# Finite Difference Method for the Reverse Parabolic Problem

Authors: Charyyar Ashyralyyev; Ayfer Dural; Yasar Sozen
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A finite difference method for the approximate solution of the reverse multidimensional parabolic differential equation with a multipoint boundary condition and Dirichlet condition is applied. Stability, almost coercive stability, and coercive stability estimates for the solution of the first and second orders of accuracy difference schemes are obtained. The theoretical statements are supported by the numerical example.


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## Body

## 1. Introduction

In the study of boundary value problems for partial differential equations, the role played by the well-posedness (coercivity inequalities) is well known (see, e.g.,  [1–3]). Well-posedness of nonlocal boundary value problems for partial differential equations of parabolic type has been studied extensively by many researchers (see, e.g., [4–15] and the references therein).In the paper [4], Ashyralyev studied the positivity of second-order differential and difference operators with nonlocal condition and the structure of interpolation spaces generated by these operators in a Banach space. Applying this result, he obtained the coercive inequalities for the solutions of the nonlocal boundary value problem for differential and difference equations.In [5], Ashyralyev et al. considered the nonlocal boundary value problem
(1.1)v′(t)+Av(t)=f(t),0<t<1,v(0)=v(λ)+μ,0<λ≤1,
in a Banach space with strongly positive operator A. They established the well-posedness of problem (1.1) in Hölder spaces. Moreover, they obtained the exact Schauder's estimates in Hölder norms of solutions of the boundary values problem for 2m-th order multidimensional parabolic equations.Ashyralyev established in [6] the well-posedness of the nonlocal boundary-value problem (1.1) in Bochner spaces. He considered the first and second order of accuracy difference schemes for the approximate solutions of problem (1.1). He also established the coercive inequalities for the solutions of these difference schemes. Moreover, in applications, he obtained the almost coercive stability and coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary-value problem for parabolic equation.Clément and Guerre-Delabriére studied in [8] maximal regularity (in the Lp-sense) for abstract Cauchy problems of order one and boundary value problems of order two. As is well-known regularity of the first problems implies regularity of the second ones; they also proved that the converse to hold if the underlying Banach space has the UMD property. A stronger notion of regularity, which is introduced by Sobolevskii, plays an important role in the proofs.In [9], Gulin et al. considered the linear heat equation:
(1.2)ut=uxx+uyy,0<x<1,0<y<1,
with Dirichlet condition u(x,0,t)=u(x,1,t)=0, 0≤x≤1 and nonlocal boundary conditions u(0,y,t)=0, ux(0,y,t)=ux(1,y,t), 0≤y≤1. They constructed an explicit difference scheme with second order of approximation with respect to the space variables and first order of approximation with respect to t. Moreover, using previous results of Ionkin and Morozova for the one-dimensional heat equation with nonlocal boundary conditions, they proved the stability of this scheme with respect to the D1-norm ∥y∥D1=(D1y,y)1/2, which is induced by the symmetric and positive-definite matrix D1.Liu et al. studied in [10] a finite difference method for multidimensional nonlinear coupled system of parabolic and hyperbolic equations. By using a variational method, they obtained an a priori estimate. They also proved that the finite difference scheme is uniquely solvable and unconditionally stable. To support the theory, they gave numerical example of two-dimensional problem.In [11, 12], Martin-Vaquero and Vigo-Aguiar provided algorithms improving the CPU time and accuracy of Crandall's formula. They studied the convergence of the algorithms and compared the efficiency of the methods with well-known numerical examples.In [13], Sapagovas applied finite