# A Numerical Method for the Variable-Order Time-Fractional Wave Equations Based on the H2N2 Approximation

Authors: Xiao Liu; Yu Bo; Yuanfeng Jin
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Aiming at the initial boundary value problem of variable-order time-fractional wave equations in one-dimensional space, a numerical method using second-order central difference in space and H2N2 approximation in time is proposed. A finite difference scheme with second-order accuracy in space and3−γ∗ order accuracy in time is obtained. The stability and convergence of the scheme are further discussed by using the discrete energy analysis method. A numerical example shows the effectiveness of the results.


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## Body

## 1. Introduction

In recent years, due to the non-locality of fractional calculus, more and more problems in physical science, electromagnetism, electrochemistry, diffusion and general transport theory can be described by the fractional calculus approach, among which the Riemann-Liouville fractional derivative and the Caputo fractional derivative are the most widely used [1–4]. At the same time, more and more researchers found that a variety of important dynamical problems exhibit fractional-order behavior that may vary with time, space, or other conditions. This phenomenon indicates that variable-order fractional calculus is a natural choice to provide an effective mathematical framework for the description of complex problems.In 2020, Shen et al. proposed a new numerical approximation method—the H2N2 approximation [5] for the numerical differential formula of the Caputo fractional derivative of γ∈1,2 and applied it for the constant-order time-fractional wave equations in the following multidimensional space
(1)0CDtγux,t=Δu+qx,t,x∈Ω,t∈0,T,ux,0=φx,utx,0=ψx,x∈Ω,ux,t=0,x∈∂Ω,t∈0,T,where qx,t,φx,ψx are given sufficiently smooth functions, Ω=∏j=1dlj,rj⊂Rd,∂Ω is the boundary of Ω,x=x1,x2,⋯,xd∈Ω, Δu=∑j=1d∂xj2u. When x∈∂Ω,φxandψx satisfy consistency conditions φx=ψx=0. It was proved that the proposed scheme has the accuracy of order of 3−γ in time and 2 in space, and it is clear that its theoretical analysis is similar to the L1 method applied in solving the constant-order time-fractional slow diffusion equations.Motivated by the above literature [6–9], in this work, we consider the numerical solution of the following variable-order time-fractional wave equations in one-dimensional space
(2)0CDtγtux,t=uxxx,t+fx,t,x∈0,L,t∈0,T.(3)ux,0=φx,utx,0=ψx,x∈0,L.(4)u0,t=0,uL,t=0,t∈0,T.where 1<γt<2,0CDtγtux,t is the variable-order Caputo fractional derivative, fx,t,φx,ψx are given suffificiently smooth functions and satisfy φ0=ψ0,φL=ψL. Suppose its solution function u∈C4,30,L×0,T.The rest of this paper is organized as follows. In the next section, some necessary notations are introduced. In Section3, the H2N2-based finite difference scheme for the variable-order time-fractional wave equations is derived. In Section 4, the stability and convergence of the difference scheme are studied. In Section 5, a numerical result is listed to verify the theoretical prediction and the effectiveness of the difference scheme. Finally, a brief conclusion is provided.


