# Stability of Difference Schemes for Fractional Parabolic PDE with the Dirichlet-Neumann Conditions

Authors: Zafer Cakir
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The stable difference schemes for the fractional parabolic equation with Dirichlet and Neumann
boundary conditions are presented. Stability estimates and almost coercive stability estimates with ln(1/(τ+|h|)) for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes of one-dimensional fractional parabolic partial differential equations.


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## Body

## 1. Introduction

Theory and applications, methods of solutions of problems for fractional differential equations have been studied extensively by many researchers [1–18]. In this study, initial-boundary-value problem for the fractional parabolic equation(1.1)∂u(t,x)∂t+Dt1/2u(t,x)-∑p=1m(ap(x)uxp)xp+σu(t,x)=f(t,x),x=(x1,…,xm)∈Ω,0<t<T,u(t,x)∣S1=0,∂u(t,x)∂n⃗|S2=0,0≤t≤T,S1∪S2=S=∂Ω¯,u(0,x)=0,x∈Ω¯,
with Dirichlet and Neumann conditions is considered. Here Dt1/2=D0+1/2 is the standard Riemann-Liouville’s derivative of order 1/2 and Ω is the open cube in the m-dimensional Euclidean space(1.2)Rm:{x∈Ω:x=(x1,…,xm);0<xj<1,1≤j≤m},
with boundary S,Ω¯=Ω∪S,ap(x)(x∈Ω) and f(t,x)(t∈(0,T),x∈Ω) are given smooth functions, ap(x)≥a>0, σ>0, and n⃗ is the normal vector to Ω.The first and second orders of accuracy stable difference schemes for the numerical solution of problem (1.1) are presented. Stability estimates and almost coercive stability estimates with ln(1/(τ+|h|)) for the solution of these difference schemes are obtained. The method is illustrated by numerical examples.


