Approximate Solution of Inverse Problem for Elliptic Equation with ...

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 548017, 11 pages http://dx.doi.org/10.1155/2013/548017

Research Article Approximate Solution of Inverse Problem for Elliptic Equation with Overdetermination Charyyar Ashyralyyev1,2 and Mutlu Dedeturk1 1 2

Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey TAU, Gerogly Street 143, 74400 Ashgabat, Turkmenistan

Correspondence should be addressed to Mutlu Dedeturk; [email protected] Received 5 July 2013; Revised 14 August 2013; Accepted 20 August 2013 Academic Editor: Abdullah Said Erdogan Copyright © 2013 C. Ashyralyyev and M. Dedeturk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 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.

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: −𝑢𝑡𝑡 (𝑡) + 𝐴𝑢 (𝑡) = 𝑓 (𝑡) + 𝑝, 𝑢 (0) = 𝜑,

𝑢 (𝑇) = 𝜓,

0 < 𝑡 < 𝑇,

𝑢 (𝜆) = 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 𝑛dimensional 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 𝑛

−𝑢𝑡𝑡 (𝑡, 𝑥) − ∑ (𝑎𝑟 (𝑥) 𝑢𝑥𝑟 )𝑥 + 𝛿𝑢 (𝑡, 𝑥) = 𝑓 (𝑡, 𝑥) + 𝑝 (𝑥) , 𝑟

𝑟=1

𝑥 = (𝑥1 , . . . , 𝑥𝑛 ) ∈ Ω, 𝑢 (0, 𝑥) = 𝜑 (𝑥) ,

0 < 𝑡 < 𝑇,

𝑢 (𝑇, 𝑥) = 𝜓 (𝑥) ,

𝑢 (𝜆, 𝑥) = 𝜉 (𝑥) , 𝑥 ∈ Ω,

𝜕𝑢 (𝑡, 𝑥) = 0, 𝜕𝑛 ⃗

𝑥 ∈ 𝑆, 0 ≤ 𝑡 ≤ 𝑇. (2)

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 DirichletNeumann 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

2

Abstract and Applied Analysis

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.

󵄩󵄩 󸀠󸀠 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝑢 󵄩󵄩 𝛼,𝛼 + 󵄩󵄩󵄩𝑝󵄩󵄩󵄩𝐿 2 (Ω) 2 󵄩 󵄩C0𝑇 (𝐿 2 (Ω)) + ‖𝑢‖C𝛼,𝛼 0𝑇 (𝑊2 (Ω)) ≤ 𝑀[

𝑛

𝐴𝑥 𝑢 (𝑥) = −∑(𝑎𝑟 (𝑥) 𝑢𝑥𝑟 )𝑥 + 𝛿𝑢 (𝑥)

(3)

𝑟

defines a self-adjoint positive definite operator 𝐴𝑥 acting on 𝐿 2 (Ω) with the domain 𝐷(𝐴𝑥 ) = {𝑢(𝑥) ∈ 𝑊22 (Ω), 𝜕𝑢/𝜕𝑛⃗ = 0 on 𝑆}. Let 𝐻 be the Hilbert space 𝐿 2 (Ω). By using abstract Theorems 2.1 and 2.2 of paper [11], we get the following theorems about well-posedness of problem (2). 𝑥

Theorem 1. Assume that 𝐴 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:

󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝑊2 (Ω) + 󵄩󵄩󵄩𝜉󵄩󵄩󵄩𝑊2 (Ω) ] 2

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 = 𝐿 2 (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]): 󵄩󵄩 𝑘 󵄩󵄩 1/2 −𝑘 󵄩󵄩𝑅 󵄩󵄩 󵄩 󵄩𝐻 → 𝐻 ≤ 𝑀(1 + 𝛿 𝜏) ,

(7)

Lemma 4. For 1 ≤ 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

󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝜉󵄩󵄩󵄩𝐿 2 (Ω) + 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐶(𝐿 2 (Ω)) ] , 󵄩 𝑥 󵄩 󵄩 𝑥 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑝󵄩󵄩𝐿 2 (Ω) ≤ 𝑀 [󵄩󵄩󵄩𝐴 𝜑󵄩󵄩󵄩𝐿 2 (Ω) + 󵄩󵄩󵄩𝐴 𝜓󵄩󵄩󵄩 𝐿 2 (Ω) 1 󵄩󵄩 󵄩󵄩 + ], 󵄩𝑓󵄩 𝛼,𝛼 𝛼 (1 − 𝛼) 󵄩 󵄩C0𝑇 (𝐿 2 (Ω)) (4)

where 𝑀 is independent of 𝛼, 𝜑(𝑥), 𝜉(𝑥), 𝜓(𝑥), and 𝑓(𝑡, 𝑥). Here, C𝛼,𝛼 0𝑇 (𝐿 2 (Ω)) is the space obtained by completion of the space of all smooth 𝐿 2 (Ω)-valued functions 𝜌 on [0, 𝑇] with the norm

‖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 − 𝑅2𝑁 − 𝑅𝑙 + 𝑅2𝑁−𝑙 − 𝑅𝑁−𝑙 + 𝑅𝑁+𝑙 = (𝐼 − 𝑅𝑁−𝑙 ) (𝐼 − 𝑅𝑁) (𝐼 − 𝑅𝑙 ) .

(9)

Lemma 5. For 1 ≤ 𝑙 ≤ 𝑁 − 1 and for the operator

󵄩󵄩 󵄩󵄩 󵄩󵄩𝜌󵄩󵄩C𝛼,𝛼 (𝐿 2 (Ω)) 0𝑇 󵄩 󵄩 = 󵄩󵄩󵄩𝜌󵄩󵄩󵄩C(𝐿 2 (Ω)) sup

𝛿 > 0,

𝑀 󵄩󵄩 𝑘 󵄩󵄩 󵄩󵄩𝐶𝑅 󵄩󵄩 󵄩 󵄩𝐻 → 𝐻 ≤ 𝑘𝜏 , 𝑘 ≥ 1, 󵄩 󵄩󵄩 󵄩󵄩(𝐼 − 𝑅2𝑁)−1 󵄩󵄩󵄩 󵄩󵄩𝐻 → 𝐻 ≤ 𝑀. 󵄩󵄩

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ [𝑀 󵄩󵄩󵄩𝜑󵄩󵄩󵄩𝐿 2 (Ω) + 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐿 2 (Ω) + 󵄩󵄩󵄩𝜉󵄩󵄩󵄩𝐿 2 (Ω) + 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐶(𝐿 2 (Ω)) ] , 󵄩󵄩 𝑥 −1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩(𝐴 ) 𝑝󵄩󵄩 󵄩 󵄩𝐿 2 (Ω) ≤ 𝑀 [󵄩󵄩𝜑󵄩󵄩𝐿 2 (Ω)

