q2Vk−1+⋯+qk−1V2+qkV1+qk+1V0+2adkτH1+bexk+1+F1k+1.Here,Vn=ui0+1n,ui0+2n,⋯,ui0+ln′,H1=φ1xi0+1,φ1xi0+2,⋯,φ1xi0+l′,F1=fi0+1,fi0+2,⋯,fi0+l′,bimk+1=rui0−1k+1,0,⋯,0,rui0+l+1k+1′,bexk+1=ru1k,0,⋯,0,rui0+l+1k′.The PASE-I scheme is as follows:(11)q0I+rG1Un+1=q1I−rG2Un+q2Un−1⋯+qn−1U2+qnU1+qn+1U0+2adkτH+b1n+1+Fk+1,q0I+rG2Un+2=q1I−rG1Un+1+q2Un⋯+qnU2+qn+1U1+qn+2U0+2adk+1τH+b2n+2+Fn+2.n=1,3,5⋯, b1n+1=ru0n,0,⋯,0,ruMn′,b2n+2=ru0n+2,0,⋯,0,ruMn+2′,Ql′ is a zero matrix of order l′,l′=l−1/l−2;G1,G2 is martix of order M−1 and the definition is as follows:
(12)G1=Ql−1Gl+2Ql−2Gl+2⋱Ql−2Gl+2Ql−1M−1×M−1,G2=G~l+1Ql−2Gl+2Ql−2⋱Gl+2Ql−2G¯l+1M−1×M−1.Here,
(13)Gl+2=00−12−1⋱⋱⋱−12−100l+2×l+2,(14)G~l+1=2−1−12−1⋱⋱⋱−12−100l+1×l+1,(15)G¯l+1=00−12−1⋱⋱⋱−12−1−12l+1×l+1.


## 2.1. Time Fractional Telegraph Equation

Consider the following time fractional telegraph equation [3–5]:
(1)∂αux,t∂tα+∂α−1ux,t∂tα−1=K∂2ux,t∂x2+fx,t,0≤x≤L,t≥0,1<α≤2.The initial conditions:ux,0=φ0x,∂ux,0/∂t=φ1x.The boundary conditions:u0,t=0,uL,t=0.