## 2. The First and Second Orders of Accuracy Stable Difference Schemes 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(2.1)Ω¯h={x=xp=(h1p1,…,hmpm),p=(p1,…,pm),0≤pj≤Mj,hjMj=1,j=1,…,m},Ωh=Ω¯h∩Ω,Sh=Ω¯h∩S.
We introduce the Hilbert space L2h=L2(Ω¯h) of the grid function φh(x)={φ(h1p1,…,hmpm)} defined on Ω¯, equipped with the norm(2.2)‖φh‖L2(Ω¯h)=(∑x∈Ω¯h|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(2.3)Ahxuh=-∑p=1m(ap(x)ux̅ph)xp,jp+σuxh,
acting in the space of grid functions uh(x), satisfying the conditions uh(x)=0 for all x∈Sh1 and Dhuh(x)=0 for all x∈Sh2. Here Dhuh(x) is the approximation of ∂u/∂n⃗. It is known that Ahx is a self-adjoint positive definite operator in L2(Ω¯h). With the help of Ahx we arrive at the initial-boundary-value problem(2.4)dvh(t,x)dt+Dt1/2vh(t,x)+Ahxvh(t,x)=fh(t,x),0<t<T,x∈Ωh,vh(0,x)=0,x∈Ω¯h,
for a finite system of ordinary fractional differential equations.In the second step, applying the first order of approximation formula(2.5)Dτ1/2uk=1π∑r=1kΓ(k-r+1/2)(k-r)!(ur-ur-1τ1/2),1≤k≤N,
for(2.6)Dt1/2u(tk)=1Γ(1/2)∫0tk(tk-s)-1/2u′(s)ds,
(see [19]) and using the first order of accuracy stable difference scheme for parabolic equations, one can present the first order of accuracy difference scheme with respect to t(2.7)ukh(x)-uk-1h(x)τ+Dτ1/2ukh+Ahxukh(x)=fkh(x),x∈Ω¯h,fkh(x)=fh(tk,x),tk=kτ,1≤k≤N,Nτ=T,u0h(x)=0,x∈Ω¯h,
for the approximate solution of problem (2.4). Here(2.8)Γ(k-r+12)=∫0∞tk-r+1/2e-tdt.
Moreover, applying the second order of approximation formula(2.9)Dτ1/2uk={[-d23]u0+[d23]u1,k=1,[2d65]u0+[d65]u1+[d65]u2,k=2,d∑m=2k-1{[(k-m)b1(k-m)+b2(k-m)]um-2+[(2m-2k-1)b1(k-m)-2b2(k-m)]um-1+[(k-m+1)b1(k-m)+b2(k-m)]um}+d62[-uk-2-4uk-1+5uk],3≤k≤N,
for(2.10)Dt1/2u(tk-τ2)=1Γ(1/2)∫0tk-τ/2(tk-τ2-s)-1/2u′(s)ds
and the Crank-Nicholson difference scheme for parabolic equations, one can present the second order of accuracy difference scheme with respect to t and to x and(2.11)ukh(x)-uk-1h(x)τ+Dτ1/2ukh(x)+12Ahx(ukh(x)+uk-1h(x))=fkh(x),x∈Ω¯h,fkh(x)=f(tk-τ2,x),tk=kτ,1≤k≤N,Nτ=T,u0h(x)=0,x∈Ω¯h,
for the approximate solution of problem (2.4). Here(2.12)d=2πτ,b1(r)=r+12-r-12,b2(r)=-13((r+12)3/2-(r-12)3/2).Theorem 2.1.
Letτ and  |h|=h12+⋯+hn2 be sufficiently small numbers. Then, the solutions of difference scheme  (2.7) and  (2.11) satisfy the following stability estimate:
(2.13)max⁡1≤k≤N‖ukh‖L2h≤C1max⁡1≤k≤N‖fkh‖L2h.
Here C1 does not depend on τ,h, and fkh, 1≤k≤N.Proof.
For the solution of difference scheme (2.7), we have the following formulas:
(2.14)ukh(x)=∑s=1kRk-s+1Fsh(x)τ,1≤k≤N,
where
(2.15)R=(I+τAhx)-1,Fkh(x)=fkh(x)-Dτ1/2ukh(x),Dτ1/2ukh(x)=1π∑m=1kΓ(k-m+1/2)(k-m)!τ-1/2[-Dτ1/2umh(x)+fmh(x)].
The proof of (2.13) for (2.7) is based on (2.14) and estimate
(2.16)‖AhxRk‖L2h⟶L2h≤1kτ,‖Rk‖Lh2⟶L2h≤1,1≤k≤N,
and the triangle inequality.
In the same manner, we can obtain (2.13) for (2.11) using the inequality
(2.17)‖AhxBkC2‖L2h⟶L2h≤1kτ,‖Bk‖L2h⟶L2h≤1,1≤k≤N.Theorem 2.2.
Letτ and  |h|=h12+⋯+hn2 be sufficiently small numbers. Then, the solutions of difference scheme  (2.7) satisfy the following almost coercive stability estimate:
(2.18)max⁡1≤k≤N⁡‖ukh-uk-1hτ‖L2h+max⁡1≤k≤N⁡∑p=1m‖(ukh)x̅pxp,jp‖L2h≤C2ln⁡⁡1τ+|h|max⁡1≤k≤N⁡‖fkh‖L2h.
Here C2 is independent of τ, h, and fkh, 1≤k≤N.Proof.
The proof of (2.18) for (2.7) is based on (2.14) and estimate (2.16) and the triangle inequality.Theorem 2.3.
Letτ and  |h|=h12+⋯+hm2 be sufficiently small numbers. Then, the solutions of difference scheme  (2.11) satisfy the following almost coercive stability estimate:
(2.19)max⁡1≤k≤N⁡‖ukh-uk-1hτ‖L2h+max⁡1≤k≤N⁡12∑p=1m‖(ukh+uk-1h)x̅pxp,jp‖L2h≤C3ln⁡⁡1τ+|h|max⁡1≤k≤N⁡‖fkh‖L2h.
Here C3 does not depend on τ,h, and fkh, 1≤k≤N.Proof.
The proof of (2.19) for (2.11) is based on (2.14) and estimate (2.17) and the triangle inequality.Remark 2.4.
The method of proofs of Theorems2.1–2.3 enables us to obtain the estimate of convergence of difference schemes of the first and second orders of accuracy for approximate solutions of the initial-boundary-value problem
(2.20)∂u(∂t,x)∂t-∑p=1nap(x)uxpxp+∑p=1nbp(x)uxp+Dtαu(t,x)=f(t,x;u(t,x),ux1(t,x),…,uxn(t,x)),x=(x1,…,xn)∈Ω,0<t<T,u(0,x)=0,x∈Ω¯,u(t,x)∣S1=0,∂u(t,x)∂n⃗|S2=0
for semilinear fractional parabolic partial differential equations.Note that one has not been able to obtain a sharp estimate for the constants figuring in the stability estimates of Theorems2.1, 2.2, and 2.3. Therefore, our interest in the present paper is studying the difference schemes (2.7) and (2.11) by numerical experiments. Applying these difference schemes, the numerical methods are proposed in the following section for solving the one-dimensional fractional parabolic partial differential equation. The method is illustrated by numerical experiments.


