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 Β© 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., [3β17] 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 [18β25] (see also the references therein). In [4], Orlovsky proved existence and uniqueness theorems for the inverse problem of finding a function π’ and an element π for the elliptic equation in an arbitrary Hilbert space π» with the self-adjoint positive definite operator A: βπ’π‘π‘ (π‘) + π΄π’ (π‘) = π (π‘) + π, π’ (0) = π,
π’ (π) = π,
0 < π‘ < π,
π’ (π) = π,
0 < π < π.
(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 Ξ© = (0, β) Γ β
β
β
Γ (0, β) be the open cube in the πdimensional Euclidean space with boundary π and Ξ© = Ξ© βͺ π. In [0, π] Γ Ξ©, we consider the inverse problem of finding functions π’(π‘, π₯) and π(π₯) for the multidimensional elliptic equation π
βπ’π‘π‘ (π‘, π₯) β β (ππ (π₯) π’π₯π )π₯ + πΏπ’ (π‘, π₯) = π (π‘, π₯) + π (π₯) , π
π=1
π₯ = (π₯1 , . . . , π₯π ) β Ξ©, π’ (0, π₯) = π (π₯) ,
0 < π‘ < π,
π’ (π, π₯) = π (π₯) ,
π’ (π, π₯) = π (π₯) , π₯ β Ξ©,
ππ’ (π‘, π₯) = 0, ππ β
π₯ β π, 0 β€ π‘ β€ π. (2)
Here, 0 < π < π and πΏ > 0 are known numbers, ππ (π₯) (π₯ β Ξ©), π(π₯), π(π₯), π(π₯) (π₯ β Ξ©), and π(π‘, π₯) (π‘ β (0, π), π₯ β Ξ©) are given smooth functions, and also ππ (π₯) β₯ π > 0 (π₯ β Ξ©). 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
π‘ 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.
σ΅©σ΅© σΈ σΈ σ΅©σ΅© σ΅© σ΅© σ΅©σ΅©π’ σ΅©σ΅© πΌ,πΌ + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΏ 2 (Ξ©) 2 σ΅© σ΅©C0π (πΏ 2 (Ξ©)) + βπ’βCπΌ,πΌ 0π (π2 (Ξ©)) β€ π[
π
π΄π₯ π’ (π₯) = ββ(ππ (π₯) π’π₯π )π₯ + πΏπ’ (π₯)
(3)
π
defines a self-adjoint positive definite operator π΄π₯ acting on πΏ 2 (Ξ©) with the domain π·(π΄π₯ ) = {π’(π₯) β π22 (Ξ©), ππ’/ππβ = 0 on π}. Let π» be the Hilbert space πΏ 2 (Ξ©). By using abstract Theorems 2.1 and 2.2 of paper [11], we get the following theorems about well-posedness of problem (2). π₯
Theorem 1. Assume that π΄ is defined by formula (3), π, π, π β π·(π΄π₯ ). Then, for the solutions (π’, π) of inverse boundary value problem (2), the stability estimates are satisfied:
σ΅© σ΅© σ΅© σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©π2 (Ξ©) + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©π2 (Ξ©) ] 2
πΆ = (1/2)(ππ΄π₯ + β4π΄π₯ + π2 (π΄π₯ )2 ) is a self-adjoint positive definite operator and π
= (πΌ + ππΆ)β1 which is defined on the whole space π» = πΏ 2 (Ξ©) is a bounded operator. Here, πΌ is the identity operator. Now we present the following lemmas, which will be used later. Lemma 3. The following estimates are satisfied (see [27]): σ΅©σ΅© π σ΅©σ΅© 1/2 βπ σ΅©σ΅©π
σ΅©σ΅© σ΅© σ΅©π» β π» β€ π(1 + πΏ π) ,
(7)
Lemma 4. For 1 β€ π β€ π β 1 and for the operator π = π
2π + π
π β π
2πβπ + π
πβπ β π
π+π , the operator πΌ β π has an inverse πΊ = (πΌ β π)β1 and the estimate
σ΅© σ΅© σ΅© σ΅© + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΏ 2 (Ξ©) + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΆ(πΏ 2 (Ξ©)) ] , σ΅© π₯ σ΅© σ΅© π₯ σ΅© σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©πσ΅©σ΅©πΏ 2 (Ξ©) β€ π [σ΅©σ΅©σ΅©π΄ πσ΅©σ΅©σ΅©πΏ 2 (Ξ©) + σ΅©σ΅©σ΅©π΄ πσ΅©σ΅©σ΅© πΏ 2 (Ξ©) 1 σ΅©σ΅© σ΅©σ΅© + ], σ΅©πσ΅© πΌ,πΌ πΌ (1 β πΌ) σ΅© σ΅©C0π (πΏ 2 (Ξ©)) (4)
where π is independent of πΌ, π(π₯), π(π₯), π(π₯), and π(π‘, π₯). Here, CπΌ,πΌ 0π (πΏ 2 (Ξ©)) is the space obtained by completion of the space of all smooth πΏ 2 (Ξ©)-valued functions π on [0, π] with the norm
βπΊβπ» β π» β€ π
(8)
is satisfied, where π does not depend on π. Proof of Lemma 4 is based on Lemma 3 and representation π = πΌ β π
2π β π
π + π
2πβπ β π
πβπ + π
π+π = (πΌ β π
πβπ ) (πΌ β π
π) (πΌ β π
π ) .