## 2.2. Construction of PASE-I Scheme

In order to obtain the PASE-I difference scheme for the fractional telegraph equation, the solution region is first meshed: the space and time steps are takenh=L/M and τ=T/N, where M and N are natural numbers; xi=ihi=1,2,⋯,M,Mh=L; tk=kτk=1,2,⋯,N,Nτ=T; and the grid nodes are xi,tk.The time fractional derivative can be approximated by L1 formula [8, 9]:
(2)∂α−1uxi,tk+1∂tα−1=1Γ2−α∑j=0kuxi,tj+1−uxi,tjτ∫jτj+1τdξtk+1−ξα−1+Oτ3−α≈τ2−α2−αΓ2−α∑j=0kuxi,tk+1−j−uxi,tk−jτj+12−α−j2−α=τ1−αΓ3−α∑j=0kcj∇tuxi,tk−j,(3)∂αuxi,tk+1∂tα=1Γ2−α∑j=0kuxi,tk+1−j−2uxi,tk−j+uxi,tk−j−1τ2∫jτj+1τdξtk+1−ξα−1+Oτ3−α≈τ−αΓ3−α∑j=0k‐1djδt2uxi,tk−j+2dkuxi,t1−uxi,t0−τutxi.Spatial second derivatives are discretized by central difference method:(4)∂2uxi,tk+1∂x2=δx2θuxi,tk+1+1−θuxi,tk+Oh2.Here,δx2uxi,tk=1/h2uxi−1,tk−2uxi,tk+uxi+1,tk,∇tuxi,tk=1/τuxi,tk+1−uxi,tk,cj=j+12−α−j2−α,dj=j+12−α−j2−α,j=0,1,2,⋯,k−1.By substituting formulas (2), (3), (4) into equation (1), the universal difference schemes are obtained as follows:
(5)τ−αΓ3−α∑j=0k‐1cjδt2uxi,tk−j+τ1−αΓ3−α∑j=0kdj∇tuxi,tk−j=Kh2δx2θuxi,tk+1+1−θuxi,tk+fxi,tk+1.Letuik denote the value of ux,t at the point xi,tk, fik denote the value of fx,t at the pointxi,tk, and a=τ−α/Γ3−α,b=τ1−α/Γ3−α,r=K/h2. The universal difference scheme can be written as follows:
(6)a∑j=0k−1djδt2uik−j+2dkuxi,t1−uxi,t0−τutxi+b∑j=0k−1cj∇tuik−j=rδx2θuik+1+1−θuik+fik+1,k=0,1,2,⋯.Whenθ=0, (6) is the explicit scheme, its advantage is explicit parallel computation, but its disadvantage is conditional stability. When θ=1, (6) is the implicit scheme, its advantage is unconditional stability. The disadvantage is that it needs to calculate tridiagonal equations, which is not easy to parallel computation and has low computational efficiency.The matrix form of universal difference scheme is as follows:(7)2a+bI+rθGU1=2a+bI−r1−θGU0+2aτH+b1+F1,AUk+1=BUk+q2Uk−1+⋯+qk−2U2+qk−1U1+qkU0+2adkτH+bk+1+Fk+1.Here,q0=ad0+bc0,q1=a2d0−d1+bc0−c1,qj=−adj−2−2dj−1+dj+bcj−1−cj,j=2,3,⋯,k−1, qk=a−dk−2+2dk−1−dk+bck−1−ck,qk+1=a−dk−1+dk+bck,bk+1=rθu1k+1+1−θu1k,0,⋯,0,rθuM+1k+1+1−θuM+1k′,Uk=u1k,u2k,⋯,uM−1k′,U0=φ0x1,φ0x2,⋯,φ0xM−1′,H=φ1x1,φ1x2,⋯,φ1xM−1′,F=f1,f2,⋯,fM−1′,A=ad0+bc0I+rθG,B=a2d0−d1+bc0−c1I−r1−θG,E is the unit matrix of order M−1,
