Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination

Abstract

A …finite 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 fi…rst 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.

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: −𝑢_𝑡𝑡(𝑡)(𝑡)(𝑡)𝑢(0)(𝑇)(𝜆)+𝐴𝑢=𝑓+𝑝,0<𝑡<𝑇,=𝜑,𝑢=𝜓,𝑢=𝜉,0<𝜆<𝑇. (1)

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.

Let Ω=(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 −𝑢_𝑡𝑡(𝑡,𝑥)−∑^𝑛_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}()()(𝑥),𝑥+𝛿𝑢𝑡,𝑥=𝑓𝑡,𝑥+𝑝𝑥=₁,…,𝑥_𝑛𝑢()(𝑥)()(𝑥)()(𝑥),∈Ω,0<𝑡<𝑇,0,𝑥=𝜑,𝑢𝑇,𝑥=𝜓,𝑢𝜆,𝑥=𝜉𝑥∈()Ω,𝜕𝑢𝑡,𝑥𝜕𝑛=0,𝑥∈𝑆,0≤𝑡≤𝑇. (2)

Here, 0<𝜆<𝑇 and 𝛿>0 are known numbers, 𝑎_𝑟(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 𝑎_𝑟  (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 Section 2, 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] 𝐴^𝑥𝑢(𝑥)=−^𝑛_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}+𝛿𝑢(𝑥) (3) defines a self-adjoint positive definite operator 𝐴^𝑥 acting on 𝐿₂(Ω) with the domain 𝐷(𝐴^𝑥)={𝑢(𝑥)∈𝑊²₂(Ω),𝜕𝑢/𝜕𝑛=0on𝑆}.

Let 𝐻 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 that 𝐴^𝑥 is defined by formula (3), 𝜑,𝜉,𝜓∈𝐷(𝐴^𝑥). Then, for the solutions (𝑢,𝑝) of inverse boundary value problem (2), the stability estimates are satisfied: ‖𝑢‖_{𝐶(𝐿2(Ω))}≤𝑀𝜑_𝐿2(Ω)+𝜓_𝐿2(Ω)+𝜉_𝐿2(Ω)+𝑓_{𝐶(𝐿2(Ω))},𝐴^𝑥−1𝑝_𝐿2(Ω)≤𝑀𝜑_𝐿2(Ω)+𝜉_𝐿2(Ω)+𝑓_{𝐶(𝐿2(Ω))},𝑝_𝐿2(Ω)≤𝑀𝐴^𝑥𝜑_𝐿2(Ω)+𝐴^𝑥𝜓_𝐿2(Ω)+𝐴^𝑥𝜉_𝐿2(Ω)+1𝛼()𝑓1−𝛼^{𝒞𝛼,𝛼0𝑇(𝐿2(Ω))}, (4) where 𝑀 is independent of 𝛼,𝜑(𝑥),𝜉(𝑥),𝜓(𝑥), and 𝑓(𝑡,𝑥).
Here, 𝒞^𝛼,𝛼_0𝑇(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 𝜌_𝒞^𝛼,𝛼_{0𝑇(𝐿2(Ω))}=𝜌_{𝒞(𝐿2(Ω))}+sup_{0≤𝑡<𝑡+𝜏≤𝑇}(𝑡+𝜏)^𝛼(𝑇−𝑡)^𝛼𝜌(𝑡+𝜏)−𝜌(𝑡)_𝐿2(Ω)𝜏^𝛼. (5)

Theorem 2. Assume that 𝐴^𝑥 is defined by formula (3), 𝜑,𝜓,𝜉∈𝐷(𝐴^𝑥). Then, for the solution of inverse boundary value problem (2), coercive stability estimate 𝑢^′′_{𝒞𝛼,𝛼0𝑇(𝐿2(Ω))}+‖𝑢‖_𝒞^𝛼,𝛼_0𝑇(𝑊²_2(Ω))+𝑝_𝐿2(Ω)1≤𝑀𝛼()𝑓1−𝛼^{𝒞𝛼,𝛼0𝑇(𝐿2(Ω))}+𝜑_𝑊22(Ω)+𝜓_𝑊22(Ω)+𝜉_𝑊22(Ω) (6) 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 that 𝐴^𝑥 is defined by formula (3). Then (see [26]), 𝐶=(1/2)(𝜏𝐴^𝑥+4𝐴^𝑥+𝜏²(𝐴^𝑥)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]): 𝑅^𝑘_𝐻→𝐻≤𝑀1+𝛿^1/2𝜏^−𝑘,𝛿>0,𝐶𝑅^𝑘_𝐻→𝐻≤𝑀𝑘𝜏,𝑘≥1,𝐼−𝑅^2𝑁−1_𝐻→𝐻≤𝑀. (7)

Lemma 4. For 1≤𝑙≤𝑁−1 and for the operator 𝑆=𝑅^2𝑁+𝑅^𝑙−𝑅^2𝑁−𝑙+𝑅^𝑁−𝑙−𝑅^𝑁+𝑙, the operator 𝐼−𝑆 has an inverse 𝐺=(𝐼−𝑆)⁻¹ and the estimate ‖𝐺‖_𝐻→𝐻≤𝑀 (8) is satisfied, where 𝑀 does not depend on 𝜏.
Proof of Lemma 4 is based on Lemma 3 and representation 𝑄=𝐼−𝑅^2𝑁−𝑅^𝑙+𝑅^2𝑁−𝑙−𝑅^𝑁−𝑙+𝑅^𝑁+𝑙=𝐼−𝑅^𝑁−𝑙𝐼−𝑅^𝑁𝐼−𝑅^𝑙. (9)

Lemma 5. For 1≤𝑙≤𝑁−1 and for the operator 𝑆₁=𝑅^2𝑁−𝜆𝜏−𝑙−1𝑅^𝑙−𝑅^2𝑁−𝑙+𝑅^𝑁−𝑙−𝑅^𝑁+𝑙+𝜆𝜏−𝑙𝑅^𝑙+1−𝑅^{2𝑁−𝑙−1}+𝑅^{𝑁−𝑙−1}−𝑅^𝑁+𝑙+1, (10) the operator 𝐼−𝑆₁ has an inverse 𝐺₁=𝐼−𝑅^2𝑁+𝜆𝜏×𝑅−𝑙−1_𝑙−𝑅^2𝑁−𝑙+𝑅^𝑁−𝑙−𝑅^𝑁+𝑙−𝜆𝜏−𝑙𝑅^𝑙+1−𝑅^{2𝑁−𝑙−1}+𝑅^{𝑁−𝑙−1}−𝑅^{𝑁+𝑙+1−1}, (11) and the estimate 𝐺_1𝐻→𝐻≤𝑀 (12) is valid, where 𝑀 is independent of 𝜏.

