# Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

Authors: Charyyar Ashyralyyev; Mutlu Dedeturk
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Afinite difference method for the approximate solution of the inverse problem for the multidimensional elliptic equation with overdetermination is applied. Stability and coercive stability estimates of the first and second orders of accuracy difference schemes for this problem are established. The algorithm for approximate solution is tested in a two-dimensional inverse problem.


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## Body

## 1. Introduction

It is well known that inverse problems arise in various branches of science (see [1, 2]). The theory and applications of well-posedness of inverse problems for partial differential equations have been studied extensively by many researchers (see, e.g., [3–17] and the references therein). One of the effective approaches for solving inverse problem is reduction to nonlocal boundary value problem (see, e.g., [6, 8, 11]). Well-posedness of the nonlocal boundary value problems of elliptic type equations was investigated in  [18–25] (see also the references therein).In [4], Orlovsky proved existence and uniqueness theorems for the inverse problem of finding a function u and an element p for the elliptic equation in an arbitrary Hilbert space H with the self-adjoint positive definite operator A:
(1)-utt(t)+Au(t)=f(t)+p,0<t<T,u(0)=φ,u(T)=ψ,u(λ)=ξ,0<λ<T.In [11], the authors established stability estimates for this problem and studied inverse problem for multidimensional elliptic equation with overdetermination in which the Dirichlet condition is required on the boundary.In present work, we study inverse problem for multidimensional elliptic equation with Dirichlet-Neumann boundary conditions.LetΩ=(0,ℓ)×⋯×(0,ℓ) be the open cube in the n-dimensional Euclidean space with boundary S and Ω¯=Ω∪S. In [0,T]×Ω, we consider the inverse problem of finding functions u(t,x) and p(x) for the multidimensional elliptic equation
(2)-utt(t,x)-∑r=1n‍(ar(x)uxr)xr+δu(t,x)=f(t,x)+p(x),x=(x1,…,xn)∈Ω,0<t<T,u(0,x)=φ(x),u(T,x)=ψ(x),u(λ,x)=ξ(x),mmmmmmmmmmmmmmmmmmmmmmmmx∈Ω¯,∂u(t,x)∂n→=0,x∈S,0≤t≤T.Here,0<λ<T and δ>0 are known numbers, ar(x)  (x∈Ω),   φ(x),  ψ(x),  ξ(x)  (x∈Ω¯), and f(t,x)  (t∈(0,T),x∈Ω) are given smooth functions, and also ar  (x)≥a>0  (x∈Ω).The aim of this paper is to investigate inverse problem (2) for multidimensional elliptic equation with Dirichlet-Neumann boundary conditions. We obtain well-posedness of problem (2). For the approximate solution of problem (2), we construct first and second order of accuracy in t and difference schemes with second order of accuracy in space variables. Stability and coercive stability estimates for these difference schemes are established by applying operator approach. The modified Gauss elimination method is applied for solving these difference schemes in a two-dimensional case.The remainder of this paper is organized as follows. In Section2, we obtain stability and coercive stability estimates for problem (2). In Section 3, we construct the difference schemes for (2) and establish their well-posedness. In Section 4, the numerical results in a two-dimensional case are presented. Section 5 is conclusion.


## 2. Well-Posedness of Inverse Problem with Overdetermination

It is known that the differential expression [26]
(3)Axu(x)=-∑r=1n‍(ar(x)uxr)xr+δu(x)
defines a self-adjoint positive definite operator Ax acting on L2(Ω¯) with the domain D(Ax)={u(x)∈W22(Ω¯),∂u/∂n→=0onS}.LetH be the Hilbert space L2(Ω¯). By using abstract Theorems 2.1 and 2.2 of paper [11], we get the following theorems about well-posedness of problem (2).Theorem 1.