(8)G=2−1−12−1⋱⋱⋱−12−1−12M−1×M−1.By alternatingly segment applying explicit and implicit schemes, the alternating segment explicit-implicit (PASE-I) scheme for fractional telegraph equation is designed as follows: LetM−1=ql, q,l∈N+, l≥3, q an odd number, and q≥3. The points to be computed in the same even time layer are divided into q segments, which are computed accordingly by the rule of “explicit segment-implicit segment-explicit segment”. Similarly, the next odd layer is also divided into q segment calculation. The calculation rule is changed to “implicit segment-explicit segment-implicit segment”. The computational lattice diagram of PASE-I scheme is shown in Figure 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.Figure 1
Computational lattice diagram of PASE-I scheme.For the PASE-I scheme andi0≥0, consider the calculation of pointsi0+i,k+1, i=1,2,⋯,l in implicit segments. The implicit segment format is as follows:
(9)2ad0+bc0I+rGVk+1=a2d0−d1+bc0−c1IVk+q2Vk−1+⋯+qk−1V2+qkV1+qk+1V0+2adkτH1+bimk+1+F1k+1.The explicit segment format is as follows:(10)2ad0+bc0IVk+1=a2d0−d1+bc0−c1I+rGVk+q2Vk−1+⋯+qk−1V2+qkV1+qk+1V0+2adkτH1+bexk+1+F1k+1.Here,Vn=ui0+1n,ui0+2n,⋯,ui0+ln′,H1=φ1xi0+1,φ1xi0+2,⋯,φ1xi0+l′,F1=fi0+1,fi0+2,⋯,fi0+l′,bimk+1=rui0−1k+1,0,⋯,0,rui0+l+1k+1′,bexk+1=ru1k,0,⋯,0,rui0+l+1k′.The PASE-I scheme is as follows:(11)q0I+rG1Un+1=q1I−rG2Un+q2Un−1⋯+qn−1U2+qnU1+qn+1U0+2adkτH+b1n+1+Fk+1,q0I+rG2Un+2=q1I−rG1Un+1+q2Un⋯+qnU2+qn+1U1+qn+2U0+2adk+1τH+b2n+2+Fn+2.n=1,3,5⋯, b1n+1=ru0n,0,⋯,0,ruMn′,b2n+2=ru0n+2,0,⋯,0,ruMn+2′,Ql′ is a zero matrix of order l′,l′=l−1/l−2;G1,G2 is martix of order M−1 and the definition is as follows:
(12)G1=Ql−1Gl+2Ql−2Gl+2⋱Ql−2Gl+2Ql−1M−1×M−1,G2=G~l+1Ql−2Gl+2Ql−2⋱Gl+2Ql−2G¯l+1M−1×M−1.Here,
(13)Gl+2=00−12−1⋱⋱⋱−12−100l+2×l+2,(14)G~l+1=2−1−12−1⋱⋱⋱−12−100l+1×l+1,(15)G¯l+1=00−12−1⋱⋱⋱−12−1−12l+1×l+1.