difference approximations to a nonhomogeneous heat equation in one space dimension, subject to nonlocal boundary conditions. He presented a stable difference approximation for the equation and a piecewise constant discretization of the integrals appearing in the boundary conditions. He discussed the stability of the complete problem with respect to two parameters included in the integral terms. He constructed a stability region in the plane of the parameters and gave practical examples with specific choices of the integral conditions. Sapagovas investigated in [14] the stability of implicit difference schemes for the equation of a thermoelastic rod, which is a parabolic equation subject to integral conditions for the boundaries.In [15], Shakhmurov dealt with a nonlocal boundary value problem for a degenerate equation in a Banach space E with unbounded operators in E. He proved the maximal Lp regularity and Fredholmness of the problem. He also applied the results to nonlocal boundary value problems for degenerate elliptic and quasielliptic differential equations and their finite or infinite systems on cylindrical domains.It is well known that reverse problems arise in various applications, for example, boundary layer problems in fluid dynamics [16, 17], plasma physics, and astrophysics in the study of propagation of an electron beam through the solar corona [18]. For further applications of such problems, we refer the reader to [19–22] and the references therein.In the paper [23], Ashyralyev et al. considered the multipoint nonlocal boundary value problem for reverse parabolic equations
(1.3)du(t)dt-Au(t)=f(t),(0≤t≤1),u(1)=∑k=1p‍αku(θk)+φ,0≤θ1<θ2<⋯<θp<1
in a Hilbert space H with self-adjoint positive definite operator A.u
(
t
) is called a solution of problem (1.3) if the following conditions hold:  (1)
u(t) is continuously differentiable on the segment [0,1]. The derivatives at the end points of the segment are understood as the appropriate unilateral derivatives.(2)
The elementu(t) belongs to D(A) for all t∈[0,1] and the function Au(t) is continuous on the segment [0,1].(3)
u(t) satisfies the equation and the nonlocal boundary conditions (1.3).A solution of problem (1.3) defined in this manner will be from now referred to as a solution of problem (1.3) in the space C(H)=C([0,1],H) of all continuous functions φ(t) defined on [0,1] with values in H equipped with the norm
(1.4)‖φ‖C(H)=max⁡0≤t≤1‖φ(t)‖H.Problem (1.3) is well posed in C(H), if for the solutions of (1.3), we have the following coercivity inequality:(1.5)‖u′‖C(H)+‖Au(t)‖C(H)≤M(‖f‖C(H)+‖Aφ‖H).
Here, 1≤M is independent of f(t)∈C(H),φ∈D(A).Throughout the paper,M indicates positive constants which can be different from time to time and we are not interested to make precise. We write M(α,β,…) to stress the fact that the constant depends only on α,β,…Under the assumption:(1.6)∑k=1p‍|αk|≤1.Ashyralyev et al. established in [23] the well-posedness of these problems in the space of smooth functions. In applications, they obtained coercivity estimates for the solution of parabolic differential equations.Moreover, in [24], Ashyralyev et al. considered the first order of accuracy Rothe difference scheme:
(1.7)τ-1(uk-uk-1)-Auk-1=φk,φk=f(tk),tk=kτ,1≤k≤N,Nτ=1,uN=∑m=1p‍αmulm+φ,lm=[θmτ],1≤m≤p,
for approximately solving problem (1.3). They established some stability estimates and almost coercivity of the solution for the difference scheme.In the present paper, multipoint nonlocal boundary value problem for the multidimensional parabolic equation with Dirichlet condition,(1.8)ut(t,x)+∑r=1n‍(ar(x)uxr)xr=f(t,x),x=(x1,…,xn)∈Ω,0<t<1,u(1,x)=∑i=1p‍αiu(θi,x)+φ(x),x∈Ω¯,0≤θ1<θ2<⋯<θp<1,u(t,x)=0,x∈S,0≤t≤1
under the condition (1.6) is considered. Here, ar(x),(x∈Ω),φ(x)(x∈Ω¯), and f(t,x)(t∈(0,1),x∈Ω) are given smooth functions and ar(x)≥a>0, and Ω=(0,ℓ)×⋯×(0,ℓ) is the open cube in the n-dimensional Euclidean space with boundary S,Ω¯=Ω∪S. In the Hilbert space H=L2(Ω¯), we introduce the self-adjoint positive definite operator A defined by
(1.9)Au(x)=-∑r=1n‍(ar(x)uxr(x))xr,
with domain
(1.10)D(A)={u,uxr,uxrxr∈L2(Ω¯),1≤r≤n:u(x)|x∈S=0}.