## 2. Preliminary Knowledge and Relevant Lemmas

Definition 1 (see [10]).
Suppose the functionft is defined on the interval 0,T,1<γt<2, then the variable-order Caputo fractional derivative is defined as
(5)0CDtγtft=1Γ2−γt∫0tf′′st−s1−γtds.
Next, mesh the solution intervals0,L and 0,T, take integers M and N, denote h=L/M,τ=T/N, h and τ are called space step and time step, respectively. Denote xi=ih0≤i≤M,tk=kτ0≤k≤N,Ωh=xi0≤i≤M,Ωτ=tk0≤k≤N. Define the following grid function spaces
(6)Uh=u∣u=u0,u1,⋯,uM,U^h=uu∈Uh,u0=uM=0.
For grid functionu=uik∣0≤i≤M,0≤k≤N defined on Ωh×Ωτ, introduce the following notations
(7)δxui−1/2k=1huik−ui−1k,δx2uik=1h2ui+1k−2uik+ui−1k,δtuik+1/2=1τuik+1−uik−1,δt2uik=1τδtuik+1/2−δtuik−1/2.
For any grid functionsu,v∈U^h, denote the following notations
(8)u,v=h∑i=1M−1uivi,u=u,u,δxu,δxv=h∑i=0M−1δxui+1/2δxvi+1/2,u∞=max0≤i≤Mui,δxu=δxu,δxu.
For any functionft defined on the interval 0,t1, using the data t0,ft0,t1,ft1,t0,f′t0 to make the quadratic Hermite interpolation polynomial of ft(9)H2,0t=ft0+f′t0t−t0+1τδtf1/2−f′t0t−t02.
Taking the twice derivative arrives at(10)H2,0′′t=2τδtf1/2−f′t0.
For any functionft defined on the interval tk−1,tk+11≤k≤N−1, using three points tk−1,ftk−1,tk,ftk,tk+1,ftk+1 to make the quadratic Newton interpolation polynomial of ft(11)N2,kt=ftk−1+δtfk−1/2t−tk−1+12δt2fkt−tk−1t−tk.
Taking the second-order derivative yields(12)N2,k′′t=δt2fk.
On the basis of the above interpolation polynomial, we next discuss the high-precision approximation formula of the variable-order Caputo fractional derivative.
Here, we denotefl=ftl,γn−1/2=γtn−1/2,tn−1/2=tn−τ/2. Suppose ft∈C3t0,tnand1<γt<2, then at the half-grid point tn−1/2, we have
(13)0CDtγn−1/2ftn−1/2=1Γ2−γn−1/2∫t0t1/2f′′ttn−1/2−t1−γn−1/2dt+∑k=1n−1∫tk−1/2tk+1/2f′′ttn−1/2−t1−γn−1/2dt≈1Γ2−γn−1/2∫t0t1/2H2,0′′ttn−1/2−t1−γn−1/2dt+∑k=1n−1∫tk−1/2tk+1/2N2,k′′ttn−1/2−t1−γn−1/2dt=1Γ2−γn−1/2∫t0t1/22τδtf1/2−f′t0tn−1/2−t1−γn−1/2dt+∑k=1n−1∫tk−1/2tk+1/2δt2fktn−1/2−t1−γn−1/2dt=1Γ2−γn−1/22τ∫t0t1/2tn−1/2−t1−γn−1/2dt⋅δtf1/2−f′t0+1τ∑k=1n−1∫tk−1/2tk+1/2tn−1/2−t1−γn−1/2dt⋅δtfk+1/2−δtfk−1/2=1Γ2−γn−1/2bn−1n,γn−1/2δtf1/2−f′t0+∑k=1n−1bn−k−1n,γn−1/2δtfk+1/2−δtfk−1/2=1Γ2−γn−1/2b0n,γn−1/2δtfn−1/2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2δtfk−1/2−bn−1n,γn−1/2f′t0≡Dtγn−1/2ftn−1/2.
Here(14)bn−1n,γn−1/2=2τ∫t0t1/2tn−1/2−t1−γn−1/2dt,(15)bn−k−1n,γn−1/2=1τ∫tk−1/2tk+1/2tn−1/2−t1−γn−1/2dt,where 1≤k≤n−1.
Then, it can be calculated that(16)bkn,γn−1/2=τ1−γn−1/22−γn−1/2k+12−γn−1/2−k2−γn−1/2,0≤k≤n−2,2τ1−γn−1/22−γn−1/2n−122−γn−1/2−n−12−γn−1/2,k=n−1.
Denote(17)rn=0CDtγn−1/2ftn−1/2−Dtγn−1/2ftn−1/2,we have
(18)rn≤C0maxt0≤t≤tnf′′′tτ3−γn−1/2.
Here,C0=1/8Γ2−γn−1/2+1/12Γ3−γn−1/2+γn−1/2−1/2Γ4−γn−1/2, the proof process is similar to Theorem 2.1 in Reference [5].Lemma 2.
For anyn≥2, according to bkn,γn−1/2 defined by (14)–(15), we have
(19)τ1−γn−1/2n−1/2γn−1/2−1<bn−1n,γn−1/2<bn−2n,γn−1/2<⋯<b1n,γn−1/2<b0n,γn−1/2=τ1−γn−1/22−γn−1/2.Proof.
According to the formula (14)–(15), we have
(20)bn−1n,γn−1/2=2τ1−γn−1/22−γn−1/2n−122−γn−1/2−n−12−γn−1/2,(21)bn−k−1n,γn−1/2=τ1−γn−1/22−γn−1/2n−k2−γn−1/2−n−k−12−γn−1/2,1≤k≤n−1.
Whenk=n−1, it can be obtained by calculation
(22)b0n,γn−1/2=τ1−γn−1/22−γn−1/2.