Proof. We have that 𝐺₁−𝐺=𝐺₁𝐺𝐾, (13) where 𝜆𝐾=−𝜏−𝑙𝑅^𝑙−𝑅^2𝑁−𝑙+𝑅^𝑁−𝑙−𝑅^𝑁+𝑙+𝜆𝜏−𝑙𝑅^𝑙+1−𝑅^{2𝑁−𝑙−1}+𝑅^{𝑁−𝑙−1}−𝑅^𝑁+𝑙+1. (14)
By using estimates of Lemma 3, we have that ‖𝐾‖_𝐻→𝐻=−𝜆𝜏−𝑙𝑅^𝑙−𝑅^2𝑁−𝑙+𝑅^𝑁−𝑙−𝑅^𝑁+𝑙+𝜆𝜏−𝑙𝑅^𝑙+1−𝑅^{2𝑁−𝑙−1}+𝑅^{𝑁−𝑙−1}−𝑅^𝑁+𝑙+1_𝐻→𝐻≤𝑀₁𝜏, (15) where 𝑀₁ is independent of 𝜏. Using the triangle inequality, formula (13), and estimates (8) and (15), we obtain 𝐺_1𝐻→𝐻=‖𝐺‖_𝐻→𝐻+𝐺_1𝐻→𝐻‖𝐺‖_𝐻→𝐻≤𝑀+𝐺_1𝐻→𝐻𝑀𝑀₁𝜏 (16) 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 Ω_ℎ=𝑥=𝑥_𝑚=ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛;𝑚=𝑚₁,…,𝑚_𝑛,𝑚_𝑟=0,…,𝑀_𝑟,ℎ_𝑟𝑀_𝑟,Ω=ℓ,𝑟=1,…,𝑛_ℎ=Ω_ℎ∩Ω,𝑆_ℎ=Ω_ℎ∩𝑆. (17)

Introduce the Hilbert space 𝐿_2ℎ=𝐿₂(Ω_ℎ) and 𝑊²_2ℎ=𝑊²₂(Ω_ℎ) of grid functions 𝜌^ℎ(𝑥)={𝜌(ℎ₁𝑚₁,…,ℎ_𝑛𝑚_𝑛)}, defined on Ω_ℎ, equipped with the norms 𝜌^ℎ_𝐿2ℎ=_𝑥∈Ω_ℎ𝜌^ℎ(𝑥)²ℎ₁,…,ℎ_𝑛^1/2,𝜌^ℎ_𝑊22ℎ=𝜌^ℎ_𝐿2ℎ+_𝑥∈Ω_ℎ^𝑛𝑟=1(𝜌^ℎ)_𝑥𝑟²ℎ₁,…,ℎ_𝑛^1/2+_𝑥∈Ω_ℎ^𝑛𝑟=1(𝜌^ℎ(𝑥))_{𝑥𝑟𝑥𝑟,𝑚𝑟}²ℎ₁,…,ℎ_𝑛^1/2, (18) respectively.

To the differential operator 𝐴^𝑥 generated by problem (2) we assign the difference operator 𝐴^𝑥_ℎ defined by formula (3), acting in the space of grid functions 𝑢^ℎ(𝑥), satisfying the condition 𝐷^ℎ𝑢^ℎ(𝑥)=0 for all 𝑥∈𝑆_ℎ. Here, 𝐷^ℎ𝑢^ℎ(𝑥) is an approximation of 𝜕𝑢/𝜕𝑛.

By using 𝐴^𝑥_ℎ, for obtaining 𝑢^ℎ(𝑡,𝑥) functions, we arrive at problem −𝑑²𝑢^ℎ(𝑡,𝑥)𝑑𝑡²+𝐴^𝑥_ℎ𝑢^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥)+𝑝^ℎ(𝑥),0<𝑡<𝑇,𝑥∈Ω_ℎ,𝑢^ℎ(0,𝑥)=𝜑^ℎ(𝑥),𝑢^ℎ(𝜆,𝑥)=𝜉^ℎ(𝑥),𝑢^ℎ(𝑇,𝑥)=𝜓^ℎ(𝑥),𝑥∈Ω_ℎ. (19) For finding a solution 𝑢^ℎ(𝑡,𝑥) of problem (19) we apply the substitution 𝑢^ℎ(𝑡,𝑥)=𝑣^ℎ(𝑡,𝑥)+𝐴^𝑥_ℎ⁻¹𝑝^ℎ(𝑥), (20) where 𝑣^ℎ(𝑡,𝑥) is the solution of nonlocal boundary value problem; a system of ordinary differential equations −𝑑²𝑣^ℎ(𝑡,𝑥)𝑑𝑡²+𝐴^𝑥_ℎ𝑣^ℎ(𝑡,𝑥)=𝑓^ℎ(𝑡,𝑥),0<𝑡<𝑇,𝑥∈Ω_ℎ,𝑣^ℎ(0,𝑥)−𝑣^ℎ(𝜆,𝑥)=𝜑^ℎ(𝑥)−𝜉^ℎ(𝑥),𝑥∈Ω_ℎ,𝑣^ℎ(𝑇,𝑥)−𝑣^ℎ(𝜆,𝑥)=𝜓^ℎ(𝑥)−𝜉^ℎ(𝑥),𝑥∈Ω_ℎ (21) and unknown function 𝑝^ℎ(𝑥) is defined by formula 𝑝^ℎ(𝑥)=𝐴^𝑥_ℎ𝜑^ℎ(𝑥)−𝐴^𝑥_ℎ𝑣^ℎ(0,𝑥),𝑥∈Ω_ℎ. (22)

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 𝑣^ℎ(𝑡,𝑥). In the second stage, we put 𝑡=0 and find 𝑣^ℎ(0,𝑥). Then, using (22), we obtain 𝑝^ℎ(𝑥). Finally, in the third stage, we use formula (20) for obtaining the solution 𝑢^ℎ(𝑡,𝑥) of problem (19).