Then, problem (1.8) can be written in the abstract form as the nonlocal boundary value problem for reverse parabolic equation (1.3).The first and second orders of accuracy int and the second order of accuracy in space variables for the approximate solution of problem (1.8) are presented. Applying the method of papers [23, 24], the stability, almost coercive stability, and coercive stability estimates for the solution of these difference schemes are obtained. The modified Gauss elimination method for solving these difference schemes in the case of one-dimensional parabolic partial differential equations is used.


## 2. Difference Schemes: Stability Estimates

We will discretize problem (1.8) in two steps. In the first step, we define the grid spaces
(2.1)Ω~h={x=xm=(h1m1,…,hnmn);m=(m1,…,mn),mr=0,…,Nr,hrNr=1,r=1,…,n},Ωh=Ω~h∩Ω,Sh=Ω~h∩S.We denote that|h|=h12+⋯+h12. Let L2h=L2(Ω~h) denote the Banach space of grid functions:
(2.2)φh(x)={φ(h1m1,…,hnmn)},
defined on Ω~h, equipped with the norm
(2.3)‖φh‖L2h=(∑x∈Ω~h‍|φh(x)|2h1⋯hn)1/2.To the differential operatorA generated by problem (1.8), we assign the second-order approximation difference operator Ahx=Chx+Bhx acting in the space of grid functions uh(x), satisfying the condition uh(x)=0 for all x∈Sh. Assume that Chx is self-adjoint, positive-definite operator in L2h and (Chx)-1Bhx is bounded operator in L2h.By usingAhx, we arrive at the multipoint nonlocal boundary value problem:
(2.4)duh(t,x)dt-Ahxuh(t,x)=fh(t,x),0<t<1,x∈Ω~h,uh(1,x)=∑m=1p‍αmuh(θm,x)+φh(x),x∈Ω~h,
for a finite system of ordinary differential equations with a fixed |h|. Note that |h|=h12+⋯+hn2→0. Therefore, we will try to obtain stability, coercivity stability, and almost coercivity estimates with constants independent of |h|.In the second step, problem (2.4) is replaced by the first order of accuracy difference scheme
(2.5)ukh(x)-uk-1h(x)τ-Ahxuk-1h(x)=fkh(x),fkh(x)=fh(tk,x),tk=kτ,1≤k≤N,x∈Ω~h,uNh(x)=∑m=1p‍αmulmh(x)+φh(x),x∈Ω~h,lm=[θmτ],m=1,…,p,
and the second order of accuracy difference scheme
(2.6)ukh(x)-uk-1h(x)τ-AhxBhxuk-1h(x)=fkh(x),Bhx=I+τAhx2,fkh(x)=Bhxfh(tk-(τ/2),x),tk=kτ,1≤k≤N,Nτ=1,x∈Ω~h,uNh(x)=∑m=1p‍αm{(I+dmAhx)ulmh(x)+dmBhxψlm+1}+φh(x),x∈Ω~h,dm=θm-[θmτ]τ,lm=[θmτ],m=1,…,p,
where [·] denotes the greatest integer function.To formulate our results, letL2h=L2(Ω~h) and W2h2=W22(Ω~h) be spaces of the grid functions φh(x)={φ(h1m1,…,hnmn)} defined on Ω~h, equipped with the norms
(2.7)‖φh‖L2h=(∑x∈Ω~h‍|φh(x)|2h1⋯hn)1/2,‖φh‖W2h2=‖φh‖L2h+(∑x∈Ω~h‍∑r=1n‍|(φh)xr|2h1⋯hn)1/2+(∑x∈Ω~h‍∑r=1n‍|(φh(x))xrxr¯,mr|2h1⋯hn)1/2.Furthermore, let[0,1]τ={tk=kτ,1≤k≤N,Nτ=1} be the uniform grid space with step size τ>0, where N is a fixed positive integer. We denote ℱτ(H)=ℱ([0,1]τ,H) for the linear space of grid functions φτ={φk}1N with values in the Hilbert space H.Forα∈[0,1], let 𝒞α(H)=𝒞α([0,1]τ,H) and 𝒞1α(H)=𝒞1α([0,1]τ,H) be, respectively, the Hölder space and the weighted Hölder space with the norms
(2.8)‖φτ‖Cα(H)=‖φτ‖Cτ(H)+max⁡1≤k<k+r≤N‖φk+r-φk‖H(rτ)α,‖φτ‖C1α(H)=‖φτ‖Cτ(H)+max⁡1≤k<k+r≤N((N-k)τ)α‖φk+r-φk‖H(rτ)α.