0≤𝑡 0, where 𝑁 is a fixed positive integer. Applying the approximate formulas 𝜆 𝑢 (𝜆, 𝑥) = 𝑢 ([ ] 𝜏, 𝑥) + 𝑜 (𝜏) , 𝜏 ℎ

ℎ

(23)

𝜆 𝜆 × (𝑢ℎ ([ ] 𝜏 + 𝜏, 𝑥) − 𝑢ℎ ([ ] 𝜏, 𝑥)) 𝜏 𝜏

ℎ

−

𝑥 ∈ Ωℎ ,

𝜆 𝜆 𝜆 𝑢ℎ (𝜆, 𝑥) = 𝑢ℎ ([ ] 𝜏, 𝑥) + ( − [ ]) 𝜏 𝜏 𝜏

+ 𝑜 (𝜏2 ) ,

For approximate solution of nonlocal problem (21), we have first order of accuracy difference scheme

=

𝑓𝑘ℎ

𝑥 ∈ Ωℎ ,

ℎ V𝑁 (𝑥) − V𝑙ℎ (𝑥) = 𝜓ℎ (𝑥) − 𝜉ℎ (𝑥) ,

̃ℎ 𝑥∈Ω

𝑓𝑘ℎ (𝑥) = 𝑓ℎ (𝑡𝑘 , 𝑥) , 𝑡𝑘 = 𝑘𝜏,

(𝑥) + 𝑝 (𝑥) ,

ℎ V𝑁 (𝑥) − (

𝑡𝑘 = 𝑘𝜏,

1 ≤ 𝑘 ≤ 𝑁 − 1,

𝑥 ∈ Ωℎ ,

𝑢0ℎ (𝑥) = 𝜑ℎ (𝑥) ,

̃ℎ, 𝑥∈Ω

𝑢𝑙ℎ (𝑥) = 𝜉ℎ (𝑥) ,

̃ℎ, 𝑥∈Ω

(24)

𝑥 ∈ Ωℎ ,

𝜆 𝜆 ℎ − 𝑙) V𝑙+1 (𝑥) + ( − 𝑙 − 1) V𝑙ℎ (𝑥) 𝜏 𝜏

= 𝜓ℎ (𝑥) − 𝜉ℎ (𝑥) ,

Theorem 6. Let 𝜏 and |ℎ| = √ℎ12 + ⋅ ⋅ ⋅ + ℎ𝑛2 be sufficiently 𝑁−1

small positive numbers. Then, for the solutions ({𝑢𝑘ℎ }𝐾−1 , 𝑝ℎ ) of difference schemes (24) and (25) the stability estimates 󵄩󵄩 ℎ 𝑁−1 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩{𝑢 } 󵄩󵄩 ≤ 𝑀 [󵄩󵄩󵄩󵄩𝜑ℎ 󵄩󵄩󵄩󵄩𝐿 + 󵄩󵄩󵄩󵄩𝜓ℎ 󵄩󵄩󵄩󵄩𝐿 + 󵄩󵄩󵄩󵄩𝜉ℎ 󵄩󵄩󵄩󵄩𝐿 󵄩󵄩 𝑘 1 󵄩󵄩 2ℎ 2ℎ 2ℎ 󵄩C𝜏 (𝐿 2ℎ ) 󵄩

ℎ 𝑢ℎ (𝑥) − 2𝑢𝑘ℎ (𝑥) + 𝑢𝑘−1 (𝑥) + 𝐴𝑥ℎ 𝑢𝑘ℎ (𝑥) − 𝑘+1 𝜏2

󵄩󵄩 𝑁−1 󵄩 󵄩 + 󵄩󵄩󵄩󵄩{𝑓𝑘ℎ }1 󵄩󵄩󵄩󵄩 ], 󵄩 󵄩C𝜏 (𝐿 2ℎ )

= 𝑓𝑘ℎ (𝑥) + 𝑝ℎ (𝑥) , 𝑡𝑘 = 𝑘𝜏,

󵄩󵄩 ℎ 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑝 󵄩󵄩 ≤ 𝑀 [󵄩󵄩󵄩𝜑ℎ 󵄩󵄩󵄩 2 + 󵄩󵄩󵄩𝜓ℎ 󵄩󵄩󵄩 2 + 󵄩󵄩󵄩𝜉ℎ 󵄩󵄩󵄩 2 󵄩 󵄩𝐿 2ℎ 󵄩 󵄩𝑊2ℎ 󵄩 󵄩𝑊2ℎ 󵄩 󵄩𝑊2ℎ

1 ≤ 𝑘 ≤ 𝑁 − 1, 𝑥 ∈ Ωℎ , 𝑢0ℎ (𝑥) = 𝜑ℎ (𝑥) ,

̃ℎ, 𝑥∈Ω

respectively.

̃ℎ, 𝑙 = [ 𝜆 ] 𝑥∈Ω 𝜏

𝑓𝑘ℎ (𝑥) = 𝑓ℎ (𝑡𝑘 , 𝑥) ,

𝜆 𝜆 ℎ − 𝑙) V𝑙+1 (𝑥) + ( − 𝑙 − 1) V𝑙ℎ (𝑥) 𝜏 𝜏 (27)

and second order of accuracy difference scheme

(25)

̃ℎ, 𝑥∈Ω

+

𝜆 ℎ − 𝑙) (𝑢𝑙+1 (𝑥) − 𝑢𝑙ℎ (𝑥)) = 𝜉ℎ (𝑥) , 𝜏

̃ℎ, 𝑙 = [ 𝜆 ] . 𝑥∈Ω 𝜏

󵄩󵄩 ℎ 𝑁−1 󵄩󵄩 1 󵄩󵄩{𝑓 } 󵄩󵄩 ] 󵄩 󵄩 𝛼 (1 − 𝛼) 󵄩󵄩 𝑘 1 󵄩󵄩C𝜏 (𝐿 2ℎ ) (28)

̃ℎ, 𝑥∈Ω ℎ 𝑢𝑁 (𝑥) = 𝜓ℎ (𝑥) ,

1 ≤ 𝑘 ≤ 𝑁,

= 𝜑ℎ (𝑥) − 𝜉ℎ (𝑥) ,

𝑓𝑘ℎ (𝑥) = 𝑓ℎ (𝑡𝑘 , 𝑥) ,

𝑢𝑙ℎ (𝑥) + (

(26)

ℎ ℎ V𝑘+1 (𝑥) − 2V𝑘ℎ (𝑥) + V𝑘−1 (𝑥) + 𝐴𝑥ℎ V𝑘ℎ (𝑥) = 𝑓𝑘ℎ (𝑥) , 𝜏2

V0ℎ (𝑥) − (

ℎ

ℎ 𝑢𝑁 (𝑥) = 𝜓ℎ (𝑥) ,

1 ≤ 𝑘 ≤ 𝑁 − 1,

̃ℎ, 𝑥∈Ω

for 𝑢 (𝜆, 𝑥) = 𝜉 (𝑥), problem (19) is replaced by first order of accuracy difference scheme ℎ ℎ 𝑢𝑘+1 (𝑥) − 2𝑢𝑘ℎ (𝑥) + 𝑢𝑘−1 (𝑥) + 𝐴𝑥ℎ 𝑢𝑘ℎ (𝑥) 𝜏2