Assume thatAx is defined by formula (3), φ,ξ,ψ∈D(Ax). Then, for the solutions (u,p) of inverse boundary value problem (2), the stability estimates are satisfied:
(4)∥u∥C(L2(Ω¯))≤[M∥φ∥L2(Ω¯)+∥ψ∥L2(Ω¯)+∥ξ∥L2(Ω¯)+∥f∥C(L2(Ω¯))],∥(Ax)-1p∥L2(Ω¯)≤M[∥φ∥L2(Ω¯)mmmmmmmmmmm+∥ξ∥L2(Ω¯)+∥f∥C(L2(Ω¯))],∥p∥L2(Ω¯)≤M[∥Axφ∥L2(Ω¯)+∥Axψ∥L2(Ω¯)mmmmmmm+∥Axξ∥L2(Ω¯)+1α(1-α)∥f∥𝒞0Tα,α(L2(Ω¯))],
where M is independent of α,φ(x),ξ(x),ψ(x), and f(t,x).
Here,𝒞0Tα,α(L2(Ω¯)) is the space obtained by completion of the space of all smooth L2(Ω¯)-valued functions ρ on [0,T] with the norm
(5)∥ρ∥𝒞0Tα,α(L2(Ω¯))=∥ρ∥𝒞(L2(Ω¯))+sup0≤t<t+τ≤T(t+τ)α(T-t)α∥ρ(t+τ)-ρ(t)∥L2(Ω¯)τα.Theorem 2.
Assume thatAx is defined by formula (3), φ,ψ,ξ∈D(Ax). Then,  for the solution of inverse boundary value problem (2), coercive stability estimate
(6)∥u′′∥𝒞0Tα,α(L2(Ω¯))+∥u∥𝒞0Tα,α(W22(Ω¯))+∥p∥L2(Ω¯)≤M[1α(1-α)∥f∥𝒞0Tα,α(L2(Ω¯))+∥φ∥W22(Ω¯)mmmm,,+∥ψ∥W22(Ω¯)+∥ξ∥W22(Ω¯)1α(1-α)]
holds, where M is independent of α,φ(x),ξ(x),ψ(x), and f(t,x).


## 3. Difference Schemes and Their Well-Posedness

Suppose thatAx is defined by formula (3). Then (see [26]), C=(1/2)(τAx+4Ax+τ2(Ax)2) is a self-adjoint positive definite operator and R=(I+τC)-1 which is defined on the whole space H=L2(Ω¯) is a bounded operator. Here, I is the identity operator.Now we present the following lemmas, which will be used later.Lemma 3.
The following estimates are satisfied (see [27]):
(7)∥Rk∥H→H≤M(1+δ1/2τ)-k,δ>0,∥CRk∥H→H≤Mkτ,k≥1,∥(I-R2N)-1∥H→H≤M.Lemma 4.
For1≤l≤N-1 and for the operator S=R2N+Rl-R2N-l+RN-l-RN+l, the operator I-S has an inverse G=(I-S)-1 and the estimate
(8)∥G∥H→H≤M
is satisfied, where M does not depend on τ.
Proof of Lemma4 is based on Lemma 3 and representation
(9)Q=I-R2N-Rl+R2N-l-RN-l+RN+l=(I-RN-l)(I-RN)(I-Rl).Lemma 5.
For1≤l≤N-1 and for the operator
(10)S1=R2N-(λτ-l-1)(Rl-R2N-l+RN-l-RN+l)+(λτ-l)(Rl+1-R2N-l-1+RN-l-1-RN+l+1),
the operator I-S1 has an inverse
(11)G1=(I-R2N+(λτ-l-1)×(Rl-R2N-l+RN-l-RN+l)-(λτ-l)(Rl+1-R2N-l-1+RN-l-1-RN+l+1))-1,
and the estimate
(12)∥G1∥H→H≤M
is valid, where M is independent of τ.Proof.
We have that(13)G1-G=G1GK,
where
(14)K=-(λτ-l)(Rl-R2N-l+RN-l-RN+l)+(λτ-l)(Rl+1-R2N-l-1+RN-l-1-RN+l+1).