From equations (20) and (21), we have
(23)bn−1n,γn−1/2=2τ1−γn−1/2∫n−1n−1/2ξ1−γn−1/2dξ,bkn,γn−1/2=τ1−γn−1/2∫kk+1ξ1−γn−1/2dξ,0≤k≤n−2.
Therefore, it can be obtained(24)bn−1n,γn−1/2<bn−2n,γn−1/2<⋯<b1n,γn−1/2<b0n,γn−1/2=τ1−γn−1/22−γn−1/2.
Whenn≥2, we have
(25)1−12n−12−γn−1/2=1−2−γn−1/22n−1+2−γn−1/21−γn−1/22!−12n−12+2−γn−1/21−γn−1/2−γn−1/23!−12n−13+⋯.
From the above formula(26)n−122−γn−1/2−n−12−γn−1/2−2−γn−1/22n−1/2γn−1/2−1=n−122−γn−1/21−2−γn−1/22n−1/2−1−12n−12−γn−1/2=n−122−γn−1/2−2−γn−1/21−γn−1/22!−12n−12−2−γn−1/21−γn−1/2−γn−1/23!−12n−13−⋯>0.
And whenn=1, we have
(27)122−γn−1/2−2−γn−1/22⋅1/2γn−1/2−1=γn−1/2−122−γn−1/2>0.
Therefore, it can be seen that(28)bn−1n,γn−1/2>2τ1−γn−1/22−γn−1/2⋅2−γn−1/22n−1/2γn−1/2−1=τ1−γn−1/2n−1/2γn−1/2−1.
To sum up, Lemma2 is proved.Lemma 3 (see [11]).
If the functionf∈C4xi−1,xi+1,λ∈xi−1,xi+1, there is
(29)f′′xi=fxi−1−2fxi+fxi+1h2−h212f4λ.Lemma 4.
For any positive integerm and any ψ,V1,V2,⋯,VN∈U^h, when
(30)tn+1/2−tγn+1/2−1≥tn−1/2−tγn−1/2−1,t∈0,tn−1/2,tn+1/2≤T,we have
(31)∑n=1mb0n,γn−1/2Vn−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2Vk−bn−1n,γn−1/2ψ,Vn≥12∑k=1mbm−km,γm−1/2Vk2−∑n=1mbn−1n,γn−1/2ψ2,where 1≤m≤N.Proof.
On the basis of [12], it can be seen from the condition
(32)∑n=1mb0n,γn−1/2Vn−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2Vk−bn−1n,γn−1/2ψ,Vn=∑n=1mb0n,γn−1/2Vn2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2Vk,Vn−bn−1n,γn−1/2ψ,Vn≥∑n=1mb0n,γn−1/2Vn2−12∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2Vk2+Vn2−12bn−1n,γn−1/2ψ2+Vn2=12∑n=1m2b0n,γn−1/2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2−bn−1n,γn−1/2Vn2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2Vk2−bn−1n,γn−1/2ψ2=12∑n=1mb0n,γn−1/2Vn2−∑k=1n−1bn−k−1n,γn−1/2Vk2+∑k=1n−1bn−kn,γn−1/2Vk2−bn−1n,γn−1/2ψ2=12∑n=1m∑k=1nbn−kn,γn−1/2Vk2−∑k=1n−1bn−k−1n,γn−1/2Vk2−bn−1n,γn−1/2ψ2=12∑k=1m∑n=kmbn−kn,γn−1/2Vk2−∑k=1m−1∑n=k+1mbn−k−1n,γn−1/2Vk2−∑n=1mbn−1n,γn−1/2ψ2=12b0m,γm−1/2Vm2+∑k=1m−1∑n=kmbn−kn,γn−1/2−∑n=k+1mbn−k−1n,γn−1/2Vk2−∑n=1mbn−1n,γn−1/2ψ2=12b0m,γm−1/2Vm2+∑k=1m−1bm−km,γm−1/2Vk2+∑k=1m−1∑n=km−1bn−kn,γn−1/2−bn−kn+1,γn+1/2Vk2−∑n=1mbn−1n,γn−1/2ψ2.
Whenγt satisfies the following condition
(33)bn−kn,γn−1/2−bn−kn+1,γn+1/2=1τ∫tk−3/2tk−1/2tn−1/2−t1−γn−1/2−tn+1/2−t1−γn+1/2dt,namely
(34)tn+1/2−tγn+1/2−1≥tn−1/2−tγn−1/2−1.
Then, we have(35)∑n=1mb0n,γn−1/2Vn−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2⋅Vk−bn−1n,γn−1/2ψ,Vn≥12∑k=1mbm−km,γm−1/2Vk2−∑n=1mbn−1n,γn−1/2ψ2.Remark 5.
Consider the function(36)gx,y=xy,x>0,y>0.
We have(37)∂gx,y∂x=yxy−1>0,∂gx,y∂y=xyInx=<0,x∈0,1,>0,x>1.
IfT≤1 and γt is an non-increasing function on 0,T, then tn+1/2−t∈0,1,tn−1/2−t∈0,1andγn+1/2≤γn−1/2, consequently
(38)tn+1/2−tγn+1/2−1≥tn−1/2−tγn−1/2−1,t∈0,tn−1/2,tn+1/2≤T.
(30) is valid.
IfT>1,γt is an non-increasing function on the interval 0,1 and γt is a constant on the interval 1,T, (30) is also valid.Lemma 6 (see [11]).
For anyε>0,a,b≥0, there is
(39)ab≤εa2+14εb2.Lemma 7 (see [11]).
For any grid functionu∈U^h, there is
(40)u∞≤L2δxu.Lemma 8 (see [13]).
SupposeFk∣k≥0,Gk∣k≥1 are two non-negative sequences, Gk does not decrease with k, if
(41)Fk≤Cτ∑l=0kFl+Gk,k=1,2,⋯,where C is an non-negative constant, when τ≤2/3C, then
(42)Fk≤exp3CkτCτF0+3Gk,k=1,2,⋯.