Here, 𝒞τ(H)=𝒞([0,1]τ,H) is the Banach space of bounded grid functions with norm:
(2.9)‖φτ‖Cτ(H)=max⁡1≤k≤N‖φk‖H.Theorem 2.1.
Letτ and |h| be sufficiently small positive numbers. Then, for the solutions of difference schemes (2.5) and (2.6), the following stability estimate holds:
(2.10)‖{ukh}0N‖Cτ(L2h)≤M(δ,θp)[‖φh‖L2h+‖{fkh}1N‖Cτ(L2h)],
where M(δ,θp) is independent of τ,h,φh(x), and fkh(x) and k=1,…,N.Proof.
The proof of Theorem2.1 is based on the formulas for the solution of difference scheme (2.5)
(2.11)ukh=RN-kuNh-∑j=k+1N‍Rj-kfjhτ,0≤k≤N-1,(2.12)uNh=Tτ(-∑k=1p‍∑j=lk+1N‍αkRj-lkfjhτ+φh),
and for the solution of difference scheme (2.6)
(2.13)ukh=DN-kuNh-∑j=k+1N‍Dj-kfjhτ,0≤k≤N-1,(2.14)uNh=Tτ′{-∑m=1p‍∑j=lm+1N‍αm(I+dmAhx)Dj-lmfjhτ+∑m=1p‍αmdm(Bhx)-1flm+1h+φh}.
Here,
(2.15)R=(I+τAhx)-1,D=(I+τAhx+(τAhx)22)-1,Tτ=(I-∑k=1p‍αkRN-[θk/τ])-1,T′τ=(I-∑k=1p‍αk(I+dkAhx)DN-[θk/τ])-1.
By the spectral representation of self-adjoint positive definite operator and the triangle inequality, we have(2.16)‖Tτ‖H→H≤sup⁡δ≤μ1|1-∑k=1pαk(1+τμ)-N+[θp/τ]|≤M(δ,θp).
Similarly, we have(2.17)‖Tτ′‖H→H≤M(δ,θp).
Estimates (2.16) and (2.17) conclude the proof of Theorem 2.1.Theorem 2.2.
Letτ and |h| be sufficiently small positive numbers. Then, for the solutions of difference problem (2.5) and (2.6), the following almost coercivity inequality
(2.18)‖{τ-1(ukh-uk-1k)}1N‖Cτ(L2h)≤M(δ,θp)[‖φh‖W2h2+‖{fkh}1N‖Cτ(L2h)ln⁡1τ+|h|]
is valid, where M(δ,θp) does not depend on τ,h,φh(x),fkh(x),k=1,…,N.Proof.
Using formulas (2.11)–(2.14), estimates (2.16) and (2.17), the triangle inequality, assumption (1.6), we obtain
(2.19)‖{τ-1(ukh-uk-1k)}1N‖Cτ(L2h)≤M(δ,θp)[‖φh‖W2h2+min⁡{ln⁡1τ,1+|ln⁡‖Axh‖L2h→L2h|}‖{fkh}1N‖Cτ(L2h)].
Since
(2.20)|ln⁡‖Axh‖L2h→L2h|≤Mln⁡1|h|,
we have that
(2.21)min⁡{ln⁡1τ,1+|ln⁡‖Axh‖L2h→L2h|}≤M1ln⁡1τ+|h|.
From that, inequality (2.19), and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h it follows inequality (2.18). Theorem 2.2 is proved.Theorem 2.3  (see [25, 26]).