By using estimates of Lemma3, we have that
(15)∥K∥H→H=∥-(λτ-l)(Rl-R2N-l+RN-l-RN+l)+(λτ-l)(Rl+1-R2N-l-1+RN-l-1mmmmmmm,m,-RN+l+1)(λτ-l)∥H→H≤M1τ,
where M1 is independent of τ. Using the triangle inequality, formula (13), and estimates (8) and (15), we obtain
(16)∥G1∥H→H=∥G∥H→H+∥G1∥H→H∥G∥H→H≤M+∥G1∥H→HMM1τ
for sufficiently small positive τ. From that it follows estimate (11). Lemma 5 is proved.Further, we discretize problem (2) in two steps. In the first step, we define the grid spaces
(17)Ω~h={x=xm=(h1m1,…,hnmn);m=(m1,…,mn),mmmr=0,…,Mr,hrMr=ℓ,r=1,…,n},Ωh=Ω~h∩Ω,Sh=Ω~h∩S.Introduce the Hilbert spaceL2h=L2(Ω~h) and W2h2=W22(Ω~h) of grid functions ρh(x)={ρ(h1m1,…,hnmn)}, defined on Ω~h, equipped with the norms
(18)∥ρh∥L2h=(∑x∈Ω~h‍|ρh(x)|2h1,…,hn)1/2,∥ρh∥W2h2=∥ρh∥L2h+(∑x∈Ω~h‍∑r=1n‍|(ρh)xr|2h1,…,hn)1/2+(∑x∈Ω~h‍∑r=1n‍|(ρh(x))xrxr¯,mr|2h1,…,hn)1/2,
respectively.To the differential operatorAx generated by problem (2) we assign the difference operator Ahx defined by formula (3), acting in the space of grid functions uh(x), satisfying the condition Dhuh(x)=0 for all x∈Sh. Here, Dhuh(x) is an approximation of ∂u/∂n→.By usingAhx, for obtaining uh(t,x) functions, we arrive at problem
(19)-d2uh(t,x)dt2+Ahxuh(t,x)=fh(t,x)+ph(x),mmmmmmmmmmmmmm,0<t<T,x∈Ωh,uh(0,x)=φh(x),uh(λ,x)=ξh(x),uh(T,x)=ψh(x),mmmmmmmmmmmmmmmmmmmmmmm,mmmx∈Ω~h.
For finding a solution uh(t,x) of problem (19) we apply the substitution
(20)uh(t,x)=vh(t,x)+(Ahx)-1ph(x),
where vh(t,x) is the solution of nonlocal boundary value problem; a system of ordinary differential equations
(21)-d2vh(t,x)dt2+Ahxvh(t,x)=fh(t,x),mmmmmm,m,,,0<t<T,x∈Ωh,vh(0,x)-vh(λ,x)=φh(x)-ξh(x),x∈Ω~h,vh(T,x)-vh(λ,x)=ψh(x)-ξh(x),x∈Ω~h
and unknown function ph(x) is defined by formula
(22)ph(x)=Ahxφh(x)-Ahxvh(0,x),x∈Ω~h.Thus, we consider the algorithm for solving problem (19) which includes three stages. In the first stage, we get the nonlocal boundary value problem (21) and obtain vh(t,x). In the second stage, we put t=0 and find vh(0,x). Then, using (22), we obtain ph(x). Finally, in the third stage, we use formula (20) for obtaining the solution uh(t,x) of problem (19).In the second step, we approximate (19) in variable t. Let [0,T]τ={tk=kτ,k=1,…,N,Nτ=T} be the uniform grid space with step size τ>0, where N is a fixed positive integer.  Applying the approximate formulas
(23)uh(λ,x)=uh([λτ]τ,x)+o(τ),x∈Ωh,uh(λ,x)=uh([λτ]τ,x)+(λτ-[λτ])mmmmm,×(uh([λτ]τ+τ,x)-uh([λτ]τ,x))mmmmm,+o(τ2),x∈Ωh
for uh(λ,x)=ξh(x), problem (19) is replaced by first order of accuracy difference scheme
(24)-uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)=fkh(x)+ph(x),fkh(x)=fh(tk,x),tk=kτ,m1≤k≤N-1,x∈Ωh,u0h(x)=φh(x),x∈Ω~h,ulh(x)=ξh(x),x∈Ω~h,uNh(x)=ψh(x),x∈Ω~h,l=[λτ]
and second order of accuracy difference scheme
(25)-uk+1h(x)-2ukh(x)+uk-1h(x)τ2+Ahxukh(x)=fkh(x)+ph(x),fkh(x)=fh(tk,x),tk=kτ,mmm1≤k≤N-1,x∈Ωh,u0h(x)=φh(x),x∈Ω~h,ulh(x)+(λτ-l)(ul+1h(x)-ulh(x))=ξh(x),mmmmmmmmmmmmmmmmmmmx∈Ω~h,uNh(x)=ψh(x),x∈Ω~h,l=[λτ].