## 3. Establishment of the Difference Scheme

DenoteUin=uxi,tn,0≤i≤M,0≤n≤N;φi=φxi,ψi=ψxi, consider (2) at the point xi,tn−1/2, we have
(43)0CDtγn−1/2uxi,tn−1/2=uxxxi,tn−1/2+fxi,tn−1/2,1≤i≤M−1,1≤n≤N.Applying (13) to approximate the temporal fractional derivative and central difference quotient (29) to approximate the spatial derivative, we can obtain
(44)1Γ2−γn−1/2b0n,γn−1/2δtUin−1/2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2δtUik−1/2−bn−1n,γn−1/2ψi=δx2Uin−1/2+fin−1/2+Rin−1/2,1≤i≤M−1,1≤n≤N.There exists a positive constantC1 such that
(45)Rin−1/2≤C1τ3−γn−1/2+h2,1≤i≤M−1,1≤n≤N.Noticing the initial and boundary value conditions (3) and (4), we have
(46)Ui0=φi,1≤i≤M−1.U0n=0,UMn=0,0≤n≤N.Omitting the small termRin−1/2 in the equation and replacing the grid function Uin by its numerical approximation uin, we construct the difference scheme for solving the problems (2)–(4) as follows
(47)1Γ2−γn−1/2b0n,γn−1/2δtuin−1/2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2δtuik−1/2−bn−1n,γn−1/2ψi=δx2uin−1/2+fin−1/2,1≤i≤M−1,1≤n≤N.(48)ui0=φi,1≤i≤M−1.(49)u0n=0,uMn=0,0≤n≤N.