## 3. Numerical Analysis of PASE-I and PASI-E Difference Schemes

### 3.1. Existence and Uniqueness of PASE-I Solution

Lemma 1.
The matricesa2d0−d1+bc0−c1I+G1 and a2d0−d1+bc0−c1I+G2 defined by PASE-I format are nonsingular matrices.Proof.
From the definition ofG1 and a2d0−d1+bc0−c1>0, we can know that a2d0−d1+bc0−c1I+G1 is a strictly diagonally dominant matrix, and the principal diagonal element is a positive real number. So a2d0−d1+bc0−c1I+G1 is a nonsingular matrix, and its inverse matrix also exists. Thus, there is the following theorem.Theorem 2.
The PASE-I difference scheme (11) for the time fractional 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 any parameterρ≥0,0≤σ1≤σ2, there is an estimate
(16)σ1I−ρCσ2I+ρC−12≤1.Proof.
σ1I−ρCσ2I+ρC−122=maxφ∈Rn,φ≠0σ1I−ρCσ2I+ρC−1φ,σ1I−ρCσ2I+ρC−1φ/φ,φ.
Make a transformationψ=σ2I+ρC−1φ, then
(17)σ1I−ρCσ2I+ρC−122=maxφ∈Rn,φ≠0σ1I−ρCψ,σ1I−ρCψσ2I+ρCψ,σ2I+ρCψ=maxψ∈Rn,ψ≠0σ12ψ,ψ−2σ1ρCψ,ψ+ρ2Cψ,Cψσ22ψ,ψ+2σ2ρCψ,ψ+ρ2Cψ,Cψ.
From0≤σ1≤σ2, we have σ1I−ρCσ2I+ρC−12≤1.
Letu~in be the approximate solution of the difference scheme, and uin be the exact solution of difference scheme. Let εin=u~in−uin,En=ε1n,ε3n,⋯,εM−1n,1≤n≤N+1,ε0n=0,εMn=0, so we have
(18)a0I+G1En+1=w1I−G2En+w2En−1⋯+wn−1E2+anE1,a0I+G2En+2=w1I−G1En+1+w2En⋯+wnE2+an+1E1.where n=1,3,⋯,w1=ad0+bc0,w2=a2d0−d1+bc0−c1.
Whenn≥3, we have
(19)En+2=q0I+G2−1q1I−G1q0I+G1−1q1I−G2En+q0I+G2−1q1I−G1q0I+G1−1q2En−1+⋯+qn−1E2+qnE1+q0I+G2−1q2En+⋯+qnE2+qn+1E1.
Take norms on both sides of formula (19):
(20)En+2=q0I+G2−1q1I−G1q0I+G1−1q1I−G2En+q0I+G2−1q1I−G1q0I+G1−1q2En−1+⋯+qn−1E2+qnE1+q0I+G2−1q2En+⋯+qnE2+qn+1E1.
The growth matrix of PASE-I scheme isT=q0I+G2−1q1I−G1q0I+G1−1q1I−G2. According to the definition of the matrix, Gl,G1, and G2 have the same eigenvalue. Let T~=q0I+G2Tq0I+G2−1, suppose the eigenvalue of G1 or G2 in the matrix is r. From the Lemma 3, we have
(21)T=T~=q1I−rG1q0I+rG1−1q1I−rG2q0I+rG2−1=maxq1−rq0+r2≤1.
Ifq1>r, let q1=μrμ>1, because q1−r/q0+r≤1, we have μ≤2.
The following is proven by mathematical inductionEn≤E1.
Whenn=1,
(22)E2=q0I+rG1−1q1I−rG2E1≤E1.
Whenn=2, case I maxq1,r=q1, and r<q1≤2r.
(23)E3≤q0I+rG2−1q1I−rG1q0I+rG1−1q1I−rG2E1+q0I+rG2−1q2E1≤maxq1−rq0+r2E1+maxq2q0+rE1≤maxr−q2q0+rE1≤E1.
Whenmaxq1,r=r,
(24)E3≤q0I+rG2−1q1I−rG1q0I+rG1−1q1I−rG2E1+q0I+rG2−1q2E1≤maxq1−rq0+rE1+maxq2q0+rE1≤maxr+q2r+q0E1≤E1.
SupposeEn≤E1 when n≤k+1. When n=k+2, we consider the following two situations.Case 1.
maxq1,r≤q1, and r<q1≤2r.
(25)Ek+2=q0I+rG¯2−1q1I−rG¯1q0I+rG¯1−1q1I−rG¯2Ek+q0I+rG¯2−1q1I−rG¯1q0I+rG¯1−1q2Ek−1+⋯+qk−1E2+qkE1+q0I+rG¯2−1q2Ek+⋯+qkE2+qk+1E1≤rq0+r2E1+rq0−q1q0+r2E1+q0−q1q0+rE1≤rq0+rE1+q0−q1q0+rE1≤E1.Case 2.
maxq1,r≤r.
(26)Ek+2=q0I+G¯2−1q1I−G¯1q0I+G¯1−1q1I−G¯2Ek+q0I+G¯2−1q1I−G¯1q0I+G¯1−1q2Ek−1+⋯+qk−1E2+qkE1+q0I+G¯2−1q2Ek+⋯+qkE2+qk+1E1≤ra0+r2E1+rq1−q0a0+r2E1+q1−q0a0+rE1=rq0+rrq0+r+q1−q0q0+rE1+q1−q0q0+rE1≤rq0+rE1+q1−q0q0+rE1≤E1.
To sum up, we have the following theorem.Theorem 4.
The PASE-I difference scheme (11) for the time fractional telegraph equation is unconditionally stable.