For approximate solution of nonlocal problem (21), we have first order of accuracy difference scheme
(26)-vk+1h(x)-2vkh(x)+vk-1h(x)τ2+Ahxvkh(x)=fkh(x),fkh(x)=fh(tk,x),tk=kτ,,,m1≤k≤N-1,x∈Ωh,v0h(x)-vlh(x)=φh(x)-ξh(x),x∈Ω~h,vNh(x)-vlh(x)=ψh(x)-ξh(x),x∈Ω~h
and second order of accuracy difference scheme
(27)-vk+1h(x)-2vkh(x)+vk-1h(x)τ2+Ahxvkh(x)=fkh(x),fkh(x)=fh(tk,x),tk=kτ,1≤k≤N,x∈Ωh,v0h(x)-(λτ-l)vl+1h(x)+(λτ-l-1)vlh(x)=φh(x)-ξh(x),vNh(x)-(λτ-l)vl+1h(x)+(λτ-l-1)vlh(x)=ψh(x)-ξh(x),x∈Ω~h,
respectively.Theorem 6.
Letτ and |h|=h12+⋯+hn2 be sufficiently small positive numbers. Then, for the solutions ({ukh}K-1N-1,ph) of difference schemes (24) and (25) the stability estimates
(28)∥{ukh}1N-1∥𝒞τ(L2h)≤M[∥{fkh}1N-1∥𝒞τ(L2h)∥φh∥L2h+∥ψh∥L2h+∥ξh∥L2hmm,m.+∥{fkh}1N-1∥𝒞τ(L2h)],∥ph∥L2h≤M[1α(1-α)∥φh∥W2h2+∥ψh∥W2h2+∥ξh∥W2h2mmm,,+1α(1-α)∥{fkh}1N-1∥𝒞τ(L2h)]
hold, where M is independent of τ,α,h,φh(x),ψh(x),  ξh(x), and {fkh(x)}1N-1.Theorem 7.
Letτ and |h|=h12+⋯+hn2 be sufficiently small positive numbers. Then, for the solutions of difference schemes (24) and (25) the following almost coercive stability estimate
(29)∥{uk+1h-2ukh+uk-1kτ2)}1N-1∥𝒞τ(L2h)+∥{ukh}1N-1∥𝒞τ(W2h2)+∥ph∥L2h≤M[1α(1-α)∥{fkh}1N-1∥𝒞τ(L2h)∥φh∥W2h2+∥ψh∥W2h2+∥ξh∥W2h2mmm,m+ln(1τ+h)∥{fkh}1N-1∥𝒞τ(L2h)]
holds, where M is independent of τ,α,h,φh(x),ψh(x),  ξh(x), and {fkh(x)}1N-1.Proofs of Theorems6 and 7 are based on the symmetry property of operator Ax, on Lemmas 3–5, the formulas
(30)ukh(x)=(I-R2N)-1×[∑i=1N-1‍((Rk-R2N-k)v0h(x)mmmm+(RN-k-RN+k)vNh(x))mmm-(RN-k-RN+k)(I+τC)(2I+τC)-1×C-1∑i=1N-1‍(RN-i-RN+i)fih(x)τ]+(I+τC)(2I+τC)-1C-1×∑i=1N-1‍(R|k-i|-Rk+i)∑i=1N-1‍×fih(x)τ+φh(x)-v0h(x),ph(x)=Ahxφh(x)-Ahxv0h(x),vNh(x)=v0h(x)+ψh(x)-φh(x),v0h(x)=-G(RN-l-RN+l)×(I+τC)(2I+τC)-1C-1×∑i=1N-1‍(RN-i-RN+i)fih(x)τ+G(I-R2N)(I+τC)×(2I+τC)-1C-1∑i=1N-1‍(R|l-i|-Rl+i)fih(x)τ+G(I-R2N)(φh(x)-ξh(x))+G(RN-l-RN+l)(ψh(x)-φh(x)),
for difference scheme (24),
(31)v0h(x)=(λτ-l-1)G1(RN-l-RN+l)×(I+τC)(2I+τC)-1C-1×∑i=1N-1‍(RN-i-RN+i)fih(x)τ-(λτ-l-1)G1(I-R2N)×(I+τC)(2I+τC)-1C-1×∑i=1N-1‍(R|l-i|-Rl+i)fih(x)τ-(λτ-l)G1(RN-l-1-RN+l+1)×(I+τC)(2I+τC)-1C-1×∑i=1N-1‍(RN-i-RN+i)fih(x)τ+(λτ-l)G1(I-R2N)×(I+τC)(2I+τC)-1C-1×∑i=1N-1‍(R|l+1-i|-Rl+1+i)fih(x)τ+G1(I-R2N)(φh(x)-ξh(x))+((λτ-l-1)G1(RN-l-RN+l)mmm+(λτ-l)G1(RN-l-1-RN+l+1))m×(ψh(x)-φh(x)),
for difference scheme (25), and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h.Theorem 8 (see [28]).
