The stable difference scheme for the numerical solution of the mixed problem for the
multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this
difference scheme and for the first and second orders difference derivatives are obtained. A procedure
of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional
fractional hyperbolic partial differential equations.

1. Introduction

It is known that various problems in fluid mechanics (dynamics, elasticity) and other areas of physics lead to fractional partial differential equations. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [1–11] and the references given therein).

The role played by stability inequalities (well posedness) in the study of boundary-value problems for hyperbolic partial differential equations is well known (see, e.g., [12–25]). In the present paper, the mixed boundary value problem for the multidimensional fractional hyperbolic equation

∂2u(t,x)∂t2-∑r=1m(ar(x)uxr)xr+Dt1/2u(t,x)=f(t,x),x=(x1,…,xm)∈Ω,0<t<1,u(0,x)=0,ut(0,x)=0,x∈Ω¯,u(t,x)=0,x∈S
is considered. Here Dt1/2=D0+1/2 is the standard Riemann-Lioville's derivative of order 1/2 and Ω is the unit open cube in the m-dimensional Euclidean space ℝm:{Ω=x=(x1,…,xm):0<xj<1,1≤j≤m} with boundary S,Ω¯=Ω∪S, ar(x),(x∈Ω) and f(t,x)(t∈(0,1),x∈Ω) are given smooth functions and ar(x)≥a>0.

The first order of accuracy in t and the second order of accuracy in space variables for the approximate solution of problem (1.1) are presented. The stability estimates for the solution of this difference scheme and its first and second ordes difference derivatives are established. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

2. The Difference Scheme and Stability Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, let us define the grid space

Ω̃h={x=xr=(h1r1,…,hmrm),r=(r1,…,rm),0≤rj≤Nj,hjNj=1,j=1,…,m},Ωh=Ω̃h∩Ω,Sh=Ω̃h∩S.
We introduce the Banach space L2h=L2(Ω̃h) of the grid functions φh(x)={φ(h1r1,…,hmrm)} defined on Ω̃h, equipped with the norm

∥φh∥L2(Ω̃h)=(∑x∈Ωh¯|φh(x)|2h1⋯hm)1/2.
To the differential operator Ax generated by problem (1.1), we assign the difference operator Ahx by the formula

Ahxuxh=-∑r=1m(ar(x)ux̅rh)xr,jr
acting in the space of grid functions uh(x), satisfying the conditions uh(x)=0 for all x∈Sh. It is known that Ahx is a self-adjoint positive definite operator in L2(Ω̃h). With the help of Ahx we arrive at the initial boundary value problem

d2vh(t,x)dt2+Ahxvh(t,x)+Dt1/2vh(t,x)=fh(t,x),0≤t≤1,x∈Ωh,vh(0,x)=0,dvh(0,x)dt=0,x∈Ω̃
for an finite system of ordinary fractional differential equations.

In the second step, applying the first order of approximation formula (1/π)∑m=1k(Γ(k-m+(1/2))/(k-m)!)((u(tk)-u(tk-1))/τ1/2) for Dt1/2u(t) (see [10]) and using the first order of accuracy stable difference scheme for hyperbolic equations (see [25]), one can present the first order of acuraccy difference scheme

uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxuk+1h+1π∑m=1kΓ(k-m+1/2)(k-m)!(umh-um-1h)τ1/2=fkh(x),x∈Ω̃h,fkh(x)=f(tk,xn),tk=kτ,1≤k≤N-1,Nτ=1,u1h(x)-u0h(x)τ=0,u0h(x)=0,x∈Ω̃h
for the approximate solution of problem (2.4). Here Γ(k-m+1/2)=∫0∞tk-m-1/2e-tdt.

Theorem 2.1.

Let τ and |h| be sufficiently small numbers. Then, the solutions of difference scheme (2.5) satisfy the following stability estimates:
max1≤k≤N∥ukh∥L2h+max1≤k≤N∥ukh-uk-1hτ∥L2h≤C1max1≤k≤N-1∥fkh∥L2h,max1≤k≤N-1∥τ-2(uk+1h-2ukh+uk-1h)∥L2h+max1≤k≤N∑r=1m∥(ukh)x̅rxr,jr∥L2h≤C2[∥f1h∥L2h+max2≤k≤N-1∥τ-1(fkh-fk-1h)∥L2h].
Here C1 and C2 do not depend on τ,h, and fkh,1≤k<N-1.