For the solution of the elliptic difference problem:(2.22)Ahxuh(x)=ωh(x),x∈Ω~h,uh(x)=0,x∈Sh,
the following coercivity inequality holds:
(2.23)∑r=1n‍‖(ukh)x¯rx¯r,jr‖L2h≤M‖ωh‖L2h,
where M does not depend on h and ωh.Theorem 2.4.
Letτ and |h| be sufficiently small positive numbers. Then, the solutions of difference problem (2.5) and (2.6) satisfy the following coercivity stability estimate:
(2.24)‖{τ-1(ukh-uk-1h)}1N‖C1α(L2h)≤M(δ,θp,α)[‖φh‖W2h2+‖{fkh}1N‖C1α(L2h)],
where M(δ,θp,α) is independent of τ,h,fkh(x), and φh(x),k=1,…,N.Theorem 2.5.
LetAhxφh(x)=fNh(x)-∑k=1p‍αmfℓmh(x). Then, for solutions of problem (2.5) and (2.6), the following coercive stability estimate holds:
(2.25)‖{τ-1(ukh-uk-1h)}1N‖Cα(L2h)+‖{ukh}1N‖Cα(W2h2)≤M(δ,θp,α)‖{fkh}1N‖Cα(L2h),
where M(δ,θp,α) does not depend on τ,h,fkh(x), and φh(x),k=1,…,N.The proofs of Theorems2.4–2.5 are based on the formulas:
(2.26)Ahxuk-1h=RN-k+1AhxuNh-∑j=k+1N‍τAhxRj-k+1(fjh-fkh)+(RN-k+1-I)fkh,AhxBhxuk-1h=DN-k+1AhxBhxuNh-∑j=k+1N‍τAhx(I+τAhx2)Dj-k+1(fjh-fkh)+(DN-k+1-I)fkh,
the self-adjoint positive definiteness of the operator Ahx in L2h, estimates (2.16) and (2.17), the triangle inequality, and assumption (1.6).


## 3. Numerical Results

For the numerical result, we consider the nonlocal boundary value problem:(3.1)∂u(t,x)∂t+(2+cos⁡(x))∂2u(t,x)∂x2-sin⁡⁡(x)∂u(t,x)∂x=f(t,x),0<x<π,0<t<1,f(t,x)=(2t-6t2+4t3)sin⁡(x)-t2(1-t)2(2+cos⁡(x))sin⁡⁡(x)-t2(1-t)2cos⁡(x)sin⁡⁡(x),u(0,x)=u(1,x),0≤x≤π,u(t,0)=u(t,π)=0,0≤t≤1,
for the reverse parabolic equation. It is easy to see that u(t,x)=t2(1-t)2sin(x) is the exact solution of (3.1).For the approximate solution of nonlocal boundary value problem (3.1), consider the set [0,1]τ×[0,π]h of a family of grid points depending on the small parameters τ and h(3.2)[0,1]τ×[0,π]h={(t,xn):tk=kτ,k=1,…,N-1,Nτ=1,xn=nh,n=1,…,M-1,Mh=π}.Applying (2.5), we get the first order of accuracy in t and the second order of accuracy in x(3.3)unk-unk-1τ+(2+cos⁡(xn))un+1k-1-2unk-1+un-1k-1h2-sin⁡(xn)un+1k-1-un-1k-12h=f(tk,xn),k=1,…,N,n=1,…,M-1,u0k=uMk=0,k=0,…,N,un0=unN,n=0,…,M,
for the approximate solutions of the nonlocal boundary value problem (3.1).Note that for difference scheme (3.3), we have that
(3.4){-(2+cos⁡(xn))un+1k-1-2unk-1+un-1k-1h2-sin⁡(xn)un+1k-1-un-1k-12h}1M-1=Chxuk-1h(x)+Bhxuk-1h(x),
where
(3.5)Chxuh(x)={-(2+cos⁡(xn+1))((un+1-un)/h)-(2+cos⁡(xn))((un-un-1)/h)h}1M-1,Bhxuh(x)={cos⁡(xn+1)-cos⁡(xn)hun-un-1h-sin⁡(xn)un+1-un2h}1M-1.It is easy to see thatChx=(Chx)* and Chx≥δIh, and
(3.6)‖(Chx)-1Bhx‖L2h→L2h≤M,
where Ih is the identity operator.So, Theorems2.1, 2.2, 2.4, and 2.5 are compatible for the solution of (3.3).We can write (3.3) as in the matrix form
(3.7)Anun+1+Bnun+Cnun-1=Iφn,n=1,…,M-1,u0=0→,uM=0→.