For the solution of the elliptic difference problem(32)Ahxuh(x)=ωh(x),x∈Ω~h,Dhuh(x)=0,x∈Sh,
the following coercivity inequality holds:
(33)∑r=1n‍∥(ukh)x¯rx¯r,jr∥L2h≤M∥ωh∥L2h,
where M does not depend on h and ωh.


## 4. Numerical Results

We have not been able to obtain a sharp estimate for the constants figuring in the stability estimates. Therefore, we will give the following results of numerical experiments of the inverse problem for the two-dimensional elliptic equation with Dirichlet-Neumann boundary conditions(34)-∂2u(t,x)∂t2-∂∂x((2+cosx)∂u(t,x)∂x)+u(t,x)=f(t,x)+p(x),0<x<π,0<t<T,f(t,x)=-exp(-t)cos(x)mmmm,,+(exp(-t)+t)(3cos(x)+cos(2x)),u(0,x)=2cos(x),0≤x≤π,u(T,x)=(exp(-T)+T+1)cos(x),0≤x≤π,u(λ,x)=(exp(-λ)+λ+1)cos(x),0≤x≤π,ux(t,0)=ux(t,π)=0,0≤t≤T,λ=3T5.
It is clear that u(t,x)=(exp(-t)+t+1)cos(x) and p(x)=sin(x)+(x+2)cos(x) are the exact solutions of (34).We can obtainu(t,x) by formula u(t,x)=v(t,x)+w(t,x), where v(t,x) is the solution of the nonlocal boundary value problem
(35)-d2v(t,x)dt2-∂∂x((2+cosx)∂v(t,x)∂x)+v(t,x)=f(t,x),0<x<π,0<t<T,v(0,x)-v(λ,x)=(1-exp(-λ)-λ)cos(x),mmmmmmmmmmmmm,,,mmm0≤x≤πv(T,x)-v(λ,x)=(exp(-T)-exp(-λ)+T-λ)cos(x),mmmmmmmmmmmmmmmm0≤x≤π,vx(t,0)=vx(t,π)=0,0≤t≤T,
and w(t,x) is the solution of the boundary value problem
(36)-d2w(t,x)dt2-∂∂x((2+cosx)∂w(t,x)∂x)+w(t,x)=p(x),0<x<π,0<t<T,w(0,x)=(exp(-λ)+λ+1)cos(x)-v(λ,x),mmmmmmmmmmmmmmmmmm0≤x≤π,w(T,x)=(exp(-λ)+λ+1)cos(x)-v(λ,x),mmmmmmmmmmmmmmmmmm0≤x≤π,wx(t,0)=wx(t,π)=0,0≤t≤T.Introduce small parametersτ and h such that Nτ=T,Mh=π. For approximate solution of nonlocal boundary value problem (35) we consider the set [0,T]τ×[0,π]h of a family of grid points
(37)[0,T]τ×[0,π]h={(tk,xn):tk=kτ,mmmk=0,…,N,xn=nh,n=0,…,M}.Applying (21), we obtain difference schemes of the first order of accuracy in t and the second order of accuracy in x(38)vnk+1-2vnk+vnk-1τ2+(2+cos(xn))vn+1k-2vnk+vn-1kh2-sin(xn)vn+1k-vn-1k2h-vnk=θnk,θnk=-f(tk,xn),k=1,…,N-1,n=1,…,M-1,v0k-v1k=vMk-vM-1k=0,k=0,…,N,vn0-vnl=(1-exp(-λ)-λ)cos(xn),n=0,…,M,vnN-vnl=(exp(-tN)-exp(-λ)+tN-λ)cos(xn),mmmmmmmmmmmmmn=0,…,M,l=[λτ],
for the approximate solutions of the nonlocal boundary value problem (35), and
(39)wnk+1-2wnk+wnk-1τ2+(2+cos(xn))wn+1k-2wnk+wn-1kh2-sin(xn)wn+1k-wn-1k2h-wnk=-pn,k=1,…,N-1,pn=p(xn),n=1,…,M-1,w0k-w1k=wMk-wM-1k=0,k=0,…,N,wn0=(exp(-λ)+λ+1)cos(xn)-vnl,mmmmmmmmn=0,…,M,l=[λτ],wnN=(exp(-λ)+λ+1)cos(xn)-vnl,mmmmmmm,mmmmmmn=0,…,M,
for the approximate solutions of the boundary value problem (36).By using (22) and second order of accuracy in x approximation of A, we get the following values of p  in grid points:
(40)pn=-(2+cos(xn))h2((φn+1-vn+10)-2(φn-vn0)+(φn-1-vn-10))+sin(xn)2h×((φn+1-vn+10)-(φn-1-vn-10))+(φn-vn0),n=1,…,M-1.We can rewrite difference scheme (38) in the matrix form
(41)Anvn+1+Bnvn+Cnvn-1=Iθnk,n=1,…,M-1,v0=v1,vM=vM-1.
Here, I is the (N+1)×(N+1) identity matrix, An,  Bn,  Cn are (N+1)×(N+1) square matrices, and θn is a (N+1)×1 column matrix which are defined by
(42)An=[0000⋯00000an00⋯000000an0⋯0000000an⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯an0000000⋯0an000000⋯00an00000⋯0000],Bn=[1000⋯-1⋯0000dbnd0⋯0⋯00000dbnd⋯0⋯000000dbn⋯0⋯0000⋮⋮⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯0⋯bnd000000⋯0⋯dbnd00000⋯0⋯0dbnd0000⋯-1⋯0001],(43)Cn=[0000⋯00000cn00⋯000000cn0⋯0000000cn⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯cn0000000⋯0cn000000⋯00cn00000⋯0000],(44)an=2+cos(xn)h2-sin(xn)2h,bn=-2τ2-2(2+cos(xn))h2-1,cn=2+cos(xn)h2+sin(xn)2h,d=1τ2,(45)θn=[θn0⋮θnN],θn0=(1-exp(-λ)-λ)cos(xn),θnN=(exp(-tN)-exp(-λ)+tN-λ)cos(xn),θnNmmm,,,,,mmmmmmm=n=1,…,M-1,θnk=-f(tk,xn),k=1,…,N-1,n=1,…,M-1,vs=[vs0⋮vsN](N+1)×1,s=n-1,n,n+1.For solving (41) we use the modified Gauss elimination method (see [29]). Namely, we seek solution of (41) by the formula
(46)vn=αn+1vn+1+βn+1,n=M-1,…,1,
where vM=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 αn+1,βn+1, we get formulas
(47)αn+1=-(Bn+Cnαn)-1An,βn+1=-(Bn+Cnαn)-1(Iθn-Cnβn),n=1,…,M-1,
where α1 is the (N+1)×(N+1) identity matrix and β1 is the (N+1)×1 zero column vector.Futher, we rewrite difference scheme (39) in the matrix form
(48)Anwn+1+Enwn+Cnwn-1=Iηnk,mmmmmmmn=1,…,M-1,w0=w1,wM=wM-1.Here,(49)En=[1000⋯0000dbnd0⋯00000dbnd⋯000000dbn⋯0000⋮⋮⋮⋮⋱⋮⋮⋮⋮0000⋯bnd000000⋯dbnd00000⋯0dbnd0000⋯0001],An and Cn are defined by (42) and (43) and (N+1)×1 column matrix ηn is defined by
(50)ηn=[ηn0⋮ηnN],ηn0=(exp(-λ)+λ+1)cos(xn)-vnl,ηnN=(exp(-λ)+λ+1)cos(xn)-vnl,n=1,…,M-1,ηnk=-pn,k=1,…,N-1,n=1,…,M-1,ws=[ws0⋮wsN](N+1)×1,s=n-1,n,n+1.Now we present second order of accuracy int and x difference schemes for problems (35) and (36). Applying (27) and formulas for sufficiently smooth function ρ(51)ρ(xn+1)-ρ(xn-1)2h-ρ′(xn)=O(h2),ρ(xn+1)-2ρ(xn)+ρ(xn-1)h2-ρ′′(xn)=O(h2),10ρ(0)-15ρ(h)+6ρ(2h)-ρ(3h)h3-ρ′′′(0)=O(h2),-3ρ(0)+4ρ(h)-ρ(2h)2h-ρ′(0)=O(h2),10ρ(π)-15ρ(π-h)+6ρ(π-2h)-ρ(π-3h)h3-ρ′′′(π)=O(h2),-3ρ(π)+4ρ(π-h)-ρ(π-2h)2h-ρ′(π)=O(h2),
we get
(52)vnk+1-2vnk+vnk-1τ2+(2+cos(xn))vn+1k-2vnk+vn-1kh2-sin(xn)vn+1k-vn-1k2h-vnk=θnk,θnk=-f(tk,xn),k=1,…,N-1,n=1,…,M-1,-3v0k+4v1k-v2k=-3vMk+4vM-1k-vM-2k=0,k=0,…,N,10v0k-15v1k+6v2k-v3k=10vMk-15vM-1k+6vM-2k-vM-3k=0,vn0+(λτ-l-1)vnl-(λτ-l)vnl+1=(1-exp(-λ)-λ)cos(xn),n=0,…,M,vnN+(λτ-l-1)vnl-(λτ-l)vnl+1=(exp(-tN)-exp(-λ)+tN-λ)cos(xn),mmmmmmmmmmmmmmm,,mn=0,…,M,