The proof of Theorem 2.1 is based on the self-adjointness and positive definitness of operator Ahx in L2h and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.

Theorem 2.2.

For the solutions of the elliptic difference problem
Ahxuh(x)=ωh(x),x∈Ωh,uh(x)=0,x∈Sh,
the following coercivity inequality holds [26]:
∑r=1m∥uxrx̅r,jrh∥L2h≤C∥ωh∥L2h.

Remark 2.3.

The stability estimates of Theorem 2.1 are satisfied in the case of operator
Au=-∑k=1nak(x)∂2u∂xk2+∑k=1nbk(x)∂u∂xk+c(x)u
with Dirichlet condition u=0 in S and Dtα=D0+α is the standard Riemann-Lioville's derivative of order α,0≤α<1. In this case, A is not self-adjoint operator in H. Nevertheless, Au=A0u+Bu and A0 is a self-adjoint positive definite operator in H and BA0-1 is bounded in H. The proof of this statement is based on the abstract results of [25] and difference analogy of integral inequality.

Remark 2.4.

The stability estimates of Theorem 2.1 permit us to obtain the estimate of convergence of difference scheme of the first order of accuracy for approximate solutions of the initial-boundary value problem
∂2u(t,x)∂t2-∑r=1nar(x)uxrxr+∑r=1nbr(x)uxr+Dtαu(t,x)=f(t,x;u(t,x),ut(t,x),ux1(t,x),…,uxn(t,x)),x=(x1,…,xn)∈Ω,0<t<1,u(0,x)=0,∂u(0,x)∂t=0,x∈Ω¯,u(t,x)=0,x∈S
for semilinear fractional hyperbolic partial differential equations.

Note that, one has not been able to obtain a sharp estimate for the constants figuring in the stability estimates of Theorem 2.1. Therefore, our interest in the present paper is studying the difference scheme (2.5) by numerical experiments. Applying this difference scheme, the numerical methods are proposed in the following section for solving the one-dimensional fractional hyperbolic partial differential equation. The method is illustrated by numerical experiments.

3. Numerical Results

For the numerical result, the mixed problem

Dt2u(t,x)-uxx(t,x)+Dt1/2u(t,x)=f(t,x),f(t,x)=(2-8t3/23π+(πt)2)sinπx,0<t,x<1,u(0,x)=0,ut(0,x)=0,0≤x≤1,u(t,0)=u(t,1)=0,0≤t≤1
for the one-dimensional fractional hyperbolic partial differential equation is considered. Applying difference scheme (2.5), we obtain

unk+1-2unk+unk-1τ2-un+1k+1-2unk+1+un-1k+1h2+1Π∑m=1kΓ(k-m+1/2)(k-m)!(unm-unm-1τ1/2)=φnk,φnk=f(tk,xn),1≤k≤N-1,1≤n≤M-1,un0=0,τ-1(un1-un0)=0,0≤n≤M,u0k=uMk=0,0≤k≤N.
We get the system of equations in the matrix form