## 4. Stability and Convergence of the Difference Scheme

Theorem 9.
Supposeuin∣0≤i≤M,0≤n≤N is the solution of the following difference scheme
(50)1Γ2−γn−1/2b0n,γn−1/2δtuin−1/2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2δtuik−1/2−bn−1n,γn−1/2ψi=δx2uin−1/2+pin−1/2,1≤i≤M−1,1≤n≤N,(51)ui0=φi,1≤i≤M−1,(52)u0n=0,uMn=0,0≤n≤N,where pin−1/2 is a given perturbation term, when τ<2/3c0, it holds that
(53)δxun2≤exp3c0Tc0τδxu02+3Qn,1≤n≤N.
c0 and Qn are given in (56) and (64), respectively.Proof.
Taking an inner product (50) with Γ2−γn−1/2δtun−1/2 and summing n from 1 to m, we have
(54)∑n=1mb0n,γn−1/2δtun−1/22−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2δtuk−1/2,δtun−1/2−bn−1n,γn−1/2ψ,δtun−1/2=∑n=1mΓ2−γn−1/2δx2un−1/2,δtun−1/2+∑n=1mΓ2−γn−1/2pn−1/2,δtun−1/2,1≤m≤N.
Noticing that(55)∑n=1mΓ2−γn−1/2δx2un−1/2,δtun−1/2=−12τ∑n=1mΓ2−γn−1/2δxun2−δxun−12=−12τ∑n=1mΓ2−γnδxun2−Γ2−γn−1δxun−12−Γ2−γn−Γ2−γn−1/2δxun2−Γ2−γn−1/2−Γ2−γn−1δxun−12≤−12τ∑n=1mΓ2−γnδxun2−Γ2−γn−1δxun−12+12τ∑n=1mΓ2−γn−Γ2−γn−1/2δxun2+12τ∑n=1mΓ2−γn−1/2−Γ2−γn−1δxun−12≤−12τ∑n=1mΓ2−γnδxun2−Γ2−γn−1δxun−12+14c0∑n=1mδxun2+δxun−12,where
(56)c0=max0≤t≤TddtΓ2−γt.
Applying Lemma4, we have
(57)12∑k=1mbm−km,γm−1/2δtuk−1/22−∑n=1mbn−1n,γn−1/2ψ2≤−12τ∑n=1mΓ2−γnδxun2−Γ2−γn−1δxun−12+14c0∑n=1mδxun2+δxun−12+∑n=1mΓ2−γn−1/2pn−1/2,δtun−1/2,1≤m≤N.
Then, we have(58)12∑k=1mbm−km,γm−1/2δtuk−1/22+12τΓ2−γmδxum2−δxu02≤12∑n=1mbn−1n,γn−1/2ψ2+∑n=1mΓ2−γn−1/2pn−1/2,δtun−1/2+12c0∑n=0mδxun2,1≤m≤N.
By Lemma2, noticing that
(59)bm−km,γm−1/2>τ1−γm−1/2m−1/2γm−1/2−1=tm−1/21−γm−1/2,bn−1n,γn−1/2<τ1−γn−1/22−γn−1/2.
Then(60)12tm−1/21−γm−1/2∑k=1mδtuk−1/22+Γ2−γm2τδxum2≤12τδxu02+12∑n=1mτ1−γn−1/22−γn−1/2ψ2+∑n=1mΓ2−γn−1/2tm−1/21−γm−1/22Γ2−γn−1/2δtun−1/22+Γ2−γn−1/22tm−1/21−γm−1/2pn−1/22+12c0∑n=0mδxun2.
We use the Cauchy inequality for the inner productpn−1/2,δtun−1/2, the above equation can be simplified
(61)Γ2−γm2τδxum2≤12τδxu02+12∑n=1mτ1−γn−1/22−γn−1/2ψ2+∑n=1mΓ22−γn−1/22tm−1/21−γm−1/2pn−1/22+12c0∑n=0mδxun2.
Multiplying by2τ/Γ2−γm, then we have
(62)δxum2≤1Γ2−γmδxu02+τ∑n=1mτ1−γn−1/22−γn−1/2ψ2+τtm−1/2γm−1/2−1∑n=1mΓ22−γn−1/2pn−1/22+τc0∑n=0mδxun2.
Note thatΓ is decreasing on the interval 0,1. Since 0<2−γt≤1, Γ2−γt−1≤1. Then
(63)δxum2≤δxu02+τ∑n=1mτ1−γn−1/22−γn−1/2ψ2+τtm−1/2γm−1/2−1∑n=1mΓ22−γn−1/2pn−1/22+τc0∑n=0mδxun2.
Let(64)Qm=δxu02+τ∑n=1mτ1−γn−1/22−γn−1/2ψ2+τtm−1/2γm−1/2−1∑n=1mΓ22−γn−1/2pn−1/22,1≤m≤N.
Then(65)δxum2≤τc0∑n=0mδxun2+Qm.
It is easy to knowQm does not decrease with m. According to Lemma 8, when τ<2/3c0, we have
(66)δxum2≤exp3c0Tc0τδxu02+3Qm,1≤m≤N.
Theorem9 is proved. We can say that the difference scheme is stable.Theorem 10.
Assumeuxi,tn and uin are solutions of problems (2)–(4) and difference scheme (47)–(49), respectively. Denote
(67)ein=uxi,tn−uin,0≤i≤M,0≤n≤N.
Then, there exists a positive constantC2, when τ<2/3c0, nτ≤T, such that
(68)en∞≤L2C2τ3−γ∗+h2,0≤n≤N,where C2=3TLC1exp3c0T/2 and γ∗=max0≤t≤Tγt.Proof.
Subtracting (44) and (46) from (47)–(49), we obtain the system of error equations
(69)1Γ2−γn−1/2b0n,γn−1/2δtein−1/2−∑k=1n−1bn−k−1n,γn−1/2−bn−kn,γn−1/2δteik−1/2=δx2ein−1/2+Rin−1/2,1≤i≤M−1,1≤n≤N,ei0=0,1≤i≤M−1,e0n=0,eMn=0,0≤n≤N.
Applying Theorem9 and (45), it yields
(70)δxen2≤C22τ3−γ∗+h22,where γ∗=max0≤t≤Tγt.
Applying Lemma7, it yields
(71)en∞≤L2δxen≤L2C2τ3−γ∗+h2.
The proof is ended.