difference scheme for nonlocal problem (35), and
(53)wnk+1-2wnk+wnk-1τ2+(2+cos(xn))wn+1k-2wnk+wn-1kh2-sin(xn)wn+1k-wn-1k2h=-pn,k=1,…,N-1,pn=p(xn),n=1,…,M-1,-3w0k+4w1k-w2k=-3wMk+4wM-1k-wM-2k=0,mmmmmmmmmmmmmmmmmmmk=0,…,N,10w0k-15w1k+6w2k-w3k=10wMk-15wM-1k+6wM-2k-wM-3k=0,wn0=(exp(-λ)+λ+1)cos(xn)+(λτ-l-1)vnl-(λτ-l)vnl+1,n=0,…,M,wnN=(exp(-λ)+λ+1)cos(xn)+(λτ-l-1)vnl-(λτ-l)vnl+1,ξn=ξ(xn),n=0,…,M,
difference scheme for boundary value problem (36).By difference scheme (52), we write in matrix form
(54)Anvn+1+Bnvn+Cnvn-1=Iθnk,n=1,…,M-1,-3v0+4v1-v2=0,-3vM+4vM-1-vM-2=0,
where An,Cn are defined by (42), (43), (44), and Bn is defined by
(55)Bn=[1000⋯0yz0⋯0000dbnd0⋯0000⋯00000dbnd⋯0000⋯000000dbn⋯0000⋯0000⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮0000⋯0000⋯bnd000000⋯0000⋯dbnd00000⋯0000⋯0dbnd0000⋯0yz0⋯0001],bn=-2τ2-2(2+cos(xn))h2-1,d=1τ2,y=(λτ-l-1),z=-(λτ-l).We seek solution of (54) by the formula
(56)vn=αnvn+1+βnvn+2+γn,n=M-2,…,0,
where αn,βn  (n=0,…,M-2) are (N+1)×(N+1) square matrices and γn(n=0,…,M-2) are (N+1)×1 column matrices. For the solution of difference equation (41) we need to use the following formulas for αn,βn:
(57)αn=-(Bn+Cnαn-1)-1(An+Cnβn-1),βn=0,γn=-(Bn+Cnαn-1)-1(Iθn-Cnγn-1),n=1,…,M-1,
where
(58)α0=43I,β0=-13I,α1=85I,β1=-35I,αM-2=4I,βM-2=-3I,αM-3=83I,βM-3=-53I,
and γ0,γ1,γM-2,γM-3 are the (N+1)×1 zero column vector. For vM and vM-1 we have
(59)vM=(Q11-Q12Q22-1Q21)-1(G1-Q12Q22-1G2),vM-1=Q22-1(G2-Q21vM),
where
(60)Q11=-3AM-2-8BM-2-8CM-2αM-3-3CM-2βM-3,Q12=4AM-2+9BM-2+9CM-2αM-3+4CM-2βM-3,Q21=-3BM-1-8CM-1,Q22=AM-1+4BM-1+9CM-1,G1=IθM-2-CM-2γM-3,G2=IθM-1.We can rewrite difference scheme (53) in the matrix form
(61)Anwn+1+Enwn+Cnwn-1=Iηnk,n=1,…,M-1,-3w0+4w1-w2=0,-3wM+4wM-1-wM-2=0,
where An,En,Cn are defined by (42), (49), (43), and (44) and ηn is defined by
(62)ηn=[ηn0⋮ηnN],ηn0=(exp(-λ)+λ+1)cos(xn)+(λτ-l-1)vnl-(λτ-l)vnl+1,ηnN=(exp(-λ)+λ+1)cos(xn)+(λτ-l-1)vnl-(λτ-l)vnl+1,n=0,…,M,ηnk=-pn,k=1,…,N-1,n=1,…,M-1.Now, we give the results of the numerical realization of finite difference method for (34) by using MATLAB programs. The numerical solutions are recorded for T=2 and different values of N=M. Grid functions vnk, unk represent the numerical solutions of difference schemes for auxiliary nonlocal problem (35) and inverse problem (34) at (tk,xn), respectively. Grid function pn calculated by (40) represents numerical solution at xn for unknown function p. The errors are computed by the norms