## 3. Numerical Applications

For numerical results we consider two examples.Example 3.1.
We consider the initial-boundary-value problem(3.1)∂u(t,x)∂t+Dt1/2u(t,x)-∂∂x((1+x)∂u(t,x)∂x)+u(t,x)=f(t,x),f(t,x)=[3+t+16t5π+(1+x)2π2t]t2sin⁡2⁡πx-(1+x)2π2t3cos⁡2⁡πx-2πt3sin⁡⁡πxcos⁡⁡πx,0<t<1,0<x<1,u(t,0)=ux(t,1)=0,0≤t≤1,u(0,x)=0,0≤x≤1,
for the one-dimensional fractional parabolic partial differential equation. The exact solution of problem (3.1) is
(3.2)u(t,x)=t3sin⁡2⁡πx.
First, applying difference scheme (2.7), we obtain
(3.3)unk-unk-1τ+1π∑r=1kΓ(k-r+1/2)(k-r)!unr-unr-1τ1/2-1h[(1+xn+1)un+1k-unkh-(1+xn)unk-un-1kh]+unk=φnk,φnk=f(tk,xn),tk=kτ,1≤k≤N,xn=nh,1≤n≤M-1,u0k=0,uM-1k=uMk,0≤k≤N,un0=0,0≤n≤M.
We get the system of equations in the matrix form
(3.4)AUn+1+BUn+CUn-1=Dφn,1≤n≤M-1,U0=0̃,UM-1=UM,
where
(3.5)A=[000⋯000an0⋯0000an⋯00⋯⋯⋯⋯⋯⋯000⋯an0000⋯0an](N+1)×(N+1),(3.6)B=[b1100⋯00b21b220⋯00b31b32b33⋯00⋯⋯⋯⋯⋯⋯bN1bN2bN3⋯bNN0bN+1,1bN+1,2bN+1,3⋯bN+1,NbN+1,N+1](N+1)×(N+1),(3.7)C=[000⋯000cn0⋯0000cn⋯00⋯⋯⋯⋯⋯⋯000⋯cn0000⋯0cn](N+1)×(N+1),(3.8)D=[000⋯00010⋯00001⋯00⋯⋯⋯⋯⋯⋯000⋯10000⋯01](N+1)×(N+1),(3.9)φn=[φn0φn1φn2⋮φnN-1φnN](N+1)×1,Uq=[Uq0Uq1Uq2⋮UqN-1UqN](N+1)×1,q=n±1,n,(3.10)an=-1+xn+1h2,cn=-1+xnh2,b11=1,b21=-1τ-1τ,b22=1τ+1τ+2+xn+1+xnh2+1,b31=-Γ(1+1/2)πτ,b32=Γ(1+1/2)-Γ(1/2)πτ-1τ,b33=1τ+1τ+2+xn+1+xnh2+1,bij={-Γ(i-2+1/2)πτ(i-2)!,j=1,πτ[Γ(i-j+1/2)(i-j)!-Γ(i-j-1+1/2)(i-j-1)!],2≤j≤i-2,πτ[Γ(1+12)-Γ(12)]-1τ,j=i-1,1τ+1τ+2+xn+1+xnh2+1,j=i,0,i<j≤N+1,
for i=4,5,…,N+1 and
(3.11)φnk=[3+kτ+16kτ5π+(1+nh)2π2kτ](kτ)2sin⁡2⁡(πnh)-(1+nh)2π2(kτ)3cos⁡2⁡(πnh)-2π(kτ)3sin⁡⁡(πnh)cos⁡⁡(πnh).
So, we have the second-order difference equation with respect ton matrix coefficients. This type system was developed by Samarskii and Nikolaev [20]. To solve this difference equation we have applied a procedure for difference equation with respect to k matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:
(3.12)Uj=αj+1Uj+1+βj+1,UM=(I-αM)-1βM,j=M-1,…,2,1,
where αj(j=1,2,…,M) are (N+1)×(N+1) square matrices and βj(j=1,2,…,M) are (N+1)×1 column matrices defined by
(3.13)αj+1=-(B+Cαj)-1A,βj+1=(B+Cαj)-1(Dφj-Cβj),j=1,2,…,M-1,
where j=1,2,…,M-1,α1 is the (N+1)×(N+1) zero matrix and β1 is the (N+1)×1 zero matrix.
Second, applying difference scheme (2.11), we obtain the second order of accuracy difference scheme in t and in x(3.14)unk-unk-1τ+Dτ1/2unk-12[(1+xn)un+1k-2unk+un-1kh2+un+1k-un-1k2h-unk+(1+xn)⋅un+1k-1-2unk-1+un-1k-1h2+un+1k-1-un-1k-12h-unk-1]=φnk,φnk=f(tk-τ2,xn),tk=kτ,xn=nh,1≤k≤N,1≤n≤M-1,u0k=0,3uMk-4uM-1k+uM-2k=0,0≤k≤N,un0=0,0≤n≤M.
Here Dτ1/2unk is defined same as in (2.9). We get the system of equations in the matrix form