Here, φn is an (N+1)×1 column matrix, An,Bn,Cn are (N+1)×(N+1) square matrices, An=anR,Cn=cnR,
(3.8)R=[0000⋯00001000⋯00000100⋯00000010⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯00000000⋯10000000⋯01000000⋯0010],Bn=[1000⋯000-1bnd00⋯00000bnd0⋯000000bnd⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯d0000000⋯bnd000000⋯0bnd00000⋯00bnd],an=2+cos⁡(xn)h2-sin⁡(xn)2h,bn=-1τ-2(2+cos⁡(xn))h2,cn=2+cos⁡(xn)h2+sin⁡(xn)2h,d=1τ.φn=[φn0⋮φnN],φn0=0,n=1,…,M-1,φnk=f(tk,xn),n=1,…,N,n=1,…,M-1,
here and in the future I is the (N+1)×(N+1) identity matrix,
(3.9)us=[us0⋮usN](N+1)×1,s=n-1,n,n+1.Samarskii and Nikolaev studied this type of system in [27] for difference equations. We seek the solution of (3.7) by the formula
(3.10)un=αn+1un+1+βn+1,n=M-1,…,1,
where uM=0→,αn(n=1,…,M-1) are (N+1)×(N+1) square matrices and βn(n=1,…,M-1) are (N+1)×1 column matrices. For the solution of difference equation (3.7) we need to use the following formulas for αn,βn:
(3.11)αn=-(Bn+Cnαn-1)-1An,βn=(Bn+Cnαn-1)-1(Iφn-Cnβn-1),n=2,…,M-1,
where α1 is the (N+1)×(N+1) zero matrix and β1 is the (N+1)×1 zero column vector.Second, we consider again the nonlocal boundary value problem (3.1). Applying (2.6) and formulas:
(3.12)u(xn+1)-u(xn-1)2h-u′(xn)=O(h2),u(xn+1)-2u(xn)+u(xn-1)h2-u′′(xn)=O(h2),u(xn+2)-4u(xn+1)+6u(xn)-4u(xn-1)+u(xn-2)h4-u(4)(xn)=O(h2),2u(0)-5u(h)+4u(2h)-u(3h)h2-u′′(0)=O(h2),2u(1)-5u(1-h)+4u(1-2h)-u(1-3h)h2-u′′(1)=O(h2),
we get the second order of accuracy in t and x(3.13)unk-unk-1τ-(sin⁡(xn)+τsin⁡(xn)+τsin⁡(xn)cos⁡(xn))un+1k-1-un-1k-12h+((2+cos⁡(xn))+τ2(6cos⁡(xn)+5cos⁡2(xn)-2))un+1k-1-2unk-1+un-1k-1h2+τ(4sin⁡(xn)+2sin⁡(xn)cos⁡(xn))un+2k-1-2un+1k-1+2un-1k-1-un-2k-12h3-τ2(2+cos⁡⁡(xn))2(un+2k-1-4un+1k-1+6unk-1-4un-1k-1+un-2k-1)h4=φnk,φnk=f(tk-τ2,xn)-τ2(1h2(2+cos⁡(xn))×(f(tk-τ2,xn+1)-2f(tk-τ2,xn)+f(tk-τ2,xn-1))-12hsin⁡(xn)(f(tk-τ2,xn+1)-f(tk-τ2,xn-1))),k=1,…,N,n=2,…,M-2,u0k=uMk=0,u1k=45u2k-15u3k,k=0,…,N,uM-1k=45uM-2k-15uM-3k,k=0,…,N,un0-unN=0,n=0,…,M,
for the approximate solutions of the nonlocal boundary value problem (3.1).We can rewrite this system in the following matrix form:(3.14)Anun+2+Bnun+1+Cnun+Dnun-1+Enun-2=Iφn,n=2,…,M-2,u0=0→,uM=0→,u1=45u2-15u3,uM-1=45uM-2-15uM-3,