In the second step, we approximate (19) in variable 𝑡. Let [0,𝑇]_𝜏={𝑡_𝑘=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 𝑢^ℎ(𝜆,𝑥)=𝑢^ℎ𝜆𝜏𝜏,𝑥+𝑜(𝜏),𝑥∈Ω_ℎ,𝑢^ℎ(𝜆,𝑥)=𝑢^ℎ𝜆𝜏+𝜆𝜏,𝑥𝜏−𝜆𝜏×𝑢^ℎ𝜆𝜏𝜏+𝜏,𝑥−𝑢^ℎ𝜆𝜏𝜏𝜏,𝑥+𝑜²,𝑥∈Ω_ℎ (23) for 𝑢^ℎ(𝜆,𝑥)=𝜉^ℎ(𝑥), problem (19) is replaced by first order of accuracy difference scheme −𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑓^ℎ_𝑘(𝑥)+𝑝^ℎ(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑥∈Ω_ℎ,𝑢^ℎ₀(𝑥)=𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ_𝑙(𝑥)=𝜉^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=𝜓^ℎ(𝑥),𝑥∈Ω_ℎ𝜆,𝑙=𝜏 (24) and second order of accuracy difference scheme −𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)=𝑓^ℎ_𝑘(𝑥)+𝑝^ℎ(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑥∈Ω_ℎ,𝑢^ℎ₀(𝑥)=𝜑^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ_𝑙(𝑥)+𝜆𝜏−𝑙𝑢^ℎ_𝑙+1(𝑥)−𝑢^ℎ_𝑙(𝑥)=𝜉^ℎ(𝑥),𝑥∈Ω_ℎ,𝑢^ℎ_𝑁(𝑥)=𝜓^ℎ(𝑥),𝑥∈Ω_ℎ𝜆,𝑙=𝜏. (25) For approximate solution of nonlocal problem (21), we have first order of accuracy difference scheme −𝑣^ℎ_𝑘+1(𝑥)−2𝑣^ℎ_𝑘(𝑥)+𝑣^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑣^ℎ_𝑘(𝑥)=𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑥,𝑡_𝑘=𝑘𝜏,1≤𝑘≤𝑁−1,𝑥∈Ω_ℎ,𝑣^ℎ₀(𝑥)−𝑣^ℎ_𝑙(𝑥)=𝜑^ℎ(𝑥)−𝜉^ℎ(𝑥),𝑥∈Ω_ℎ,𝑣^ℎ_𝑁(𝑥)−𝑣^ℎ_𝑙(𝑥)=𝜓^ℎ(𝑥)−𝜉^ℎ(𝑥),𝑥∈Ω_ℎ (26) and second order of accuracy difference scheme −𝑣^ℎ_𝑘+1(𝑥)−2𝑣^ℎ_𝑘(𝑥)+𝑣^ℎ_𝑘−1(𝑥)𝜏²+𝐴^𝑥_ℎ𝑣^ℎ_𝑘(𝑥)=𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=𝑓^ℎ𝑡_𝑘,𝑡,𝑥_𝑘=𝑘𝜏,1≤𝑘≤𝑁,𝑥∈Ω_ℎ,𝑣^ℎ₀(𝑥)−𝜆𝜏−𝑙𝑣^ℎ_𝑙+1(𝑥)+𝜆𝜏−𝑙−1𝑣^ℎ_𝑙(𝑥)=𝜑^ℎ(𝑥)−𝜉^ℎ(𝑥),𝑣^ℎ_𝑁(𝑥)−𝜆𝜏−𝑙𝑣^ℎ_𝑙+1(𝑥)+𝜆𝜏−𝑙−1𝑣^ℎ_𝑙(𝑥)=𝜓^ℎ(𝑥)−𝜉^ℎ(𝑥),𝑥∈Ω_ℎ, (27) respectively.

Theorem 6. Let 𝜏 and |ℎ|=ℎ²₁+⋯+ℎ²_𝑛 be sufficiently small positive numbers. Then, for the solutions ({𝑢^ℎ_𝑘}^𝑁−1_𝐾−1,𝑝^ℎ) of difference schemes (24) and (25) the stability estimates 𝑢^ℎ_𝑘1^𝑁−1_{𝒞𝜏(𝐿2ℎ)}≤𝑀𝜑^ℎ_𝐿2ℎ+𝜓^ℎ_𝐿2ℎ+𝜉^ℎ_𝐿2ℎ+𝑓^ℎ_𝑘1^𝑁−1_{𝒞𝜏(𝐿2ℎ)},𝑝^ℎ_𝐿2ℎ≤𝑀𝜑^ℎ_𝑊22ℎ+𝜓^ℎ_𝑊22ℎ+𝜉^ℎ_𝑊22ℎ+1𝛼()𝑓1−𝛼^{ℎ𝑘1𝑁−1}_{𝒞𝜏(𝐿2ℎ)} (28) hold, where 𝑀 is independent of 𝜏,𝛼,ℎ,𝜑^ℎ(𝑥),𝜓^ℎ(𝑥),  𝜉^ℎ(𝑥), and {𝑓^ℎ_𝑘(𝑥)}₁^𝑁−1.