𝑡𝑘 = 𝑘𝜏,

V0ℎ (𝑥) − V𝑙ℎ (𝑥) = 𝜑ℎ (𝑥) − 𝜉ℎ (𝑥) ,

ℎ

−

𝑓𝑘ℎ (𝑥) = 𝑓ℎ (𝑡𝑘 , 𝑥) ,

and second order of accuracy difference scheme −

𝑥 ∈ Ωℎ

ℎ ℎ V𝑘+1 (𝑥) − 2V𝑘ℎ (𝑥) + V𝑘−1 (𝑥) + 𝐴𝑥ℎ V𝑘ℎ (𝑥) = 𝑓𝑘ℎ (𝑥) , 2 𝜏

hold, where 𝑀 is independent of 𝜏, 𝛼, ℎ, 𝜑ℎ (𝑥), 𝜓ℎ (𝑥), 𝜉ℎ (𝑥), 𝑁−1

and {𝑓𝑘ℎ (𝑥)}1

.

Abstract and Applied Analysis

5

Theorem 7. Let 𝜏 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

V0ℎ (𝑥) = −𝐺 (𝑅𝑁−𝑙 − 𝑅𝑁+𝑙 ) × (𝐼 + 𝜏𝐶) (2𝐼 + 𝜏𝐶)−1 𝐶−1 𝑁−1

× ∑ (𝑅𝑁−𝑖 − 𝑅𝑁+𝑖 ) 𝑓𝑖ℎ (𝑥) 𝜏 𝑖=1

+ 𝐺 (𝐼 − 𝑅2𝑁) (𝐼 + 𝜏𝐶)

󵄩󵄩 ℎ 𝑁−1 󵄩 󵄩󵄩 𝑘 󵄩󵄩 𝑢𝑘+1 − 2𝑢𝑘ℎ + 𝑢𝑘−1 󵄩󵄩 󵄩󵄩{ 󵄩󵄩 )} 󵄩󵄩 2 󵄩󵄩 𝜏 󵄩󵄩 󵄩󵄩C𝜏 (𝐿 2ℎ ) 1 󵄩

𝑁−1

× (2𝐼 + 𝜏𝐶)−1 𝐶−1 ∑ (𝑅|𝑙−𝑖| − 𝑅𝑙+𝑖 ) 𝑓𝑖ℎ (𝑥) 𝜏 𝑖=1

󵄩󵄩 𝑁−1 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩󵄩{𝑢𝑘ℎ }1 󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑝ℎ 󵄩󵄩󵄩 󵄩C𝜏 (𝑊2 ) 󵄩 󵄩𝐿 2ℎ 󵄩 2ℎ

(29)

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 𝑀 [󵄩󵄩󵄩󵄩𝜑ℎ 󵄩󵄩󵄩󵄩𝑊2 + 󵄩󵄩󵄩󵄩𝜓ℎ 󵄩󵄩󵄩󵄩𝑊2 + 󵄩󵄩󵄩󵄩𝜉ℎ 󵄩󵄩󵄩󵄩𝑊2 2ℎ 2ℎ 2ℎ

+ 𝐺 (𝐼 − 𝑅2𝑁) (𝜑ℎ (𝑥) − 𝜉ℎ (𝑥)) + 𝐺 (𝑅𝑁−𝑙 − 𝑅𝑁+𝑙 ) (𝜓ℎ (𝑥) − 𝜑ℎ (𝑥)) ,

󵄩󵄩 𝑁−1 󵄩 1 󵄩 + ln ( ] ) 󵄩󵄩󵄩󵄩{𝑓𝑘ℎ }1 󵄩󵄩󵄩󵄩 𝜏+ℎ 󵄩 󵄩C𝜏 (𝐿 2ℎ )

(30) for difference scheme (24), V0ℎ (𝑥) = (

holds, where 𝑀 is independent of 𝜏, 𝛼, ℎ, 𝜑ℎ (𝑥),𝜓ℎ (𝑥), 𝜉ℎ (𝑥), 𝑁−1

and {𝑓𝑘ℎ (𝑥)}1

.

Proofs of Theorems 6 and 7 are based on the symmetry property of operator 𝐴𝑥 , on Lemmas 3–5, the formulas

𝑢𝑘ℎ (𝑥) = (𝐼 − 𝑅2𝑁)

−1

−1

𝑁−1

𝑁−𝑖

× 𝐶 ∑ (𝑅

𝜆 − 𝑙 − 1) 𝐺1 (𝐼 − 𝑅2𝑁) 𝜏

𝑁−1

𝑁+𝑖

−𝑅

𝑖=1

) 𝑓𝑖ℎ

𝑖=1

𝑁−1 𝑖=1

(𝑥) 𝜏 + 𝜑 (𝑥) − 𝐴𝑥ℎ 𝜑ℎ

(𝑥) −

× (𝐼 + 𝜏𝐶) (2𝐼 + 𝜏𝐶)−1 𝐶−1 𝑁−1

+(

V0ℎ

𝐴𝑥ℎ V0ℎ

𝜆 − 𝑙) 𝐺1 (𝑅𝑁−𝑙−1 − 𝑅𝑁+𝑙+1 ) 𝜏

𝑖=1

× ∑ (𝑅|𝑘−𝑖| − 𝑅𝑘+𝑖 ) ℎ

(𝑥) 𝜏]

−(

× ∑ (𝑅𝑁−𝑖 − 𝑅𝑁+𝑖 ) 𝑓𝑖ℎ (𝑥) 𝜏

+ (𝐼 + 𝜏𝐶) (2𝐼 + 𝜏𝐶)−1 𝐶−1

𝑝 (𝑥) =

𝑖=1

× ∑ (𝑅|𝑙−𝑖| − 𝑅𝑙+𝑖 ) 𝑓𝑖ℎ (𝑥) 𝜏

− (𝑅𝑁−𝑘 − 𝑅𝑁+𝑘 ) (𝐼 + 𝜏𝐶) (2𝐼 + 𝜏𝐶)−1

ℎ

𝑁−1

× ∑ (𝑅𝑁−𝑖 − 𝑅𝑁+𝑖 ) 𝑓𝑖ℎ (𝑥) 𝜏

× (𝐼 + 𝜏𝐶) (2𝐼 + 𝜏𝐶)−1 𝐶−1

ℎ + (𝑅𝑁−𝑘 − 𝑅𝑁+𝑘 ) V𝑁 (𝑥))