where φn is an (N+1)×1 column matrix, An,  Bn,Cn,Dn,En are (N+1)×(N+1) square matrices
(3.15)φn=[φn0⋮φnN],An=vnR,Bn=ynR,Dn=znR,En=wnR,
(3.16)Cn=[100⋯0-1rnp0⋯000rnp⋯00⋮⋮⋮⋯⋮⋮000⋯p0000⋯rnp],
where p=1/τ,
(3.17)vn=τ2h3(4sin⁡(xn)+2sin⁡(xn)cos⁡(xn))-τ2h4(2+cos⁡(xn))2,yn=-12h(sin⁡(xn)+τsin⁡(xn)+τsin⁡(xn)cos⁡(xn))+1h2((2+cos⁡(xn))+τ2(6cos⁡⁡(xn)+5cos⁡2(xn)-2))-τh3(4sin⁡(xn)+2sin⁡(xn)cos⁡(xn))+2τh4(2+cos⁡(xn))2,rn=-1τ-2h2((2+cos⁡(xn))+τ2(6cos⁡(xn)+5cos⁡2(xn)-2))-3τh4(2+cos⁡(xn))2,zn=12h(sin⁡(xn)+τsin⁡(xn)+τsin⁡(xn)cos⁡(xn))+1h2((2+cos⁡(xn))+τ2(6cos⁡(xn)+5cos⁡2(xn)-2))+τh3(4sin⁡(xn)+2sin⁡(xn)cos⁡(xn))+2τh4(2+cos⁡(xn))2,wn=-τ2h4(2+cos⁡(xn))2-τ2h3(4sin⁡(xn)+2sin⁡(xn)cos⁡(xn)).For the solution of the last matrix equation, we use the modified variant of Gauss elimination method. We seek a solution of the matrix equation of the matrix equation in the following form:(3.18)un=αn+1un+1+βn+1un+2+γn+1,n=M-2,…,0,uM=0→,DM=(βM-2+5I)-(4I-αM-2)αM-1,uM-1=DM-1[(4I-αM-2)γM-1-γM-2],
where γ1=γ2=0→,α1=β1 are (N+1)×(N+1) zero matrices, α2=-4β2=(4/5)I, and
(3.19)βn+1=-Fn-1An,αn+1=-Fn-1(Bn+Dnβn+Enαn-1βn),γn+1=-Fn-1(Iφn-Dnγn-Enαn-1γn-Enγn-1),Fn=(Cn+Dnαn+Enβn-1+Enαn-1αn),n=2,…,M-2.Now, let us give the results of the numerical analysis. In order to get the solution, we used MATLAB programs. The numerical solutions are recorded for different values ofN=M and unk represents the numerical solutions of these difference schemes at (tk,xn). For their comparison, the errors are computed by
(3.20)EMN=max⁡-N≤k≤N,1≤n≤M-1|u(tk,xn)-unk|.
Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes. Table 1 is constructed for N=M=20, 40, and 60, respectively. Hence, the second order of accuracy difference scheme is more accurate compared with the first order of accuracy difference scheme.Table 1
Error analysis.
Difference schemes
N
=
M
=
20
N
=
M
=
40
N
=
M
=
60
Difference scheme (3.3)
5.693
×
10
-
3
2.865
×
10
-
3
1.914
×
10
-
3
Difference scheme (3.13)
2.390
×
10
-
4
6.408
×
10
-
5
2.901
×
10
-
5
Table1 is the error analysis between the exact solution and solutions derived by difference schemes.



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*Source: 294154-2012-07-31.xml*