(63)EvMN=max1≤k≤N-1(∑n=1M-1‍|v(tk,xn)-vnk|2h)1/2,EuMN=max1≤k≤N-1(∑n=1M-1‍|u(tk,xn)-unk|2h)1/2,EpM=(∑n=1M-1‍|p(xn)-pn|2h)1/2.Tables1–3 present the error between the exact solution and numerical solutions derived by corresponding difference schemes. The results are recorded for N=M=20,40,80 and 160, respectively. The tables show that the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme for both auxiliary nonlocal and inverse problems. Table 1 contains error between the exact and approximate solutions v of auxiliary nonlocal boundary value problem (35). Table 2 includes error between the exact and approximate solutions p of inverse problem (34). Table 3 represents error between the exact solution u of inverse problem (34) and approximate solution which is derived by the first and second orders accuracy of difference schemes.Table 1
Error analysis for nonlocal problem.
N
=
M
=
20
N
=
M
=
40
N
=
M
=
80
N
=
M
=
160
Difference  scheme (38)
0.30522
0.14933
0.073953
0.036814
Difference  scheme (52)
0.024714
0.0031054
4.36
×
10
-
4
7.52
×
10
-
5Table 2
Error analysis forp.
N
=
M
=
20
N
=
M
=
40
N
=
M
=
80
N
=
M
=
160
Difference  scheme (38), (40)
0.57878
0.33755
0.20387
0.12905
Difference  scheme (52), (40)
0.058201
0.010646
0.0020228
4.03
×
10
-
4Table 3
Error analysis foru.
N
=
M
=
20
N
=
M
=
40
N
=
M
=
80
N
=
M
=
160
Difference  scheme (38), (40), (39)
0.088225
0.038586
0.01815
0.008818
Difference  scheme (52), (40), (53)
0.017034
0.0020225
2.47
×
10
-
4
3.08
×
10
-
5


## 5. Conclusion

In this paper, inverse problem for multidimensional elliptic equation with Dirichlet-Neumann conditions is considered. The stability and coercive stability estimates for solution of this problem are established. First and second order of accuracy difference schemes are presented for approximate solutions of inverse problem. Theorems on the stability and coercive stability inequalities for difference schemes are proved. The theoretical statements for the solution of these difference schemes are supported by the results of numerical example in a two-dimensional case. As it can be seen from Tables1–3, second order of accuracy difference scheme is more accurate compared with the first order of accuracy difference scheme. Moreover, applying the result of the monograph [29] the high order of accuracy difference schemes for the numerical solution of the boundary value problem (2) can be presented.



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*Source: 548017-2013-10-08.xml*