Theorem 7. Let 𝜏 and |ℎ|=ℎ²₁+⋯+ℎ²_𝑛 be sufficiently small positive numbers. Then, for the solutions of difference schemes (24) and (25) the following almost coercive stability estimate 𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^𝑘_𝑘−1𝜏²)₁^𝑁−1_{𝒞𝜏(𝐿2ℎ)}+𝑢^ℎ_𝑘1^𝑁−1_{𝒞𝜏(𝑊22ℎ)}+𝑝^ℎ_𝐿2ℎ≤𝑀𝜑^ℎ_𝑊22ℎ+𝜓^ℎ_𝑊22ℎ+𝜉^ℎ_𝑊22ℎ1+ln𝑓𝜏+ℎ^{ℎ𝑘1𝑁−1}_{𝒞𝜏(𝐿2ℎ)} (29) holds, where 𝑀 is independent of 𝜏,𝛼,ℎ,𝜑^ℎ(𝑥),𝜓^ℎ(𝑥),  𝜉^ℎ(𝑥), and {𝑓^ℎ_𝑘(𝑥)}₁^𝑁−1.

Proofs of Theorems 6 and 7 are based on the symmetry property of operator 𝐴^𝑥, on Lemmas 3–5, the formulas 𝑢^ℎ_𝑘(𝑥)=𝐼−𝑅^2𝑁−1×𝑅^𝑘−𝑅^2𝑁−𝑘𝑣^ℎ₀(𝑥)+𝑅^𝑁−𝑘−𝑅^𝑁+𝑘𝑣^ℎ_𝑁(𝑥)−𝑅^𝑁−𝑘−𝑅^𝑁+𝑘(𝐼+𝜏𝐶)(2𝐼+𝜏𝐶)⁻¹×𝐶^−1𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+𝑖𝑓^ℎ_𝑖(𝑥)𝜏+(𝐼+𝜏𝐶)(2𝐼+𝜏𝐶)⁻¹𝐶⁻¹×^𝑁−1_𝑖=1𝑅^{|𝑘−𝑖|}−𝑅^𝑘+𝑖×𝑓^ℎ_𝑖(𝑥)𝜏+𝜑^ℎ(𝑥)−𝑣^ℎ₀(𝑥),𝑝^ℎ(𝑥)=𝐴^𝑥_ℎ𝜑^ℎ(𝑥)−𝐴^𝑥_ℎ𝑣^ℎ₀(𝑥),𝑣^ℎ_𝑁(𝑥)=𝑣^ℎ₀(𝑥)+𝜓^ℎ(𝑥)−𝜑^ℎ(𝑥),𝑣^ℎ₀(𝑥)=−𝐺𝑅^𝑁−𝑙−𝑅^𝑁+𝑙×(𝐼+𝜏𝐶)(2𝐼+𝜏𝐶)⁻¹𝐶⁻¹×^𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+𝑖𝑓^ℎ_𝑖(𝑥)𝜏+𝐺𝐼−𝑅^2𝑁()×()𝐼+𝜏𝐶2𝐼+𝜏𝐶⁻¹𝐶^−1𝑁−1_𝑖=1𝑅^{|𝑙−𝑖|}−𝑅^𝑙+𝑖𝑓^ℎ_𝑖(𝑥)𝜏+𝐺𝐼−𝑅^2𝑁𝜑^ℎ(𝑥)−𝜉^ℎ(𝑥)+𝐺𝑅^𝑁−𝑙−𝑅^𝑁+𝑙𝜓^ℎ(𝑥)−𝜑^ℎ(𝑥), (30) for difference scheme (24), 𝑣^ℎ₀(𝑥)=𝜆𝜏−𝑙−1𝐺₁𝑅^𝑁−𝑙−𝑅^𝑁+𝑙×(𝐼+𝜏𝐶)(2𝐼+𝜏𝐶)⁻¹𝐶⁻¹×^𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+𝑖𝑓^ℎ_𝑖(𝑥)𝜏−𝜆𝜏−𝑙−1𝐺₁𝐼−𝑅^2𝑁×(𝐼+𝜏𝐶)(2𝐼+𝜏𝐶)⁻¹𝐶⁻¹×^𝑁−1_𝑖=1𝑅^{|𝑙−𝑖|}−𝑅^𝑙+𝑖𝑓^ℎ_𝑖(𝑥)𝜏−𝜆𝜏−𝑙𝐺₁𝑅^{𝑁−𝑙−1}−𝑅^𝑁+𝑙+1×(𝐼+𝜏𝐶)(2𝐼+𝜏𝐶)⁻¹𝐶⁻¹×^𝑁−1_𝑖=1𝑅^𝑁−𝑖−𝑅^𝑁+𝑖𝑓^ℎ_𝑖(𝑥)𝜏+𝜆𝜏−𝑙𝐺₁𝐼−𝑅^2𝑁×(𝐼+𝜏𝐶)(2𝐼+𝜏𝐶)⁻¹𝐶⁻¹×^𝑁−1_𝑖=1𝑅^{|𝑙+1−𝑖|}−𝑅^𝑙+1+𝑖𝑓^ℎ_𝑖(𝑥)𝜏+𝐺₁𝐼−𝑅^2𝑁𝜑^ℎ(𝑥)−𝜉^ℎ(𝑥+𝜆)𝜏−𝑙−1𝐺₁𝑅^𝑁−𝑙−𝑅^𝑁+𝑙+𝜆𝜏−𝑙𝐺₁𝑅^{𝑁−𝑙−1}−𝑅^𝑁+𝑙+1×𝜓^ℎ(𝑥)−𝜑^ℎ(𝑥), (31) for difference scheme (25), and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in 𝐿_2ℎ.