(3.15)AUn+1+BUn+CUn-1=Dφn,1≤n≤M-1,U0=0̃,3UM-4UM-1+UM-2=0,
where
(3.16)A=[000⋯00anan0⋯000anan⋯00⋯⋯⋯⋯⋯⋯000⋯an0000⋯anan](N+1)×(N+1),(3.17)B=[b1100⋯00b21b220⋯00b31b32b33⋯00⋯⋯⋯⋯⋯⋯bN1bN2bN3⋯bNN0bN+1,1bN+1,2bN+1,3⋯bN+1,NbN+1,N+1](N+1)×(N+1),(3.18)C=[000⋯00cncn0⋯000cncn⋯00⋯⋯⋯⋯⋯⋯000⋯cn0000⋯cncn](N+1)×(N+1),(3.19)D=[000⋯00010⋯00001⋯00⋯⋯⋯⋯⋯⋯000⋯10000⋯01](N+1)×(N+1),(3.20)φn=[φn0φn1φn2⋮φnN-1φnN](N+1)×1,Uq=[Uq0Uq1Uq2⋮UqN-1UqN](N+1)×1,q=n±1,n,(3.21)an=-12(1+xnh2+12h),cn=-12(1+xnh2-12h),d=2πτ,b1(r)=r+12-r-12,b2(r)=-13[(r+12)3/2-(r-12)3/2],b11=1,b21=-d23-1τ+1+xnh2+12,b22=d23+1τ+1+xnh2+12,b31=d265,b32=d65-1τ+1+xnh2+12,b33=d65+1τ+1+xnh2+12,b41=d[1b1(1)+b2(1)],b42=d[-3b1(1)-2b2(1)]-d62,b43=d[2b1(1)+b2(1)]-4d62-1τ+1+xnh2+12,b44=5d62+1τ+1+xnh2+12,b51=d[2b1(2)+b2(2)],b52=d[-5b1(2)-2b2(2)+1b1(1)+b2(1)],b53=d[3b1(2)+b2(2)-3b1(1)-2b2(1)]-d62,b54=d[2b1(1)+b2(1)]-4d62-1τ+1+xnh2+12,b55=5d62+1τ+1+xnh2+12,bij={d[(i-3)b1(i-3)+b2(i-3)],j=1,d[(5-2i)b1(i-3)-2b2(i-3)+(i-4)b1(i-4)+b2(i-4)],j=2,d[(i-j+1)b1(i-j)+b2(i-j)+(2j-2i+1)b1(i-j-1)-2b2(i-j-1)+(i-j-2)b1(i-j-2)+b2(i-j-2)],3≤j≤i-3,d[3b1(2)+b2(2)-3b1(1)-2b2(1)]-d62,j=i-2,d[2b1(1)+b2(1)]-4d62-1τ+1+xnh2+12,j=i-1,5d62+1τ+1+xnh2+12,j=i,0,i<j≤N+1,
for i=6,7,…,N+1 and
(3.22)φnk=[3+kτ+16kτ5π+(1+nh)2π2kτ](kτ)2sin⁡2⁡(πnh)-(1+nh)2π2(kτ)3cos⁡2⁡(πnh)-2π(kτ)3sin⁡⁡(πnh)cos⁡⁡(πnh).
For the solution of the matrix equation (3.15), we use the same algorithm as in the first order of accuracy difference scheme, where uM=[3I-4αM+αM-1αM]-1*[(4I-αM-1)βM-βM-1].Example 3.2.
We consider the initial-boundary-value problem(3.23)∂u(t,x)∂t+Dt1/2u(t,x)-∂∂x((1+x)∂u(t,x)∂x)+u(t,x)=f(t,x),f(t,x)=[3+t+16t5π+(1+x)2π2t]t2sin⁡2⁡πx-(1+x)2π2t3cos⁡2⁡πx-2πt3sin⁡⁡πxcos⁡⁡πx,0<t<1,0<x<1,ux(t,0)=0,u(t,1)=0,0≤t≤1,u(0,x)=0,0≤x≤1,
for the one-dimensional fractional parabolic partial differential equation. The exact solution of problem (3.23) is
(3.24)u(t,x)=t3sin⁡2⁡πx.
First, applying difference scheme (2.7), we obtain
(3.25)unk-unk-1τ+1π∑r=1kΓ(k-r+1/2)(k-r)!unr-unr-1τ1/2-1h[(1+xn+1)un+1k-unkh-(1+xn)unk-un-1kh]+unk=φnk,φnk=f(tk,xn),tk=kτ,1≤k≤N,xn=nh,1≤n≤M-1,u0k=u1k,uMk=0,0≤k≤N,un0=0,0≤n≤M.
We get the system of equations in the matrix form
(3.26)AUn+1+BUn+CUn-1=Dφn,1≤n≤M-1,U0=U1,UM=0̃,
where matrices A,B,C,D,φn,Uq(q=n∓1,n) are defined same as in (3.5), (3.6), (3.7), (3.8), (3.9), respectively.
So, we have the second-order difference equation with respect ton matrix coefficients. To solve this difference equation we have applied a procedure for difference equation with respect to k matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:
(3.27)Uj=αj+1Uj+1+βj+1,UM=0,j=M-1,…,2,1,
where αj(j=1,2,…,M) are (N+1)×(N+1) square matrices and βj(j=1,2,…,M) are (N+1)×1 column matrices defined by
(3.28)αj+1=-(B+Cαj)-1A,βj+1=(B+Cαj)-1(Dφj-Cβj),j=1,2,…,M-1,
where j=1,2,…,M-1,α1 is the (N+1)×(N+1) identity matrix and β1 is the (N+1)×1 zero matrix.
Second, applying the formulas(3.29)ux(tk,0)=u1k-u0kh-h2uxx(tk,0)+o(h2),0≤k≤N,ux(tk,M)=3uMk-4uM-1k+uM-2k2h+o(h2),0≤k≤N,ut(tk,0)=u0k+1-u0k-12τ+o(τ2),1≤k≤N-1,ut(tN,0)=3u0N-4u0N-1+u0N-22τ+o(τ2),k=N,
and applying difference scheme (2.11), we obtain the second order of accuracy difference scheme in t and in x(3.30)unk-unk-1τ+Dτ1/2unk-12[(1+xn)un+1k-2unk+un-1kh2+un+1k-un-1k2h-unk+(1+xn)un+1k-1-2unk-1+un-1k-1h2+un+1k-1-un-1k-12h-unk-1]=φnk,φnk=f(tk-τ2,xn),tk=kτ,xn=nh,1≤k≤N,1≤n≤M-1,u00=0,k=0,-h4τu0k-1+[1h+h2Dt1/2+h2]u0k+h4τu0k+1=1hu1k+h2φ0k,1≤k≤N-1,h4τu0N-2-hτu0N-1+[1h+3h4τ+h2Dt1/2+h2]u0N=1hu1N+h2φ0N,k=N,uMk=0,0≤k≤N,un0=0,0≤n≤M.
Here Dτ1/2unk is defined similar to (2.9). We get the system of equations in the matrix form
(3.31)AUn+1+BUn+CUn-1=Dφn,1≤n≤M-1,EU0=FU1+Rφ0,UM=0̃,
where matrices A,B,C,D,φn,Uq(q=n∓1,n) are defined same as in (3.16), (3.17), (3.18), (3.19), (3.20), respectively.
For the solution of the matrix equation (3.31), we use the same algorithm as in the first order of accuracy difference scheme, where
(3.32)uM=0̃,α1=E-1F,β1=E-1Rφ0,F=[000⋯0001h0⋯00001h⋯00⋯⋯⋯⋯⋯⋯000⋯1h0000⋯01h](N+1)×(N+1),R=[00⋯001⋯0⋯⋯⋯⋯00⋯1](N+1)×(N+1),E=[e1100⋯00e21e220⋯00e31e32e33⋯00⋯⋯⋯⋯⋯⋯eN1eN2eN3⋯eNN0eN+1,1eN+1,2eN+1,3⋯eN+1,NeN+1,N+1](N+1)×(N+1),e11=1,e21=-h4τ-4h3πτ,e22=1h+h2+4h3πτ,e23=h4τ,e31=22h15πτ,e32=-162h15πτ-h4τ,e33=1h+h2+142h15πτ,e34=h4τ,e41=dh2[(1+12)b1(1)+b2(1)],e42=dh2[-4b1(1)-2b2(1)+12b1(0)+b2(0)],e43=dh2[(2+12)b1(1)+b2(1)-2-2(-13)]-h4τ,e44=1h+h2+dh2[(1+12)b1(0)+b2(0)],e45=h4τ,e51=dh2[(2+12)b1(2)+b2(2)],e52=dh2[-23b1(2)-2b2(2)+(1+12)b1(1)+b2(1)],e53=dh2[(2+1+12)b1(2)+b2(2)-22b1(1)-2b2(1)+12b1(0)+b2(0)],e54=dh2[(1+1+12)b1(1)+b2(1)-2b1(0)-2b2(0)]-h4τ,e55=1h+h2+dh2[(1+12)b1(0)+b2(0)],e56=h4τ,eij={dh2[(i-3+12)b1(i-3)+b2(i-3)],j=1,dh2[-2(i-2)b1(i-3)-2b2(i-3)+(i-4+12)b1(i-4)+b2(i-4)],j=2,dh2[(i-j+1+12)b1(i-j)+b2(i-j)-2(i-j)b1(i-j-1)-2b2(i-j-1)+(i-j-2+12)b1(i-j-2)+b2(i-j-2)],3≤j≤i-2,-h4τ+dh2[(2+12)b1(1)+b2(1)-2b1(0)-2b2(0)],j=i-1,1h+h2+dh2[(1+12)b1(0)+b2(0)],j=i,h4τ,j=i+1,h4τ+dh2[(i-N+2+12)b1(i-N+1)+b2(i-N+1)-2(i-N+1)b1(i-N)-2b2(i-N)+(i-N-1+12)b1(i-N-1)+b2(i-N-1)],j=N-1,-hτ+dh2[(2+12)b1(1)+b2(1)-2b1(0)-2b2(0)],j=N,1h+h2+3h4τ+dh2[(1+12)b1(0)+b2(0)],j=N+1,0,j>i+1,
for i=6,7,…,N+1 and
(3.33)φ0k=-2π2(kτ)3.