×

× (𝐼 + 𝜏𝐶) (2𝐼 + 𝜏𝐶)−1 𝐶−1

−(

× [ ((𝑅𝑘 − 𝑅2𝑁−𝑘 ) V0ℎ (𝑥)

𝑓𝑖ℎ

𝜆 − 𝑙 − 1) 𝐺1 (𝑅𝑁−𝑙 − 𝑅𝑁+𝑙 ) 𝜏

(𝑥) ,

𝜆 − 𝑙) 𝐺1 (𝐼 − 𝑅2𝑁) 𝜏

× (𝐼 + 𝜏𝐶) (2𝐼 + 𝜏𝐶)−1 𝐶−1 𝑁−1

(𝑥) ,

ℎ V𝑁 (𝑥) = V0ℎ (𝑥) + 𝜓ℎ (𝑥) − 𝜑ℎ (𝑥) ,

× ∑ (𝑅|𝑙+1−𝑖| − 𝑅𝑙+1+𝑖 ) 𝑓𝑖ℎ (𝑥) 𝜏 𝑖=1

+ 𝐺1 (𝐼 − 𝑅2𝑁) (𝜑ℎ (𝑥) − 𝜉ℎ (𝑥))

6

Abstract and Applied Analysis + ((

𝜆 − 𝑙 − 1) 𝐺1 (𝑅𝑁−𝑙 − 𝑅𝑁+𝑙 ) 𝜏

+(

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

𝜆 − 𝑙) 𝐺1 (𝑅𝑁−𝑙−1 − 𝑅𝑁+𝑙+1 )) 𝜏

× (𝜓ℎ (𝑥) − 𝜑ℎ (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 .

= 𝑓 (𝑡, 𝑥) ,

𝐷ℎ 𝑢ℎ (𝑥) = 0,

̃ℎ, 𝑥∈Ω

0≤𝑥≤𝜋 V (𝑇, 𝑥) − V (𝜆, 𝑥) = (exp (−𝑇) − exp (−𝜆) + 𝑇 − 𝜆) cos (𝑥) ,

(32)

𝑥 ∈ 𝑆ℎ ,

0 ≤ 𝑥 ≤ 𝜋, V𝑥 (𝑡, 0) = V𝑥 (𝑡, 𝜋) = 0,

the following coercivity inequality holds:

2ℎ

0 ≤ 𝑡 ≤ 𝑇, (35)

𝑛 󵄩 󵄩 󵄩 󵄩 ∑󵄩󵄩󵄩󵄩(𝑢𝑘ℎ )𝑥𝑟 𝑥𝑟 ,𝑗𝑟 󵄩󵄩󵄩󵄩𝐿 ≤ 𝑀󵄩󵄩󵄩󵄩𝜔ℎ 󵄩󵄩󵄩󵄩𝐿 ,

𝑟=1

0 < 𝑥 < 𝜋, 0 < 𝑡 < 𝑇,

V (0, 𝑥) − V (𝜆, 𝑥) = (1 − exp (−𝜆) − 𝜆) cos (𝑥) ,

Theorem 8 (see [28]). For the solution of the elliptic difference problem 𝐴𝑥ℎ 𝑢ℎ (𝑥) = 𝜔ℎ (𝑥) ,

𝜕V (𝑡, 𝑥) 𝜕 𝑑2 V (𝑡, 𝑥) − ((2 + cos 𝑥) ) + V (𝑡, 𝑥) 2 𝑑𝑡 𝜕𝑥 𝜕𝑥

(33)

2ℎ

and 𝑤(𝑡, 𝑥) is the solution of the boundary value problem

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 −

𝜕𝑢 (𝑡, 𝑥) 𝜕 𝜕2 𝑢 (𝑡, 𝑥) − ((2 + cos 𝑥) ) + 𝑢 (𝑡, 𝑥) 𝜕𝑡2 𝜕𝑥 𝜕𝑥 = 𝑓 (𝑡, 𝑥) + 𝑝 (𝑥) ,

𝑑2 𝑤 (𝑡, 𝑥) 𝜕𝑤 (𝑡, 𝑥) 𝜕 − ((2 + cos 𝑥) ) + 𝑤 (𝑡, 𝑥) 𝑑𝑡2 𝜕𝑥 𝜕𝑥 = 𝑝 (𝑥) ,

0 < 𝑥 < 𝜋, 0 < 𝑡 < 𝑇,

𝑤 (0, 𝑥) = (exp (−𝜆) + 𝜆 + 1) cos (𝑥) − V (𝜆, 𝑥) , 0 ≤ 𝑥 ≤ 𝜋, 𝑤 (𝑇, 𝑥) = (exp (−𝜆) + 𝜆 + 1) cos (𝑥) − V (𝜆, 𝑥) , 0 ≤ 𝑥 ≤ 𝜋, 𝑤𝑥 (𝑡, 0) = 𝑤𝑥 (𝑡, 𝜋) = 0,

0 < 𝑥 < 𝜋, 0 < 𝑡 < 𝑇,

0 ≤ 𝑡 ≤ 𝑇. (36)

𝑓 (𝑡, 𝑥) = − exp (−𝑡) cos (𝑥) + (exp (−𝑡) + 𝑡) (3 cos (𝑥) + cos (2𝑥)) , 𝑢 (0, 𝑥) = 2 cos (𝑥) ,

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,

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 ≤ 𝑥 ≤ 𝜋,

0 ≤ 𝑡 ≤ 𝑇, 𝜆 =

3𝑇 . 5

(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).

[0, 𝑇]𝜏 × [0, 𝜋]ℎ = {(𝑡𝑘 , 𝑥𝑛 ) : 𝑡𝑘 = 𝑘𝜏, 𝑘 = 0, . . . , 𝑁, 𝑥𝑛 = 𝑛ℎ, 𝑛 = 0, . . . , 𝑀} .

(37)

Abstract and Applied Analysis

7

Applying (21), we obtain difference schemes of the first order of accuracy in 𝑡 and the second order of accuracy in 𝑥 𝑘 𝑘 V𝑛+1 − 2V𝑛𝑘 + V𝑛−1 V𝑛𝑘+1 − 2V𝑛𝑘 + V𝑛𝑘−1 + (2 + cos (𝑥 )) 𝑛 𝜏2 ℎ2

− sin (𝑥𝑛 )

𝑘 V𝑛+1

𝜃𝑛𝑘 = −𝑓 (𝑡𝑘 , 𝑥𝑛 ) ,

− 2ℎ

𝑘 V𝑛−1

𝐴 𝑛 V𝑛+1 + 𝐵𝑛 V𝑛 + 𝐶𝑛 V𝑛−1 = 𝐼𝜃𝑛𝑘 ,

− V𝑛𝑘 = 𝜃𝑛𝑘 ,

V0 = V 1 ,

𝑘 = 0, . . . , 𝑁,

V𝑛0 − V𝑛𝑙 = (1 − exp (−𝜆) − 𝜆) cos (𝑥𝑛 ) ,

𝑛 = 0, . . . , 𝑀,

V𝑛𝑁 − V𝑛𝑙 = (exp (−𝑡𝑁) − exp (−𝜆) + 𝑡𝑁 − 𝜆) cos (𝑥𝑛 ) , 𝜆 𝑛 = 0, . . . , 𝑀, 𝑙 = [ ] , 𝜏