### 3.3. Convergence of PASE-I Scheme

Firstly, the accuracy of explicit and implicit schemes is analyzed, respectively, and Taylor expansion will be carried out atxi,tn+1. The truncation errors are recorded as T1τ,h and T2τ,h. It is known that 0CDtαuxi,tn+1 has second-order accuracy. We have
(27)T1τ,h=Dtαu+Dtα−1u−Kuxx+τuxxt−τ22uxxtt−h212uxxxx+τh212uxxxxt+Oτ3−α+h2,T2τ,h=Dtαu+Dtα−1u−Kuxx−τuxxt−τ22uxxtt−h212uxxxx−τh212uxxxxt+Oτ3−α+h2.For the PASE-I scheme, the explicit and implicit schemes are alternately used for each grid point in spatial direction. ForT1τ,h and T2τ,h, the coefficients of uxxt and uxxxxt are equal and symbolically opposite. When the explicit and implicit schemes are alternately used, the errors of the two terms will be offset. Therefore, the accuracy of the PASE-I scheme is second order in spatial direction and 3−α order in time direction.Letuxi,tn,i=1,2,⋯,M;n=1,2,⋯,N be the exact solution of equation (1) at grid point xi,tn. Let ein=uxi,tn−uin,en=e1n,e3n,⋯,eM−1nT. ein=uxi,tn−uin is introduced into the PASE-I scheme,
(28)a0I+G1en+1=q1I−G2en+q2en−1⋯+qn−1e2+qne1+ταRn+1,a0I+G2en+2=q1I−G1en+1+q2en⋯+qne2+qn+1e1+ταRn+2.Obviously,e1=0,Rn=O≤τ3−α+h2, there exists a positive constant C, such that ταRn=ταCτ3−α+h2.Whenn=1,
(29)e2=I+G1−1I−G2e1+I+G1−1R2=I+G1−1R.Combination Lemma3 has
(30)e2∞=I+G1−1R1∞≤R∞≤Cτ3−α+h2=q1−1Cτ3−α+h2.Whenn=2,
(31)a0I+G2e3=q1I−G1e2+q2e1+R3,e3=q0I+G2−1q1I−G1e2−q2e1+R3≤q0I+G2−1q1I−G1−q2−q3q3−1R.Case 3.
maxw1,r≤w1,
(32)e3≤q0I+G2−1q1I−G1−q2−q3q3−1R≤1q0+rq1+q1−q0q3−1R≤q3−1R≤q3−1Cτ3−α+h2.Case 4.
maxw1,r≤r,
(33)e3=q0I+G2−1q1I−G1e2−q2e1+R3≤q0I+G2−1w1I−G1−q2−q3q3−1R≤1q0+rr−q1−q0−q1q3−1R≤2q1−q0+rq0+rq3−1R≤q3−1R≤q3−1Cτ3−α+h2.
Suppose the inequalityen≤e1 holds, when n≤k+1. When n=k+2, reference to the proof process of stability, have
(34)ek+2=q0I+G2−1q1I−G1q0I+G1−1q1I−G2ek+q0I+G2−1q1I−G1q0I+G1−1q2ek−1+⋯+qk−1e2+qke1+q0I+G2−1q2ek+⋯+qke2+qk+1e1+q0I+G−1Rk+2≤rq0+r2R+r1−w1q0+r2R+1q0+rq2+⋯+qk+qk+1qk+1−1R≤r2q0+r2+rq0−q1q0+r2+q0−q1q0+rak+1−1R≤rq0+rrq0+r+q0−q1q0+r+q0−q1q0+rqk+1−1R≤rq0+r+q0−q1q0+rqk+1−1R≤qk+1−1R≤qk+1−1Cτ3−α+h2.
Becauseqk=−adk−1−dk+bck, then there is a positive finite constant C1, makes the inequality en+1≤qn+1−1C1τ3−α+h2 hold. We have the following conclusions:
(35)uxi,tn−uin∞≤C1τ3−α+h2.
In summary, the theorem is obtained.Theorem 5.
The PASE-I difference scheme (11) for the time fractional telegraph equation is convergent, and uxi,tn−uin∞≤Cτ3−α+h2, C is a positive number.