AUn+1+BUn+CUn-1=Dφn,1≤n≤M-1,U0=0̃,UM=0,̃
where

0̃=[0000⋯00](N+1)×(1),A=[0000⋯000000⋯0000a0⋯00000a⋯00⋯⋯⋯⋯⋯⋯⋯0000⋯a00000⋯00](N+1)×(N+1),B=[b1,1000⋯00b2,1b2,200⋯00b3,1b3,2b3,30⋯00b4,1b4,2b4,3b4,4⋯00⋯⋯⋯⋯⋯⋯⋯bN,1bN,2bN,3bN,4⋯bN,N0bN+1,1bN+1,2bN+1,3bN+1,4⋯bN+1,NbN+1,N+1](N+1)×(N+1),C=A,D=[0000⋯000000⋯000010⋯000001⋯00⋯⋯⋯⋯⋯⋯⋯0000⋯100000⋯01](N+1)×(N+1),Us=[Us0Us1Us2Us3⋯UsN-1UsN](N+1)×(1),s=n-1,n,n+1,a=-1h2,b1,1=1,b2,1=-1τ,b2,2=1τ,b3,1=1τ2+1τ1/2,b3,2=-2τ2-1τ1/2,b3,2=1τ2+2h1/2,bk+2,1=1πΓ(k-1+1/2)Γ(k)τ1/2,2≤k≤N-1,bk+2,k+1=-2τ2-1τ1/2,1≤k≤N-1,bk+2,k=1τ2+1π(-Γ(3/2)Γ(2)+Γ(1/2)Γ(1))1τ1/2,2≤k≤N-1,bk+2,k+2=1τ2+2h2,1≤k≤N-1,bk+2,i+1=1π(-Γ(k-i+1/2)Γ(k-(i-1))+Γ(k-(i+1)+1/2)Γ(k-(i-1)-1))1τ1/2,3≤k≤N-1,1≤i≤k-2,φnk=(2-8(kτ)3/23π+(πkτ)2)sinπ(nh)),φn=[φn0φn1φn2⋯φnN](N+1)×1.
So, we have the second-order difference equation with respect to n matrix coefficients. This type system was developed by Samarskii and Nikolaev [27]. To solve this difference equation we have applied a procedure of modified Gauss elimination method for difference equation with respect to k matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:

Uj=αj+1Uj+1+βj+1,n=M-1,…,2,1, where αj(j=1,…,M) are (N+1)×(N+1) square matrices and βj(j=1,…,M) are (N+1)×1 column matrices defined by

αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(Dφn-Cβn),n=1,2,…,M-1,
where

α1=[000⋯0000⋯0000⋯0⋯⋯⋯⋯⋯000⋯0](N+1)×(N+1),β1=[000⋯0](N+1)×1.
Now, we will give the results of the numerical analysis. First, as we noted above one can not obtain a sharp estimate for the constants C1 and C2 figuring in the stability estimates of Theorem 2.1. We have

C1=maxf,u(Ct1),C2=maxf,u(Ct2),
where

Ct1=max1≤k≤N∥ukh∥L2h+max1≤k≤N∥τ-1(ukh-uk-1h)∥L2h(max1≤k≤N-1∥fkh∥L2h)-1,Ct2=[max1≤k≤N-1∥τ-2(uk+1h-2ukh+uk-1h)∥L2h+max1≤k≤N-1∑r=1n∥(uk+1h)x¯r,xr,jr∥L2h]×(max2≤k≤N-1∥τ-1(fkh-fk-1h)∥L2h+∥f1h∥L2h)-1.
The constants Ct1 and Ct2 in the case of numerical solution of initial-boundary value problem (3.1) are computed.

The numerical solutions are recorded for different values of N and M,unk represents the numerical solutions of this difference scheme at (tk,xn). The constants Ct1 and Ct2 are given in Table 1 for N=20,40,80, and M=80, respectively.

The difference scheme.

M=80

M=80

M=80

N=20

N=40

N=80

The values of Ct1

1.0379

1.0667

1.0800

The values of Ct2

0.6265

0.6186

0.6140

Recall that we have not been able to obtain a sharp estimate for the constants C1 and C2 figuring in the stability estimates. The numerical results in the Tables 1 and 2 give Ct1≅1.00 and Ct2≅0.62, respectively. That means the constants C1 and C2 figuring in the stability estimates in the case of numerical solution of initial-boundary value problem (3.1) of this difference scheme is stable with no large constants.

The difference scheme.

Method

M=80

M=80

M=80

N=20

N=40

N=80

Comparison of errors (E0) for approximate solutions

0.0452

0.0235

0.0125

Comparison of errors (E1) for approximate solutions

0.2702

0.1452

0.0779

Second, for the accurate comparison of the difference scheme considered, the errors computed by

E0=max1≤k≤N-1(∑n=1M-1|u(tk,xn)-unk|2h)1/2,E1=max1≤k≤N-1(∑n=1M-1|utt(tk,xn)-unk+1-2unk+unk-1τ2|2h)1/2
of the numerical solution are recorded for higher values of N and M, where u(tk,xn) represents the exact solution and unk represents the numerical solution at (tk,xn). The errors E0 and E1 results are shown in Table 2 for N=20,40,80 and M=80, respectively.

The exact solution.

Difference Scheme.

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