(38)

for the approximate solutions of the nonlocal boundary value problem (35), and 𝑘 𝑘 𝑤𝑛+1 − 2𝑤𝑛𝑘 + 𝑤𝑛−1 𝑤𝑛𝑘+1 − 2𝑤𝑛𝑘 + 𝑤𝑛𝑘−1 +(2 + cos (𝑥 )) 𝑛 𝜏2 ℎ2

= −𝑝𝑛 ,

𝑘 𝑘 𝑤𝑛+1 − 𝑤𝑛−1 − 𝑤𝑛𝑘 2ℎ

𝑘 = 1, . . . , 𝑁 − 1,

𝑘 𝑘 − 𝑤𝑀−1 = 0, 𝑤0𝑘 − 𝑤1𝑘 = 𝑤𝑀

𝑘 = 0, . . . , 𝑁,

= (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) −

V𝑛𝑙 ,

𝜆 𝑛 = 0, . . . , 𝑀, 𝑙 = [ ] , 𝜏 𝑤𝑛𝑁 = (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) − V𝑛𝑙 , 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: 𝑝𝑛 = −

0 [0 [ [0 [ [0 [ [ 𝐴 𝑛 = [ ... [ [0 [ [0 [ [0 [0

0 𝑎𝑛 0 0 .. .

0 0 𝑎𝑛 0 .. .

0 0 0 𝑎𝑛 .. .

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 𝑑 𝑏𝑛 𝑑 .. .

0 0 𝑑 𝑏𝑛 .. .

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. .

−1 0 0 0 .. .

0 [0 [ [0 [ [0 [ [ 𝐶𝑛 = [ ... [ [0 [ [0 [ [0 [0

0 𝑐𝑛 0 0 .. .

0 0 𝑐𝑛 0 .. .

(2 + cos (𝑥𝑛 )) 0 ((𝜑𝑛+1 − V𝑛+1 ) − 2 (𝜑𝑛 − V𝑛0 ) ℎ2

0 + (𝜑𝑛−1 − V𝑛−1 )) +

× ((𝜑𝑛+1 −

(41)

V𝑀 = V𝑀−1 .

Here, 𝐼 is the (𝑁 + 1) × (𝑁 + 1) identity matrix, 𝐴 𝑛 , 𝐵𝑛 , 𝐶𝑛 are (𝑁 + 1) × (𝑁 + 1) square matrices, and 𝜃𝑛 is a (𝑁 + 1) × 1 column matrix which are defined by

1 [𝑑 [ [0 [ [0 [ [ 𝐵𝑛 = [ ... [ [0 [ [0 [ [0 [0

𝑝𝑛 = 𝑝 (𝑥𝑛 ) , 𝑛 = 1, . . . , 𝑀 − 1,

𝑤𝑛0

𝑛 = 1, . . . , 𝑀 − 1,

𝑘 = 1, . . . , 𝑁 − 1, 𝑛 = 1, . . . , 𝑀 − 1,

𝑘 𝑘 − V𝑀−1 = 0, V0𝑘 − V1𝑘 = V𝑀

− sin (𝑥𝑛 )

We can rewrite difference scheme (38) in the matrix form

0 V𝑛+1 )

+ (𝜑𝑛 − V𝑛0 ) ,

sin (𝑥𝑛 ) 2ℎ

− (𝜑𝑛−1 −

0 0 0 0

0 0 0 0

𝑎𝑛 = (40)

𝑏𝑛 = −

0 V𝑛−1 ))

𝑛 = 1, . . . , 𝑀 − 1.

0 𝑏𝑛 𝑑 0 .. .

0 0 0 0

𝑐𝑛 =

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 0 0

0 0 0 0 . ⋅ ⋅ ⋅ .. ⋅ ⋅ ⋅ 𝑎𝑛 ⋅⋅⋅ 0 ⋅⋅⋅ 0 ⋅⋅⋅ 0

0 0 0 −1

0 0 0 𝑐𝑛 .. . 0 0 0 0

0 0 0 0 .. .

0 𝑎𝑛 0 0

0 0 0 0 .. .

0 0 𝑎𝑛 0

0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] 0] 0]

0 0 0 0 . ⋅ ⋅ ⋅ .. ⋅ ⋅ ⋅ 𝑏𝑛 ⋅⋅⋅ 𝑑 ⋅⋅⋅ 0 ⋅⋅⋅ 0

0 0 0 0 .. .

0 0 0 0 .. .

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 0 0 . ⋅ ⋅ ⋅ .. ⋅ ⋅ ⋅ 𝑐𝑛 ⋅⋅⋅ 0 ⋅⋅⋅ 0 ⋅⋅⋅ 0

0 0 0 0 .. .

0 𝑐𝑛 0 0

𝑑 𝑏𝑛 𝑑 0

0 𝑑 𝑏𝑛 0

0 0 0 0 .. .

0 0 𝑐𝑛 0

(42)

0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] 𝑑] 1]

0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] 0] 0]

(43)

2 + cos (𝑥𝑛 ) sin (𝑥𝑛 ) − , ℎ2 2ℎ 2 2 (2 + cos (𝑥𝑛 )) − − 1, 𝜏2 ℎ2

2 + cos (𝑥𝑛 ) sin (𝑥𝑛 ) + , ℎ2 2ℎ

𝑑=

(44) 1 , 𝜏2

8

Abstract and Applied Analysis 𝜃𝑛0 [ ] 𝜃𝑛 = [ ... ] ,

𝐴 𝑛 and 𝐶𝑛 are defined by (42) and (43) and (𝑁+1)×1 column matrix 𝜂𝑛 is defined by

𝑁

[𝜃𝑛 ] 𝜂𝑛0 [ .. ] 𝜂𝑛 = [ . ] , 𝑁 [𝜂𝑛 ]

𝜃𝑛0 = (1 − exp (−𝜆) − 𝜆) cos (𝑥𝑛 ) , 𝜃𝑛𝑁 = (exp (−𝑡𝑁) − exp (−𝜆) + 𝑡𝑁 − 𝜆) cos (𝑥𝑛 ) , 𝑛 = 1, . . . , 𝑀 − 1, 𝜃𝑛𝑘 = −𝑓 (𝑡𝑘 , 𝑥𝑛 ) ,

𝜂𝑛0 = (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) − V𝑛𝑙 ,

𝑘 = 1, . . . , 𝑁 − 1, 𝑛 = 1, . . . , 𝑀 − 1,

𝜂𝑛𝑁 = (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) − V𝑛𝑙 ,

V𝑠0 [ .. ] V𝑠 = [ . ]

,

𝑠 = 𝑛 − 1, 𝑛, 𝑛 + 1.