### 3.4. PASI-E Parallel Difference Scheme

The PASI-E scheme of the time fractional telegraph equation can be obtained by changing the computational order of explicit segment and implicit segment. In the even time layer, the PASI-E scheme is obtained by using the rule “implicit segment-explicit segment-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 telegraph equation is as follows:(36)q0I+G2Vn+1=q1I−G1Vn+q2Vn−1⋯+qn−1V2+qnV1+2adkτH+b1n+1+Fk+1,q0I+G1Vn+2=q1I−G2Vn+1+q2Vn⋯+qnV2+qn+1V1+2adk+1τH+b1n+2+Fn+2.Here,n=1,3,5⋯, the definition of G1,G2,H,Fn,b1n is defined as before.Since the difference between PASE-I format (11) and PASI-E (36) lies only in the order of using explicit format and implicit format, the computational complexity of the two formats should be equal in theory.By the same proof process, the following theorem can be obtained.Theorem 6.
The PASI-E difference scheme (36) of the time fractional telegraph equation is unconditionally stable and convergent. Moreover, uxi,tn−uin∞≤Cτ3−α+h2, C is a positive number.


## 3.1. Existence and Uniqueness of PASE-I Solution

Lemma 1.
The matricesa2d0−d1+bc0−c1I+G1 and a2d0−d1+bc0−c1I+G2 defined by PASE-I format are nonsingular matrices.Proof.
From the definition ofG1 and a2d0−d1+bc0−c1>0, we can know that a2d0−d1+bc0−c1I+G1 is a strictly diagonally dominant matrix, and the principal diagonal element is a positive real number. So a2d0−d1+bc0−c1I+G1 is a nonsingular matrix, and its inverse matrix also exists. Thus, there is the following theorem.Theorem 2.
The PASE-I difference scheme (11) for the time fractional 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 any parameterρ≥0,0≤σ1≤σ2, there is an estimate
(16)σ1I−ρCσ2I+ρC−12≤1.Proof.
σ1I−ρCσ2I+ρC−122=maxφ∈Rn,φ≠0σ1I−ρCσ2I+ρC−1φ,σ1I−ρCσ2I+ρC−1φ/φ,φ.
Make a transformationψ=σ2I+ρC−1φ, then
(17)σ1I−ρCσ2I+ρC−122=maxφ∈Rn,φ≠0σ1I−ρCψ,σ1I−ρCψσ2I+ρCψ,σ2I+ρCψ=maxψ∈Rn,ψ≠0σ12ψ,ψ−2σ1ρCψ,ψ+ρ2Cψ,Cψσ22ψ,ψ+2σ2ρCψ,ψ+ρ2Cψ,Cψ.
From0≤σ1≤σ2, we have σ1I−ρCσ2I+ρC−12≤1.
Letu~in be the approximate solution of the difference scheme, and uin be the exact solution of difference scheme. Let εin=u~in−uin,En=ε1n,ε3n,⋯,εM−1n,1≤n≤N+1,ε0n=0,εMn=0, so we have
(18)a0I+G1En+1=w1I−G2En+w2En−1⋯+wn−1E2+anE1,a0I+G2En+2=w1I−G1En+1+w2En⋯+wnE2+an+1E1.where n=1,3,⋯,w1=ad0+bc0,w2=a2d0−d1+bc0−c1.
Whenn≥3, we have
(19)En+2=q0I+G2−1q1I−G1q0I+G1−1q1I−G2En+q0I+G2−1q1I−G1q0I+G1−1q2En−1+⋯+qn−1E2+qnE1+q0I+G2−1q2En+⋯+qnE2+qn+1E1.
Take norms on both sides of formula (19):
(20)En+2=q0I+G2−1q1I−G1q0I+G1−1q1I−G2En+q0I+G2−1q1I−G1q0I+G1−1q2En−1+⋯+qn−1E2+qnE1+q0I+G2−1q2En+⋯+qnE2+qn+1E1.
The growth matrix of PASE-I scheme isT=q0I+G2−1q1I−G1q0I+G1−1q1I−G2. According to the definition of the matrix, Gl,G1, and G2 have the same eigenvalue. Let T~=q0I+G2Tq0I+G2−1, suppose the eigenvalue of G1 or G2 in the matrix is r. From the Lemma 3, we have
(21)T=T~=q1I−rG1q0I+rG1−1q1I−rG2q0I+rG2−1=maxq1−rq0+r2≤1.
Ifq1>r, let q1=μrμ>1, because q1−r/q0+r≤1, we have μ≤2.
The following is proven by mathematical inductionEn≤E1.
Whenn=1,
(22)E2=q0I+rG1−1q1I−rG2E1≤E1.
Whenn=2, case I maxq1,r=q1, and r<q1≤2r.
(23)E3≤q0I+rG2−1q1I−rG1q0I+rG1−1q1I−rG2E1+q0I+rG2−1q2E1≤maxq1−rq0+r2E1+maxq2q0+rE1≤maxr−q2q0+rE1≤E1.
Whenmaxq1,r=r,
(24)E3≤q0I+rG2−1q1I−rG1q0I+rG1−1q1I−rG2E1+q0I+rG2−1q2E1≤maxq1−rq0+rE1+maxq2q0+rE1≤maxr+q2r+q0E1≤E1.
SupposeEn≤E1 when n≤k+1. When n=k+2, we consider the following two situations.Case 1.
maxq1,r≤q1, and r<q1≤2r.
(25)Ek+2=q0I+rG¯2−1q1I−rG¯1q0I+rG¯1−1q1I−rG¯2Ek+q0I+rG¯2−1q1I−rG¯1q0I+rG¯1−1q2Ek−1+⋯+qk−1E2+qkE1+q0I+rG¯2−1q2Ek+⋯+qkE2+qk+1E1≤rq0+r2E1+rq0−q1q0+r2E1+q0−q1q0+rE1≤rq0+rE1+q0−q1q0+rE1≤E1.Case 2.
maxq1,r≤r.
(26)Ek+2=q0I+G¯2−1q1I−G¯1q0I+G¯1−1q1I−G¯2Ek+q0I+G¯2−1q1I−G¯1q0I+G¯1−1q2Ek−1+⋯+qk−1E2+qkE1+q0I+G¯2−1q2Ek+⋯+qkE2+qk+1E1≤ra0+r2E1+rq1−q0a0+r2E1+q1−q0a0+rE1=rq0+rrq0+r+q1−q0q0+rE1+q1−q0q0+rE1≤rq0+rE1+q1−q0q0+rE1≤E1.
To sum up, we have the following theorem.Theorem 4.
The PASE-I difference scheme (11) for the time fractional telegraph equation is unconditionally stable.