Theorem 8 (see [28]). For the solution of the elliptic difference problem 𝐴^𝑥_ℎ𝑢^ℎ(𝑥)=𝜔^ℎ(𝑥),𝑥∈Ω_ℎ,𝐷^ℎ𝑢^ℎ(𝑥)=0,𝑥∈𝑆_ℎ, (32) the following coercivity inequality holds: ^𝑛_𝑟=1(𝑢^ℎ_𝑘)_{𝑥𝑟𝑥𝑟,𝑗𝑟𝐿2ℎ}≤𝑀𝜔^ℎ_𝐿2ℎ, (33) where 𝑀 does not depend on ℎ and 𝜔^ℎ.

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 −𝜕²𝑢(𝑡,𝑥)𝜕𝑡²−𝜕()()𝜕𝑥2+cos𝑥𝜕𝑢𝑡,𝑥()()(𝑥)𝑓()()(𝑥)+()((𝑥)(,𝑢()(𝑥)𝑢()=()(𝑥)𝑢()=()(𝑥)𝑢𝜕𝑥+𝑢𝑡,𝑥=𝑓𝑡,𝑥+𝑝,0<𝑥<𝜋,0<𝑡<𝑇,𝑡,𝑥=−exp−𝑡cosexp−𝑡+𝑡3cos+cos2𝑥))0,𝑥=2cos,0≤𝑥≤𝜋,𝑇,𝑥exp−𝑇+𝑇+1cos,0≤𝑥≤𝜋,𝜆,𝑥exp−𝜆+𝜆+1cos,0≤𝑥≤𝜋,_𝑥(𝑡,0)=𝑢_𝑥(𝑡,𝜋)=0,0≤𝑡≤𝑇,𝜆=3𝑇5. (34) 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 obtain 𝑢(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 −𝑑²𝑣(𝑡,𝑥)𝑑𝑡²−𝜕()()𝜕𝑥2+cos𝑥𝜕𝑣𝑡,𝑥()()𝑣()()=()(𝑥),𝑣()()=()()(𝑥),𝑣𝜕𝑥+𝑣𝑡,𝑥=𝑓𝑡,𝑥,0<𝑥<𝜋,0<𝑡<𝑇,0,𝑥−𝑣𝜆,𝑥1−exp−𝜆−𝜆cos0≤𝑥≤𝜋𝑇,𝑥−𝑣𝜆,𝑥exp−𝑇−exp−𝜆+𝑇−𝜆cos0≤𝑥≤𝜋,_𝑥(𝑡,0)=𝑣_𝑥(𝑡,𝜋)=0,0≤𝑡≤𝑇, (35) and 𝑤(𝑡,𝑥) is the solution of the boundary value problem −𝑑²𝑤(𝑡,𝑥)𝑑𝑡²−𝜕()()𝜕𝑥2+cos𝑥𝜕𝑤𝑡,𝑥()(𝑥)𝑤()=()(𝑥)(),𝑤()=()(𝑥)(),𝑤𝜕𝑥+𝑤𝑡,𝑥=𝑝,0<𝑥<𝜋,0<𝑡<𝑇,0,𝑥exp−𝜆+𝜆+1cos−𝑣𝜆,𝑥0≤𝑥≤𝜋,𝑇,𝑥exp−𝜆+𝜆+1cos−𝑣𝜆,𝑥0≤𝑥≤𝜋,_𝑥(𝑡,0)=𝑤_𝑥(𝑡,𝜋)=0,0≤𝑡≤𝑇. (36)

Introduce small parameters 𝜏 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 [0,𝑇]_𝜏×[0,𝜋]_ℎ=𝑡_𝑘,𝑥_𝑛∶𝑡_𝑘=𝑘𝜏,𝑘=0,…,𝑁,𝑥_𝑛=𝑛ℎ,𝑛=0,…,𝑀. (37)

Applying (21), we obtain difference schemes of the first order of accuracy in 𝑡 and the second order of accuracy in 𝑥𝑣_𝑛^𝑘+1−2𝑣^𝑘_𝑛+𝑣_𝑛^𝑘−1𝜏²+2+cos𝑥_𝑛𝑣^𝑘_𝑛+1−2𝑣^𝑘_𝑛+𝑣^𝑘_𝑛−1ℎ²−sin𝑥_𝑛𝑣^𝑘_𝑛+1−𝑣^𝑘_𝑛−12ℎ−𝑣^𝑘_𝑛=𝜃^𝑘_𝑛,𝜃^𝑘_𝑛=−𝑓𝑡_𝑘,𝑥_𝑛𝑣,𝑘=1,…,𝑁−1,𝑛=1,…,𝑀−1,_𝑘0−𝑣^𝑘₁=𝑣^𝑘_𝑀−𝑣^𝑘_𝑀−1𝑣=0,𝑘=0,…,𝑁,_0𝑛−𝑣^𝑙_𝑛=1−exp(−𝜆)−𝜆cos𝑥_𝑛𝑣,𝑛=0,…,𝑀,_𝑁𝑛−𝑣^𝑙_𝑛=exp−𝑡_𝑁−exp(−𝜆)+𝑡_𝑁−𝜆cos𝑥_𝑛,𝜆𝑛=0,…,𝑀,𝑙=𝜏, (38) for the approximate solutions of the nonlocal boundary value problem (35), and 𝑤_𝑛^𝑘+1−2𝑤^𝑘_𝑛+𝑤_𝑛^𝑘−1𝜏²+2+cos𝑥_𝑛𝑤^𝑘_𝑛+1−2𝑤^𝑘_𝑛+𝑤^𝑘_𝑛−1ℎ²−sin𝑥_𝑛𝑤^𝑘_𝑛+1−𝑤^𝑘_𝑛−12ℎ−𝑤^𝑘_𝑛=−𝑝_𝑛𝑝,𝑘=1,…,𝑁−1,_𝑛=𝑝𝑥_𝑛𝑤,𝑛=1,…,𝑀−1,_𝑘0−𝑤^𝑘₁=𝑤^𝑘_𝑀−𝑤^𝑘_𝑀−1𝑤=0,𝑘=0,…,𝑁,_0𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛−𝑣^𝑙_𝑛,𝜆𝑛=0,…,𝑀,𝑙=𝜏,𝑤^𝑁_𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛−𝑣^𝑙_𝑛,𝑛=0,…,𝑀, (39) 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: 𝑝_𝑛𝑥=−2+cos_𝑛ℎ²𝜑_𝑛+1−𝑣⁰_𝑛+1−2𝜑_𝑛−𝑣⁰_𝑛+𝜑_𝑛−1−𝑣⁰_𝑛−1+sin𝑥_𝑛×𝜑2ℎ_𝑛+1−𝑣⁰_𝑛+1−𝜑_𝑛−1−𝑣⁰_𝑛−1+𝜑_𝑛−𝑣⁰_𝑛,𝑛=1,…,𝑀−1. (40)