𝜂𝑛𝑘 = −𝑝𝑛 ,

𝑁 [V𝑠 ](𝑁+1)×1

(45)

V𝑛 = 𝛼𝑛+1 V𝑛+1 + 𝛽𝑛+1 ,

𝑛 = 𝑀 − 1, . . . , 1,

(46)

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

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

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 𝜌 𝜌 (𝑥𝑛+1 ) − 𝜌 (𝑥𝑛−1 ) − 𝜌󸀠 (𝑥𝑛 ) = 𝑂 (ℎ2 ) , 2ℎ 𝜌 (𝑥𝑛+1 ) − 2𝜌 (𝑥𝑛 ) + 𝜌 (𝑥𝑛−1 ) − 𝜌󸀠󸀠 (𝑥𝑛 ) = 𝑂 (ℎ2 ) , ℎ2 10𝜌 (0) − 15𝜌 (ℎ) + 6𝜌 (2ℎ) − 𝜌 (3ℎ) − 𝜌󸀠󸀠󸀠 (0) = 𝑂 (ℎ2 ) , ℎ3 −3𝜌 (0) + 4𝜌 (ℎ) − 𝜌 (2ℎ) − 𝜌󸀠 (0) = 𝑂 (ℎ2 ) , 2ℎ

𝐴 𝑛 𝑤𝑛+1 + 𝐸𝑛 𝑤𝑛 + 𝐶𝑛 𝑤𝑛−1 = 𝐼𝜂𝑛𝑘 , 𝑛 = 1, . . . , 𝑀 − 1, 𝑤0 = 𝑤1 ,

𝑠 = 𝑛 − 1, 𝑛, 𝑛 + 1. (50)

𝛼𝑛+1 = −(𝐵𝑛 + 𝐶𝑛 𝛼𝑛 ) 𝐴 𝑛 , −1

,

𝑁 [𝑤𝑠 ](𝑁+1)×1

−1

𝛽𝑛+1 = −(𝐵𝑛 + 𝐶𝑛 𝛼𝑛 ) (𝐼𝜃𝑛 − 𝐶𝑛 𝛽𝑛 ) ,

𝑘 = 1, . . . , 𝑁 − 1, 𝑛 = 1, . . . , 𝑀 − 1,

𝑤𝑠0 [ ] 𝑤𝑠 = [ ... ]

For solving (41) we use the modified Gauss elimination method (see [29]). Namely, we seek solution of (41) by the formula

𝑛 = 1, . . . , 𝑀 − 1,

10𝜌 (𝜋) − 15𝜌 (𝜋 − ℎ) + 6𝜌 (𝜋 − 2ℎ) − 𝜌 (𝜋 − 3ℎ) ℎ3

(48)

− 𝜌󸀠󸀠󸀠 (𝜋) = 𝑂 (ℎ2 ) ,

𝑤𝑀 = 𝑤𝑀−1 .

−3𝜌 (𝜋) + 4𝜌 (𝜋 − ℎ) − 𝜌 (𝜋 − 2ℎ) − 𝜌󸀠 (𝜋) = 𝑂 (ℎ2 ) , 2ℎ

Here, 1 [𝑑 [ [0 [ [0 [ [ 𝐸𝑛 = [ ... [ [0 [ [0 [ [0 [0

0 𝑏𝑛 𝑑 0 .. . 0 0 0 0

0 𝑑 𝑏𝑛 𝑑 .. . 0 0 0 0

0 0 𝑑 𝑏𝑛 .. . 0 0 0 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 0 0 . d .. ⋅ ⋅ ⋅ 𝑏𝑛 ⋅⋅⋅ 𝑑 ⋅⋅⋅ 0 ⋅⋅⋅ 0

0 0 0 0 .. .

𝑑 𝑏𝑛 𝑑 0

0 0 0 0 .. .

0 𝑑 𝑏𝑛 0

0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] 𝑑] 1]

we get (49) 𝑘 𝑘 V𝑛+1 − 2V𝑛𝑘 + V𝑛−1 V𝑛𝑘+1 − 2V𝑛𝑘 + V𝑛𝑘−1 + (2 + cos (𝑥 )) 𝑛 𝜏2 ℎ2 𝑘 𝑘 V𝑛+1 − V𝑛−1 − sin (𝑥𝑛 ) − V𝑛𝑘 = 𝜃𝑛𝑘 , 2ℎ

(51)

Abstract and Applied Analysis 𝜃𝑛𝑘 = −𝑓 (𝑡𝑘 , 𝑥𝑛 ) ,

9

𝑘 = 1, . . . , 𝑁 − 1,

𝑛 = 1, . . . , 𝑀 − 1,

𝐴 𝑛 V𝑛+1 + 𝐵𝑛 V𝑛 + 𝐶𝑛 V𝑛−1 = 𝐼𝜃𝑛𝑘 ,

− 3V0𝑘 + 4V1𝑘 − V2𝑘 𝑘 𝑘 𝑘 = −3V𝑀 + 4V𝑀−1 − V𝑀−2 = 0,

where 𝐴 𝑛 , 𝐶𝑛 are defined by (42), (43), (44), and 𝐵𝑛 is defined by

𝜆 𝜆 − 𝑙 − 1) V𝑛𝑙 − ( − 𝑙) V𝑛𝑙+1 𝜏 𝜏

= (1 − exp (−𝜆) − 𝜆) cos (𝑥𝑛 ) ,

1 [𝑑 [ [0 [ [0 [ [ 𝐵𝑛 = [ ... [ [0 [ [0 [ [0 [0

𝑛 = 0, . . . , 𝑀,

𝜆 𝜆 − 𝑙 − 1) V𝑛𝑙 − ( − 𝑙) V𝑛𝑙+1 𝜏 𝜏

= (exp (−𝑡𝑁) − exp (−𝜆) + 𝑡𝑁 − 𝜆) cos (𝑥𝑛 ) , 𝑛 = 0, . . . , 𝑀,

0 𝑏𝑛 𝑑 0 .. . 0 0 0 0

0 𝑑 𝑏𝑛 𝑑 .. . 0 0 0 0

0 0 𝑑 𝑏𝑛 .. . 0 0 0 0

(52) 𝑏𝑛 = − difference scheme for nonlocal problem (35), and 𝑤𝑛𝑘+1

−

2𝑤𝑛𝑘 𝜏2

+

𝑤𝑛𝑘−1

+ (2 + cos (𝑥𝑛 ))

𝑘 𝑤𝑛+1

−

2𝑤𝑛𝑘 ℎ2

+

𝑘 𝑤𝑛−1

𝑝𝑛 = 𝑝 (𝑥𝑛 ) ,

𝑑=

1 , 𝜏2

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. .