### 3.1. Error Analysis

Finally, we give the results of the numerical analysis. The error is computed by the following formula:(3.34)EMN=max⁡1≤k≤N,1≤n≤M-1⁡|u(tk,xn)-unk|,
where u(tk,xn) represents the exact solution and unk represents the numerical solutions of these difference schemes at (tk,xn). The numerical solutions are recorded for different values of N and M. Tables 1 and 2 are constructed for N=M=25,50, and 100, respectively.Table 1
Error analysis of first and second order of accuracy difference schemes for Example3.1.
MethodN=M=25N=M=50N=M=1001st order of accuracy0.25530.12560.06222nd order of accuracy0.00629.771×10-41.963×10-4Table 2
Error analysis of first and second order of accuracy difference schemes for Example3.2.
MethodN=M=25N=M=50N=M=1001st order of accuracy0.16530.08070.03992nd order of accuracy0.00255.943×10-41.453×10-4The results in Tables1 and 2 show that, by using the Crank-Nicholson difference scheme, more accurate approximate results can be obtained rather than the first order of accuracy difference scheme.


## 3.1. Error Analysis

Finally, we give the results of the numerical analysis. The error is computed by the following formula:(3.34)EMN=max⁡1≤k≤N,1≤n≤M-1⁡|u(tk,xn)-unk|,
where u(tk,xn) represents the exact solution and unk represents the numerical solutions of these difference schemes at (tk,xn). The numerical solutions are recorded for different values of N and M. Tables 1 and 2 are constructed for N=M=25,50, and 100, respectively.Table 1
Error analysis of first and second order of accuracy difference schemes for Example3.1.
MethodN=M=25N=M=50N=M=1001st order of accuracy0.25530.12560.06222nd order of accuracy0.00629.771×10-41.963×10-4Table 2
Error analysis of first and second order of accuracy difference schemes for Example3.2.
MethodN=M=25N=M=50N=M=1001st order of accuracy0.16530.08070.03992nd order of accuracy0.00255.943×10-41.453×10-4The results in Tables1 and 2 show that, by using the Crank-Nicholson difference scheme, more accurate approximate results can be obtained rather than the first order of accuracy difference scheme.



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*Source: 463746-2012-07-25.xml*