We can rewrite difference scheme (38) in the matrix form 𝐴_𝑛𝑣_𝑛+1+𝐵_𝑛𝑣_𝑛+𝐶_𝑛𝑣_𝑛−1=𝐼𝜃^𝑘_𝑛𝑣,𝑛=1,…,𝑀−1,₀=𝑣₁,𝑣_𝑀=𝑣_𝑀−1. (41) Here, 𝐼 is the (𝑁+1)×(𝑁+1) identity matrix, 𝐴_𝑛, 𝐵_𝑛, 𝐶_𝑛 are (𝑁+1)×(𝑁+1) square matrices, and 𝜃_𝑛 is a (𝑁+1)×1 column matrix which are defined by 𝐴_𝑛=0000⋯00000𝑎_𝑛00⋯000000𝑎_𝑛0⋯0000000𝑎_𝑛⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯𝑎_𝑛0000000⋯0𝑎_𝑛000000⋯00𝑎_𝑛0,𝐵0000⋯0000_𝑛=1000⋯−1⋯0000𝑑𝑏_𝑛𝑑0⋯0⋯00000𝑑𝑏_𝑛𝑑⋯0⋯000000𝑑𝑏_𝑛⋯0⋯0000⋮⋮⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯0⋯𝑏_𝑛𝑑000000⋯0⋯𝑑𝑏_𝑛𝑑00000⋯0⋯0𝑑𝑏_𝑛𝑑,𝐶0000⋯−1⋯0001_𝑛=0000⋯00000𝑐_𝑛00⋯000000𝑐_𝑛0⋯0000000𝑐_𝑛⋯0000⋮⋮⋮⋮⋯⋮⋮⋮⋮0000⋯𝑐_𝑛0000000⋯0𝑐_𝑛000000⋯00𝑐_𝑛0𝑎0000⋯0000_𝑛,=2+cos𝑥_𝑛ℎ²−sin𝑥_𝑛,𝑏2ℎ_𝑛2=−𝜏²−22+cos𝑥_𝑛ℎ²𝑐−1,_𝑛=2+cos𝑥_𝑛ℎ²+sin𝑥_𝑛12ℎ,𝑑=𝜏²,𝜃_𝑛=𝜃⁰_𝑛⋮𝜃^𝑁_𝑛,𝜃⁰_𝑛=1−exp(−𝜆)−𝜆cos𝑥_𝑛,𝜃^𝑁_𝑛=exp−𝑡_𝑁−exp(−𝜆)+𝑡_𝑁−𝜆cos𝑥_𝑛,𝜃𝑛=1,…,𝑀−1,_𝑘𝑛=−𝑓𝑡_𝑘,𝑥_𝑛𝑣,𝑘=1,…,𝑁−1,𝑛=1,…,𝑀−1,_𝑠=𝑣⁰_𝑠⋮𝑣^𝑁_{𝑠(𝑁+1)×1},𝑠=𝑛−1,𝑛,𝑛+1. (42)(43)(44)(45)

For solving (41) we use the modified Gauss elimination method (see [29]). Namely, we seek solution of (41) by the formula 𝑣_𝑛=𝛼_𝑛+1𝑣_𝑛+1+𝛽_𝑛+1,𝑛=𝑀−1,…,1, (46) where 𝑣_𝑀=0,  𝛼_𝑛(𝑛=1,…,𝑀−1) are (𝑁+1)×(𝑁+1) square matrices and 𝛽_𝑛  (𝑛=1,…,𝑀−1) are (𝑁+1)×1 column matrices. For 𝛼_𝑛+1,𝛽_𝑛+1, we get formulas 𝛼_𝑛+1=−𝐵_𝑛+𝐶_𝑛𝛼_𝑛⁻¹𝐴_𝑛,𝛽_𝑛+1=−𝐵_𝑛+𝐶_𝑛𝛼_𝑛⁻¹𝐼𝜃_𝑛−𝐶_𝑛𝛽_𝑛,𝑛=1,…,𝑀−1, (47) 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 𝐴_𝑛𝑤_𝑛+1+𝐸_𝑛𝑤_𝑛+𝐶_𝑛𝑤_𝑛−1=𝐼𝜂^𝑘_𝑛,𝑤𝑛=1,…,𝑀−1,₀=𝑤₁,𝑤_𝑀=𝑤_𝑀−1. (48)

Here, 𝐸_𝑛=1000⋯0000𝑑𝑏_𝑛𝑑0⋯00000𝑑𝑏_𝑛𝑑⋯000000𝑑𝑏_𝑛⋯0000⋮⋮⋮⋮⋱⋮⋮⋮⋮0000⋯𝑏_𝑛𝑑000000⋯𝑑𝑏_𝑛𝑑00000⋯0𝑑𝑏_𝑛𝑑0000⋯0001, (49)𝐴_𝑛 and 𝐶_𝑛 are defined by (42) and (43) and (𝑁+1)×1 column matrix 𝜂_𝑛 is defined by 𝜂_𝑛=𝜂⁰_𝑛⋮𝜂^𝑁_𝑛,𝜂⁰_𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛−𝑣^𝑙_𝑛,𝜂^𝑁_𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛−𝑣^𝑙_𝑛𝜂,𝑛=1,…,𝑀−1,_𝑘𝑛=−𝑝_𝑛𝑤,𝑘=1,…,𝑁−1,𝑛=1,…,𝑀−1,_𝑠=𝑤⁰_𝑠⋮𝑤^𝑁_{𝑠(𝑁+1)×1},𝑠=𝑛−1,𝑛,𝑛+1. (50)