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 0 0 .. .

0 0 0 0

𝑦 0 0 0 .. .

𝑧 0 0 0 .. .

0 0 0 𝑦

0 0 0 𝑧

0 0 0 0 .. .

0 0 0 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 0 0 .. .

𝑏𝑛 𝑑 0 0

0 0 0 0 .. .

0 0 0 0 .. .

𝑑 𝑏𝑛 𝑑 0

0 𝑑 𝑏𝑛 0

0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] 𝑑] 1]

2 2 (2 + cos (𝑥𝑛 )) − − 1, 𝜏2 ℎ2

𝑦=(

𝜆 − 𝑙 − 1) , 𝜏

𝑧 = −(

𝜆 − 𝑙) . 𝜏

(55)

We seek solution of (54) by the formula

𝑘 𝑤𝑘 − 𝑤𝑛−1 − sin (𝑥𝑛 ) 𝑛+1 = −𝑝𝑛 , 2ℎ

𝑘 = 1, . . . , 𝑁 − 1,

(54)

−3V𝑀 + 4V𝑀−1 − V𝑀−2 = 0,

𝑘 𝑘 𝑘 𝑘 = 10V𝑀 − 15V𝑀−1 + 6V𝑀−2 − V𝑀−3 = 0,

V𝑛𝑁 + (

𝑛 = 1, . . . , 𝑀 − 1,

−3V0 + 4V1 − V2 = 0,

𝑘 = 0, . . . , 𝑁,

10V0𝑘 − 15V1𝑘 + 6V2𝑘 − V3𝑘

V𝑛0 + (

By difference scheme (52), we write in matrix form

V𝑛 = 𝛼𝑛 V𝑛+1 + 𝛽𝑛 V𝑛+2 + 𝛾𝑛 ,

𝑛 = 𝑀 − 2, . . . , 0,

(56)

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 :

𝑛 = 1, . . . , 𝑀 − 1,

𝑘 𝑘 𝑘 + 4𝑤𝑀−1 − 𝑤𝑀−2 = 0, − 3𝑤0𝑘 + 4𝑤1𝑘 − 𝑤2𝑘 = −3𝑤𝑀

−1

𝛼𝑛 = −(𝐵𝑛 + 𝐶𝑛 𝛼𝑛−1 ) (𝐴 𝑛 + 𝐶𝑛 𝛽𝑛−1 ) ,

𝑘 = 0, . . . , 𝑁,

𝛽𝑛 = 0,

10𝑤0𝑘 − 15𝑤1𝑘 + 6𝑤2𝑘 − 𝑤3𝑘 =

𝑘 10𝑤𝑀

−

𝑘 15𝑤𝑀−1

+

𝑘 6𝑤𝑀−2

−1

−

𝑘 𝑤𝑀−3

𝛾𝑛 = −(𝐵𝑛 + 𝐶𝑛 𝛼𝑛−1 ) (𝐼𝜃𝑛 − 𝐶𝑛 𝛾𝑛−1 ) ,

= 0,

𝑛 = 1, . . . , 𝑀 − 1, (57)

where 𝑤𝑛0 = (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) +(

𝜆 𝜆 − 𝑙 − 1) V𝑛𝑙 − ( − 𝑙) V𝑛𝑙+1 , 𝜏 𝜏

𝑛 = 0, . . . , 𝑀,

𝑤𝑛𝑁 = (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) 𝜆 𝜆 + ( − 𝑙 − 1) V𝑛𝑙 − ( − 𝑙) V𝑛𝑙+1 , 𝜏 𝜏 𝜉𝑛 = 𝜉 (𝑥𝑛 ) ,

1 𝛽0 = − 𝐼, 3

8 𝛼1 = 𝐼, 5

3 𝛽1 = − 𝐼, 5

𝛼𝑀−2 = 4𝐼,

𝛽𝑀−2 = −3𝐼,

8 𝛼𝑀−3 = 𝐼, 3

5 𝛽𝑀−3 = − 𝐼, 3

(58)

and 𝛾0 , 𝛾1 , 𝛾𝑀−2 , 𝛾𝑀−3 are the (𝑁 + 1) × 1 zero column vector. For V𝑀 and V𝑀−1 we have

𝑛 = 0, . . . , 𝑀, (53)

difference scheme for boundary value problem (36).

4 𝛼0 = 𝐼, 3

−1

−1 −1 V𝑀 = (𝑄11 − 𝑄12 𝑄22 𝑄21 ) (𝐺1 − 𝑄12 𝑄22 𝐺2 ) , −1 (𝐺2 − 𝑄21 V𝑀) , V𝑀−1 = 𝑄22

(59)

10

Abstract and Applied Analysis

where

Table 1: Error analysis for nonlocal problem.

𝑄11 = −3𝐴 𝑀−2 − 8𝐵𝑀−2 − 8𝐶𝑀−2 𝛼𝑀−3 − 3𝐶𝑀−2 𝛽𝑀−3 , 𝑄12 = 4𝐴 𝑀−2 + 9𝐵𝑀−2 + 9𝐶𝑀−2 𝛼𝑀−3 + 4𝐶𝑀−2 𝛽𝑀−3 ,

Difference scheme (38) Difference scheme (52)

𝑄21 = −3𝐵𝑀−1 − 8𝐶𝑀−1 , 𝑄22 = 𝐴 𝑀−1 + 4𝐵𝑀−1 + 9𝐶𝑀−1 , 𝐺1 = 𝐼𝜃𝑀−2 − 𝐶𝑀−2 𝛾𝑀−3 ,

𝐺2 = 𝐼𝜃𝑀−1 .

We can rewrite difference scheme (53) in the matrix form 𝑛 = 1, . . . , 𝑀 − 1,

−3𝑤0 + 4𝑤1 − 𝑤2 = 0,

(61)

−3𝑤𝑀 + 4𝑤𝑀−1 − 𝑤𝑀−2 = 0,

Difference scheme (38), (40) Difference scheme (52), (40)

where 𝐴 𝑛 , 𝐸𝑛 , 𝐶𝑛 are defined by (42), (49), (43), and (44) and 𝜂𝑛 is defined by 𝜂𝑛0 [ ] 𝜂𝑛 = [ ... ] , 𝜂𝑛0 = (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) 𝜆 𝜆 − 𝑙 − 1) V𝑛𝑙 − ( − 𝑙) V𝑛𝑙+1 , 𝜏 𝜏

𝜂𝑛𝑁 = (exp (−𝜆) + 𝜆 + 1) cos (𝑥𝑛 ) + ( −( 𝜂𝑛𝑘 = −𝑝𝑛 ,

𝜆 − 𝑙) V𝑛𝑙+1 , 𝜏

𝜆 − 𝑙 − 1) V𝑛𝑙 𝜏

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 𝑀−1

󵄨 󵄨2 𝑁 𝐸V𝑀 = max ( ∑ 󵄨󵄨󵄨󵄨V (𝑡𝑘 , 𝑥𝑛 ) − V𝑛𝑘 󵄨󵄨󵄨󵄨 ℎ) 1≤𝑘≤𝑁−1

𝑁 𝐸𝑢𝑀

1/2

󵄨 󵄨2 = max ( ∑ 󵄨󵄨󵄨󵄨𝑢 (𝑡𝑘 , 𝑥𝑛 ) − 𝑢𝑛𝑘 󵄨󵄨󵄨󵄨 ℎ) 1≤𝑘≤𝑁−1 𝑛=1

𝐸𝑝𝑀

,

𝑛=1

𝑀−1

𝑀−1

󵄨 󵄨2 = ( ∑ 󵄨󵄨󵄨𝑝 (𝑥𝑛 ) − 𝑝𝑛 󵄨󵄨󵄨 ℎ) 𝑛=1

1/2

.