## 3.3. Convergence of PASE-I Scheme

Firstly, the accuracy of explicit and implicit schemes is analyzed, respectively, and Taylor expansion will be carried out atxi,tn+1. The truncation errors are recorded as T1τ,h and T2τ,h. It is known that 0CDtαuxi,tn+1 has second-order accuracy. We have
(27)T1τ,h=Dtαu+Dtα−1u−Kuxx+τuxxt−τ22uxxtt−h212uxxxx+τh212uxxxxt+Oτ3−α+h2,T2τ,h=Dtαu+Dtα−1u−Kuxx−τuxxt−τ22uxxtt−h212uxxxx−τh212uxxxxt+Oτ3−α+h2.For the PASE-I scheme, the explicit and implicit schemes are alternately used for each grid point in spatial direction. ForT1τ,h and T2τ,h, the coefficients of uxxt and uxxxxt are equal and symbolically opposite. When the explicit and implicit schemes are alternately used, the errors of the two terms will be offset. Therefore, the accuracy of the PASE-I scheme is second order in spatial direction and 3−α order in time direction.Letuxi,tn,i=1,2,⋯,M;n=1,2,⋯,N be the exact solution of equation (1) at grid point xi,tn. Let ein=uxi,tn−uin,en=e1n,e3n,⋯,eM−1nT. ein=uxi,tn−uin is introduced into the PASE-I scheme,
(28)a0I+G1en+1=q1I−G2en+q2en−1⋯+qn−1e2+qne1+ταRn+1,a0I+G2en+2=q1I−G1en+1+q2en⋯+qne2+qn+1e1+ταRn+2.Obviously,e1=0,Rn=O≤τ3−α+h2, there exists a positive constant C, such that ταRn=ταCτ3−α+h2.Whenn=1,
(29)e2=I+G1−1I−G2e1+I+G1−1R2=I+G1−1R.Combination Lemma3 has
(30)e2∞=I+G1−1R1∞≤R∞≤Cτ3−α+h2=q1−1Cτ3−α+h2.Whenn=2,
(31)a0I+G2e3=q1I−G1e2+q2e1+R3,e3=q0I+G2−1q1I−G1e2−q2e1+R3≤q0I+G2−1q1I−G1−q2−q3q3−1R.Case 3.
maxw1,r≤w1,
(32)e3≤q0I+G2−1q1I−G1−q2−q3q3−1R≤1q0+rq1+q1−q0q3−1R≤q3−1R≤q3−1Cτ3−α+h2.Case 4.
maxw1,r≤r,
(33)e3=q0I+G2−1q1I−G1e2−q2e1+R3≤q0I+G2−1w1I−G1−q2−q3q3−1R≤1q0+rr−q1−q0−q1q3−1R≤2q1−q0+rq0+rq3−1R≤q3−1R≤q3−1Cτ3−α+h2.
Suppose the inequalityen≤e1 holds, when n≤k+1. When n=k+2, reference to the proof process of stability, have
(34)ek+2=q0I+G2−1q1I−G1q0I+G1−1q1I−G2ek+q0I+G2−1q1I−G1q0I+G1−1q2ek−1+⋯+qk−1e2+qke1+q0I+G2−1q2ek+⋯+qke2+qk+1e1+q0I+G−1Rk+2≤rq0+r2R+r1−w1q0+r2R+1q0+rq2+⋯+qk+qk+1qk+1−1R≤r2q0+r2+rq0−q1q0+r2+q0−q1q0+rak+1−1R≤rq0+rrq0+r+q0−q1q0+r+q0−q1q0+rqk+1−1R≤rq0+r+q0−q1q0+rqk+1−1R≤qk+1−1R≤qk+1−1Cτ3−α+h2.
Becauseqk=−adk−1−dk+bck, then there is a positive finite constant C1, makes the inequality en+1≤qn+1−1C1τ3−α+h2 hold. We have the following conclusions:
(35)uxi,tn−uin∞≤C1τ3−α+h2.
In summary, the theorem is obtained.Theorem 5.
The PASE-I difference scheme (11) for the time fractional telegraph equation is convergent, and uxi,tn−uin∞≤Cτ3−α+h2, C is a positive number.


## 3.4. PASI-E Parallel Difference Scheme

The PASI-E scheme of the time fractional telegraph equation can be obtained by changing the computational order of explicit segment and implicit segment. In the even time layer, the PASI-E scheme is obtained by using the rule “implicit segment-explicit segment-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 telegraph equation is as follows:(36)q0I+G2Vn+1=q1I−G1Vn+q2Vn−1⋯+qn−1V2+qnV1+2adkτH+b1n+1+Fk+1,q0I+G1Vn+2=q1I−G2Vn+1+q2Vn⋯+qnV2+qn+1V1+2adk+1τH+b1n+2+Fn+2.Here,n=1,3,5⋯, the definition of G1,G2,H,Fn,b1n is defined as before.Since the difference between PASE-I format (11) and PASI-E (36) lies only in the order of using explicit format and implicit format, the computational complexity of the two formats should be equal in theory.By the same proof process, the following theorem can be obtained.Theorem 6.
The PASI-E difference scheme (36) of the time fractional telegraph equation is unconditionally stable and convergent. Moreover, uxi,tn−uin∞≤Cτ3−α+h2, C is a positive number.