Now we present second order of accuracy in 𝑡 and 𝑥 difference schemes for problems (35) and (36). Applying (27) and formulas for sufficiently smooth function 𝜌𝜌𝑥_𝑛+1−𝜌𝑥_𝑛−12ℎ−𝜌^′𝑥_𝑛=𝑂ℎ²,𝜌𝑥_𝑛+1−2𝜌𝑥_𝑛+𝜌𝑥_𝑛−1ℎ²−𝜌^′′𝑥_𝑛=𝑂ℎ²,10𝜌(0)−15𝜌(ℎ)+6𝜌(2ℎ)−𝜌(3ℎ)ℎ³−𝜌^′′′(0)=𝑂ℎ²,−3𝜌(0)+4𝜌(ℎ)−𝜌(2ℎ)2ℎ−𝜌^′(0)=𝑂ℎ²,10𝜌(𝜋)−15𝜌(𝜋−ℎ)+6𝜌(𝜋−2ℎ)−𝜌(𝜋−3ℎ)ℎ³−𝜌^′′′(𝜋)=𝑂ℎ²,−3𝜌(𝜋)+4𝜌(𝜋−ℎ)−𝜌(𝜋−2ℎ)2ℎ−𝜌^′(𝜋)=𝑂ℎ², (51) we get 𝑣_𝑛^𝑘+1−2𝑣^𝑘_𝑛+𝑣_𝑛^𝑘−1𝜏²+2+cos𝑥_𝑛𝑣^𝑘_𝑛+1−2𝑣^𝑘_𝑛+𝑣^𝑘_𝑛−1ℎ²−sin𝑥_𝑛𝑣^𝑘_𝑛+1−𝑣^𝑘_𝑛−12ℎ−𝑣^𝑘_𝑛=𝜃^𝑘_𝑛,𝜃^𝑘_𝑛=−𝑓𝑡_𝑘,𝑥_𝑛,𝑘=1,…,𝑁−1,𝑛=1,…,𝑀−1,−3𝑣^𝑘₀+4𝑣^𝑘₁−𝑣^𝑘₂=−3𝑣^𝑘_𝑀+4𝑣^𝑘_𝑀−1−𝑣^𝑘_𝑀−2=0,𝑘=0,…,𝑁,10𝑣^𝑘₀−15𝑣^𝑘₁+6𝑣^𝑘₂−𝑣^𝑘₃=10𝑣^𝑘_𝑀−15𝑣^𝑘_𝑀−1+6𝑣^𝑘_𝑀−2−𝑣^𝑘_𝑀−3𝑣=0,_0𝑛+𝜆𝜏−𝑙−1𝑣^𝑙_𝑛−𝜆𝜏−𝑙𝑣_𝑛^𝑙+1=1−exp(−𝜆)−𝜆cos𝑥_𝑛𝑣,𝑛=0,…,𝑀,_𝑁𝑛+𝜆𝜏−𝑙−1𝑣^𝑙_𝑛−𝜆𝜏−𝑙𝑣_𝑛^𝑙+1=exp−𝑡_𝑁−exp(−𝜆)+𝑡_𝑁−𝜆cos𝑥_𝑛,𝑛=0,…,𝑀, (52) difference scheme for nonlocal problem (35), and 𝑤_𝑛^𝑘+1−2𝑤^𝑘_𝑛+𝑤_𝑛^𝑘−1𝜏²+2+cos𝑥_𝑛𝑤^𝑘_𝑛+1−2𝑤^𝑘_𝑛+𝑤^𝑘_𝑛−1ℎ²−sin𝑥_𝑛𝑤^𝑘_𝑛+1−𝑤^𝑘_𝑛−12ℎ=−𝑝_𝑛,𝑘=1,…,𝑁−1,𝑝_𝑛=𝑝𝑥_𝑛,𝑛=1,…,𝑀−1,−3𝑤^𝑘₀+4𝑤^𝑘₁−𝑤^𝑘₂=−3𝑤^𝑘_𝑀+4𝑤^𝑘_𝑀−1−𝑤^𝑘_𝑀−2=0,𝑘=0,…,𝑁,10𝑤^𝑘₀−15𝑤^𝑘₁+6𝑤^𝑘₂−𝑤^𝑘₃=10𝑤^𝑘_𝑀−15𝑤^𝑘_𝑀−1+6𝑤^𝑘_𝑀−2−𝑤^𝑘_𝑀−3𝑤=0,_0𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛+𝜆𝜏−𝑙−1𝑣^𝑙_𝑛−𝜆𝜏−𝑙𝑣_𝑛^𝑙+1𝑤,𝑛=0,…,𝑀,_𝑁𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛+𝜆𝜏−𝑙−1𝑣^𝑙_𝑛−𝜆𝜏−𝑙𝑣_𝑛^𝑙+1,𝜉_𝑛=𝜉𝑥_𝑛,𝑛=0,…,𝑀, (53) difference scheme for boundary value problem (36).