𝑁=𝑀= 160

0.30522

0.14933

0.073953

0.036814

0.024714

0.0031054

4.36 × 10−4 7.52 × 10−5

𝑁=𝑀= 20

𝑁=𝑀= 40

𝑁=𝑀= 80

𝑁=𝑀= 160

0.57878

0.33755

0.20387

0.12905

0.058201

0.010646

0.0020228

4.03 × 10−4

𝑁=𝑀= 20

𝑁=𝑀= 40

𝑁=𝑀= 80

𝑁=𝑀= 160

0.088225

0.038586

0.01815

0.008818

0.017034

0.0020225

2.47 × 10−4 3.08 × 10−5

(62)

𝑛 = 0, . . . , 𝑀,

𝑘 = 1, . . . , 𝑁 − 1,

𝑁=𝑀= 80

Table 3: Error analysis for u.

Difference scheme (38), (40), (39) Difference scheme (52), (40), (53)

𝑁 [𝜂𝑛 ]

+(

𝑁=𝑀= 40

Table 2: Error analysis for p.

(60)

𝐴 𝑛 𝑤𝑛+1 + 𝐸𝑛 𝑤𝑛 + 𝐶𝑛 𝑤𝑛−1 = 𝐼𝜂𝑛𝑘 ,

𝑁=𝑀= 20

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.

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

Abstract and Applied Analysis 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.

References [1] A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, vol. 231 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000. [2] A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse and Ill-posed Problems Series, Walter de Gruyter GmbH & Company, Berlin, Germany, 2007. [3] V. V. Soloviev, “Inverse problems of source determination for the Poisson equation on the plane,” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, vol. 44, no. 5, pp. 862–871, 2004 (Russian). [4] D. G. Orlovski˘ı, “Inverse Dirichlet problem for an equation of elliptic type,” Differential Equations, vol. 44, no. 1, pp. 124–134, 2008. [5] D. Orlovsky and S. Piskarev, “On approximation of inverse problems for abstract elliptic problems,” Journal of Inverse and Ill-Posed Problems, vol. 17, no. 8, pp. 765–782, 2009. [6] A. Ashyralyev and A. S. Erdogan, “Well-posedness of the inverse problem of a multidimensional parabolic equation,” Vestnik of Odessa National University, Mathematics and Mechanics, vol. 15, no. 18, pp. 129–135, 2010. [7] A. Ashyralyev, “On the problem of determining the parameter of a parabolic equation,” Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1397–1408, 2011. [8] A. Ashyralyev and A. S. Erdo˘gan, “On the numerical solution of a parabolic inverse problem with the Dirichlet condition,” International Journal of Mathematics and Computation, vol. 11, no. J11, pp. 73–81, 2011. [9] C. Ashyralyyev, A. Dural, and Y. Sozen, “Finite difference method for the reverse parabolic problem,” Abstract and Applied Analysis, vol. 2012, Article ID 294154, 17 pages, 2012. [10] C. Ashyralyyev and O. Demirdag, “The difference problem of obtaining the parameter of a parabolic equation,” Abstract and Applied Analysis, vol. 2012, Article ID 603018, 14 pages, 2012. [11] Ch. Ashyralyyev and M. Dedeturk, “A finite difference method for the inverse elliptic problem with the Dirichlet condition,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 132–155, 2013. [12] A. Ashyralyev and M. Urun, “Determination of a control parameter for the Schrodinger equation,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 156–166, 2013. [13] M. Dehghan, “Determination of a control parameter in the twodimensional diffusion equation,” Applied Numerical Mathematics, vol. 37, no. 4, pp. 489–502, 2001.

11 [14] V. G. Romanov, “A three-dimensional inverse problem of viscoelasticity,” Doklady Mathematics, vol. 84, no. 3, pp. 833– 836, 2011. [15] S. I. Kabanikhin, “Definitions and examples of inverse and illposed problems,” Journal of Inverse and Ill-Posed Problems, vol. 16, no. 4, pp. 317–357, 2008. ´ [16] Y. S. Eidel’man, “An inverse problem for an evolution equation,” Mathematical Notes, vol. 49, no. 5, pp. 535–540, 1991. [17] K. Sakamoto and M. Yamamoto, “Inverse heat source problem from time distributing overdetermination,” Applicable Analysis, vol. 88, no. 5, pp. 735–748, 2009. [18] A. V. Gulin, N. I. Ionkin, and V. A. Morozova, “On the stability of a nonlocal two-dimensional finite-difference problem,” Differential Equations, vol. 37, no. 7, pp. 970–978, 2001. [19] G. Berikelashvili, “On a nonlocal boundary-value problem for two-dimensional elliptic equation,” Computational Methods in Applied Mathematics, vol. 3, no. 1, pp. 35–44, 2003. [20] M. P. Sapagovas, “Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions,” Differential Equations, vol. 44, no. 7, pp. 1018–1028, 2008. [21] A. Ashyralyev, “A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 557–573, 2008. [22] D. Orlovsky and S. Piskarev, “The approximation of BitzadzeSamarsky type inverse problem for elliptic equations with Neumann conditions,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 118–131, 2013. [23] A. Ashyralyev and F. S. O. Tetikoglu, “FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 454831, 22 pages, 2012. [24] A. Ashyralyev and E. Ozturk, “The numerical solution of the Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition,” Abstract and Applied Analysis, vol. 2013, Article ID 730804, 13 pages, 2012. [25] A. Ashyralyev, “Well-posedness of the difference schemes for elliptic equations in 𝐶𝜏𝛽,𝛾 (𝐸) spaces,” Applied Mathematics Letters, vol. 22, no. 3, pp. 390–395, 2009. [26] S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, Russia, 1966. [27] A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkh¨auser, Basel, Switzerland, 2004. [28] P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975. [29] A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations, vol. 2, Birkh¨auser, Basel, Switzerland, 1989.

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