## 4. Numerical Experiments

The numerical experiment is based on Intel Core i5-2400 CPU@3.10GHz and is carried out under the environment of MATLAB R2014a. We will verify the theoretical analysis by numerical experiments.Example 1.
Consider the following time fractional telegraph equation, takeα=1.8 [6, 7].
(37)∂1.8ux,t∂t1.8+∂0.8ux,t∂t0.8=∂2ux,t∂x2+fx,t,0≤x≤1,0≤t≤1.
Here,fx,t=x1−x2/Γ1.2t0.2+2/Γ2.2t1.2+21+t2.
The initial condition:ux,0=x1−x,utx,0=0,0<x<1.
The boundary condition:u0,t=0, u1,t=0.
Then, the equation has exact solutions:ux,t=1+t2x1−x.WhenM=50, N=100, the analytical solution surface and numerical solution surface of the three difference schemes of the fractional telegraph equation are shown in Figure 2. From Figure 2, we can see that the PASE-I and PASI-E difference schemes and implicit schemes are smooth and can approximate analytical solutions very well. The error surfaces of the three difference schemes are shown in Figure 3. From Figure 3, we can see that the error of implicit difference schemes is less than 2.5e-3, and the error limits of PASE-I and PASI-E schemes are less than 3e-4. The accuracy of PASE-I or PASI-E scheme is better than the implicit difference scheme.Figure 2
Exact and numerical solutions of surfaces (M=50, N=100).Figure 3
The error surface of numerical solutions.Because the accuracy of PASE-I and PASI-E are similar, we take PASE-I as an example to investigate the variation of relative error (RE) of PASE-I in time direction. The relative error is defined as follows. The analytical solution is regarded as the control solution, and the solution of PASE-I scheme is regarded as the perturbation solution. The formula for relative error is(38)REi=∑j=1Muji−u¯jiuji.The space steph is 1/50 and the time step is 1/100, respectively. The RE of PASE-I is shown in Figure 4. From Figure 4, we can know that the RE of PASE-I scheme is less than 0.16. The RE decreases with the advance of time step, which indicates that the PASE-I scheme of time fractional telegraph equation is computationally stable.Figure 4
The curve of RE changes with time.Next, time convergence order (order1) and space convergence order (order2) are defined by the maximum modulus error. The theoretical analysis is validated by the following numerical experiments. With fixed space step h, the order1 is defined as follows [8, 9]:
(39)order1=log2E∞τ3−α+h2E∞τ/23−α+h2≈3−α.Letτ3−α=h2, the order2 is defined as follows:
(40)order2=log2E∞τ13−α+h2E∞τ23−α+h/22=log2E∞h2E∞h/22≈2.From Table1, the convergence order of PASE-I and PASI-E schemes in time direction is 3−α order, and the accuracy of implicit schemes is equal to that of PASE-I and PASI-E schemes. From Table 2, we can see the spatial convergence orders of PASE-I and PASI-E schemes are both of second order, which is consistent with the theoretical analysis.Example 2.
Consider the following time fractional telegraph equation [1, 5].
(41)∂1.8ux,t∂t1.8+∂0.8ux,t∂t0.8=∂2ux,t∂x2,0≤x≤1,0≤t≤1.
The initial conditions:ux,0=x1−x,utx,0=0,0<x<1.
The boundary conditions:u0,t=0, u1,t=0.
When the time division is 500 and the space mesh points is 50, the numerical solution surfaces of the three schemes are shown in Figure5. From Figure 5, it can be seen that the numerical solution surfaces of the PASE-I and PASI-E schemes are the same and smooth as those of the classical implicit scheme.We define the speedup asSp=T1/Tp (T1 is the CPU time of implicit, Tp is the CPU time of parallel scheme) and the efficiency as Ep=Sp/p (p is the number of processors in parallel processor) [30, 31]. We use four cores for this numerical experiment. When the number of spatial grid points is 100, 200, 400, 800, 1600, and 3200, respectively, the CPU time required for the three schemes is shown in Table 3. From Table 3, it can be seen that the CPU time of serial difference increases exponentially, and the parallel difference schemes increase relatively slowly, with the increase of the number of spatial grid points. Compared with serial difference schemes, the computational efficiency of the PASE-I and PASI-E parallel schemes are greatly improved with the refinement of the spatial mesh. When the number of spatial grids is small (100), the computing time of parallel difference scheme is almost the same as that of the serial implicit scheme, because the communication between modules consumes a lot of CPU time. With the increase of computational domain, the parallel computing characteristics of PASE-I and PASI-E schemes will become more prominent. When the number of grid points is 1600, the efficiency of parallel difference schemes is optimal in this example. The linear acceleration ratio can be achieved when the number of space grid points is more than 800. Compared with the serial difference scheme, the computation time of PASE-I and PASI-E scheme can save about 80% when the number of grid points is great 800.Table 1
Time order (order1) of difference schemes (h=1/200).
τ
ImplicitE∞
Implicitorder1
PASE-IE∞
PASE-Iorder1
PASI-EE∞
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
1.231069Table 2
Space order (order2) of difference schemes.
h
ImplicitE∞
Implicitorder2
PASE-IE∞
PASE-Iorder2
PASI-EE∞
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
1.982850Figure 5
Surfaces of three difference schemes.Table 3
Comparison of the three difference schemes’ CPU time.
Grid numbers
Implicit
PASE-I
PASI-E
Sp
Ep
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


## 5. Conclusion

In this paper, a class of alternating segment pure explicit-implicit (PASE-I) and implicit-explicit (PASI-E) parallel difference methods are constructed for the time fractional telegraph equations. From theoretical analysis, it can be concluded that the parallel difference methods are unconditionally convergent and stable, with second-order convergence in spatial direction and3−α order in temporal direction. The numerical experiments show that the parallel difference schemes are more efficient than the serial difference scheme with the increase of space mesh generation. The numerical experiments are in agreement with the theoretical analysis, which show that the numerical algorithm is efficient and feasible for solving the time fractional telegraph equation.



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