By difference scheme (52), we write in matrix form 𝐴_𝑛𝑣_𝑛+1+𝐵_𝑛𝑣_𝑛+𝐶_𝑛𝑣_𝑛−1=𝐼𝜃^𝑘_𝑛,𝑛=1,…,𝑀−1,−3𝑣₀+4𝑣₁−𝑣₂=0,−3𝑣_𝑀+4𝑣_𝑀−1−𝑣_𝑀−2=0, (54) where 𝐴_𝑛,𝐶_𝑛 are defined by (42), (43), (44), and 𝐵_𝑛 is defined by 𝐵_𝑛=1000⋯0𝑦𝑧0⋯0000𝑑𝑏_𝑛𝑑0⋯0000⋯00000𝑑𝑏_𝑛𝑑⋯0000⋯000000𝑑𝑏_𝑛⋯0000⋯0000⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮0000⋯0000⋯𝑏_𝑛𝑑000000⋯0000⋯𝑑𝑏_𝑛𝑑00000⋯0000⋯0𝑑𝑏_𝑛𝑑,𝑏0000⋯0𝑦𝑧0⋯0001_𝑛2=−𝜏²−22+cos𝑥_𝑛ℎ²1−1,𝑑=𝜏²𝜆,𝑦=𝜏𝜆−𝑙−1,𝑧=−𝜏−𝑙. (55)

We seek solution of (54) by the formula 𝑣_𝑛=𝛼_𝑛𝑣_𝑛+1+𝛽_𝑛𝑣_𝑛+2+𝛾_𝑛,𝑛=𝑀−2,…,0, (56) where 𝛼_𝑛,𝛽_𝑛  (𝑛=0,…,𝑀−2) are (𝑁+1)×(𝑁+1) square matrices and 𝛾_𝑛(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 𝛼_𝑛,𝛽_𝑛: 𝛼_𝑛=−𝐵_𝑛+𝐶_𝑛𝛼_𝑛−1⁻¹𝐴_𝑛+𝐶_𝑛𝛽_𝑛−1,𝛽_𝑛𝛾=0,_𝑛=−𝐵_𝑛+𝐶_𝑛𝛼_𝑛−1⁻¹𝐼𝜃_𝑛−𝐶_𝑛𝛾_𝑛−1,𝑛=1,…,𝑀−1, (57) where 𝛼₀=43𝐼,𝛽₀1=−3𝛼𝐼,₁=85𝐼,𝛽₁3=−5𝛼𝐼,_𝑀−2=4𝐼,𝛽_𝑀−2𝛼=−3𝐼,_𝑀−3=83𝐼,𝛽_𝑀−35=−3𝐼, (58) and 𝛾₀,𝛾₁,𝛾_𝑀−2,𝛾_𝑀−3 are the (𝑁+1)×1 zero column vector. For 𝑣_𝑀 and 𝑣_𝑀−1 we have 𝑣_𝑀=𝑄₁₁−𝑄₁₂𝑄⁻¹₂₂𝑄₂₁⁻¹𝐺₁−𝑄₁₂𝑄⁻¹₂₂𝐺₂,𝑣_𝑀−1=𝑄⁻¹₂₂𝐺₂−𝑄₂₁𝑣_𝑀, (59) where 𝑄₁₁=−3𝐴_𝑀−2−8𝐵_𝑀−2−8𝐶_𝑀−2𝛼_𝑀−3−3𝐶_𝑀−2𝛽_𝑀−3,𝑄₁₂=4𝐴_𝑀−2+9𝐵_𝑀−2+9𝐶_𝑀−2𝛼_𝑀−3+4𝐶_𝑀−2𝛽_𝑀−3,𝑄₂₁=−3𝐵_𝑀−1−8𝐶_𝑀−1,𝑄₂₂=𝐴_𝑀−1+4𝐵_𝑀−1+9𝐶_𝑀−1,𝐺₁=𝐼𝜃_𝑀−2−𝐶_𝑀−2𝛾_𝑀−3,𝐺₂=𝐼𝜃_𝑀−1. (60)

We can rewrite difference scheme (53) in the matrix form 𝐴_𝑛𝑤_𝑛+1+𝐸_𝑛𝑤_𝑛+𝐶_𝑛𝑤_𝑛−1=𝐼𝜂^𝑘_𝑛,𝑛=1,…,𝑀−1,−3𝑤₀+4𝑤₁−𝑤₂=0,−3𝑤_𝑀+4𝑤_𝑀−1−𝑤_𝑀−2=0, (61) where 𝐴_𝑛,𝐸_𝑛,𝐶_𝑛 are defined by (42), (49), (43), and (44) and 𝜂_𝑛 is defined by 𝜂_𝑛=𝜂⁰_𝑛⋮𝜂^𝑁_𝑛,𝜂⁰_𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛+𝜆𝜏−𝑙−1𝑣^𝑙_𝑛−𝜆𝜏−𝑙𝑣_𝑛^𝑙+1,𝜂^𝑁_𝑛=exp(−𝜆)+𝜆+1cos𝑥_𝑛+𝜆𝜏−𝑙−1𝑣^𝑙_𝑛−𝜆𝜏−𝑙𝑣_𝑛^𝑙+1𝜂,𝑛=0,…,𝑀,_𝑘𝑛=−𝑝_𝑛,𝑘=1,…,𝑁−1,𝑛=1,…,𝑀−1. (62)

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 𝑣^𝑘_𝑛, 𝑢^𝑘_𝑛 represent the numerical solutions of difference schemes for auxiliary nonlocal problem (35) and inverse problem (34) at (𝑡_𝑘,𝑥_𝑛), respectively. Grid function 𝑝_𝑛 calculated by (40) represents numerical solution at 𝑥_𝑛 for unknown function 𝑝. The errors are computed by the norms 𝐸𝑣^𝑁_𝑀=max_{1≤𝑘≤𝑁−1}^𝑀−1_𝑛=1𝑣𝑡_𝑘,𝑥_𝑛−𝑣^𝑘_𝑛²ℎ^1/2,𝐸𝑢^𝑁_𝑀=max_{1≤𝑘≤𝑁−1}^𝑀−1_𝑛=1𝑢𝑡_𝑘,𝑥_𝑛−𝑢^𝑘_𝑛²ℎ^1/2,𝐸𝑝_𝑀=^𝑀−1_𝑛=1𝑝𝑥_𝑛−𝑝_𝑛²ℎ^1/2. (63)

Tables 1–3 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.

Table 2 

Error analysis for p.

Table 3 

Error analysis for u.

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 Tables 1–3, 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.

Acknowledgment

The authors would like to thank Professor Dr. Allaberen Ashyralyev (Fatih University, Turkey) for his helpful suggestions aimed at the improvement of the present paper.