## 5. Numerical Example

In order to verify the accuracy of the finite difference scheme, several different types of variable-order indexγt∈1,2 are used to solve the variable-order fractional wave equations (2)–(4) in 1D case. The scheme is implemented in MATLAB (R2019a).Here, we takeL=π, T=1. The source term of equation (2)
(72)fx,t=6Γ4−γtt3−γt+6Γ3−γtt2−γt+t3+3t2+1sinx,the initial value
(73)ux,0=sinx,x∈0,π,the boundary value
(74)u0,t=uπ,t=0,t∈0,1,the exact solution is given by
(75)ux,t=t3+3t2+1sinx.Define the error of the numerical solution(76)Eh,τ=max0≤k≤NUk−uk∞,the temporal convergence order
(77)Orderτ=log2Eh,2τEh,τ,the spatial convergence order
(78)Orderh=log2E2h,τEh,τ.DenoteM=1000, for different γt=2−t2, 1+e−t, 6+cost/4. The time step τ is varied from 1/512 to 1/4096, where N=512, 1024, 2048, 4096. Table 1 shows the errors and temporal convergence orders of the difference scheme (47)–(49). It can be seen from Table 1 that the difference scheme (47)–(49) has a precision of approximately 3−γ∗ order in time. The computational results are in good agreement with theoretical results.Table 1
Errors and temporal convergence orders,M=1000.
γtτEh,τOrderτ2−t21/5128.7295e−401/10244.2265e−41.051/20482.0502e−41.041/40969.9746e−51.041+e−t1/5124.0772e−401/10241.8602e−41.131/20488.5528e−51.121/40963.9764e−51.106+cost41/5123.8173e−401/10241.5945e−41.261/20486.6746e−51.261/40962.8186e−51.24Take a fixed and sufficiently small time stepτ=1/20000, for different γt=2−t2, 1+e−t, 6+cost/4, verify space step h from π/5 to π/40, where N=20000, M=5,10,20,40. Table 2 shows the errors and spatial convergence orders of the difference scheme (47)–(49). It can be seen from Table 2 that the difference scheme (47)–(49) in the space has an accuracy of approximately 2 order, which is consistent with the theoretical results.Table 2
Errors and spatial convergence orders,N=20000.
γthEh,τOrderh2−t2π/53.2641e−20π/108.6512e−31.92π/202.1801e−31.99π/405.5869e−41.961+e−tπ/53.1901e−20π/108.4489e−31.92π/202.1209e−31.99π/405.3548e−41.996+cost4π/53.0356e−20π/108.0401e−31.92π/202.0164e−32.00π/405.0707e−41.99


## 6. Conclusions

In this paper, we consider a numerical approximation method for the variable-order Caputo fractional derivative—H2N2 approximation, and give the corresponding calculation formula. Secondly, we use this formula to solve the one-dimensional variable-order time-fractional wave equations and discuss the stability and convergence of the equations by the discrete energy analysis method. Finally, a numerical example verifies the effectiveness of the scheme.



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