(9)
Lemma 5. For 1 β€ π β€ π β 1 and for the operator
σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©πσ΅©σ΅©CπΌ,πΌ (πΏ 2 (Ξ©)) 0π σ΅© σ΅© = σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©C(πΏ 2 (Ξ©)) sup
πΏ > 0,
π σ΅©σ΅© π σ΅©σ΅© σ΅©σ΅©πΆπ
σ΅©σ΅© σ΅© σ΅©π» β π» β€ ππ , π β₯ 1, σ΅© σ΅©σ΅© σ΅©σ΅©(πΌ β π
2π)β1 σ΅©σ΅©σ΅© σ΅©σ΅©π» β π» β€ π. σ΅©σ΅©
σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© β€ [π σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΏ 2 (Ξ©) + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΏ 2 (Ξ©) + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΏ 2 (Ξ©) + σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©πΆ(πΏ 2 (Ξ©)) ] , σ΅©σ΅© π₯ β1 σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© σ΅©σ΅©(π΄ ) πσ΅©σ΅© σ΅© σ΅©πΏ 2 (Ξ©) β€ π [σ΅©σ΅©πσ΅©σ΅©πΏ 2 (Ξ©)
0β€π‘ 0, where π is a fixed positive integer. Applying the approximate formulas π π’ (π, π₯) = π’ ([ ] π, π₯) + π (π) , π β
β
(23)
π π Γ (π’β ([ ] π + π, π₯) β π’β ([ ] π, π₯)) π π
β
β
π₯ β Ξ©β ,
π π π π’β (π, π₯) = π’β ([ ] π, π₯) + ( β [ ]) π π π
+ π (π2 ) ,
For approximate solution of nonlocal problem (21), we have first order of accuracy difference scheme
=
ππβ
π₯ β Ξ©β ,
β Vπ (π₯) β Vπβ (π₯) = πβ (π₯) β πβ (π₯) ,
Μβ π₯βΞ©
ππβ (π₯) = πβ (π‘π , π₯) , π‘π = ππ,
(π₯) + π (π₯) ,
β Vπ (π₯) β (
π‘π = ππ,
1 β€ π β€ π β 1,
π₯ β Ξ©β ,
π’0β (π₯) = πβ (π₯) ,
Μβ, π₯βΞ©
π’πβ (π₯) = πβ (π₯) ,
Μβ, π₯βΞ©
(24)
π₯ β Ξ©β ,
π π β β π) Vπ+1 (π₯) + ( β π β 1) Vπβ (π₯) π π
= πβ (π₯) β πβ (π₯) ,
Theorem 6. Let π and |β| = ββ12 + β
β
β
+ βπ2 be sufficiently πβ1
small positive numbers. Then, for the solutions ({π’πβ }πΎβ1 , πβ ) of difference schemes (24) and (25) the stability estimates σ΅©σ΅© β πβ1 σ΅©σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© σ΅©σ΅©{π’ } σ΅©σ΅© β€ π [σ΅©σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅©σ΅©πΏ + σ΅©σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅©σ΅©πΏ + σ΅©σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅©σ΅©πΏ σ΅©σ΅© π 1 σ΅©σ΅© 2β 2β 2β σ΅©Cπ (πΏ 2β ) σ΅©
β π’β (π₯) β 2π’πβ (π₯) + π’πβ1 (π₯) + π΄π₯β π’πβ (π₯) β π+1 π2
σ΅©σ΅© πβ1 σ΅© σ΅© + σ΅©σ΅©σ΅©σ΅©{ππβ }1 σ΅©σ΅©σ΅©σ΅© ], σ΅© σ΅©Cπ (πΏ 2β )
= ππβ (π₯) + πβ (π₯) , π‘π = ππ,
σ΅©σ΅© β σ΅©σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© σ΅©σ΅©π σ΅©σ΅© β€ π [σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅© 2 + σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅© 2 + σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅© 2 σ΅© σ΅©πΏ 2β σ΅© σ΅©π2β σ΅© σ΅©π2β σ΅© σ΅©π2β
1 β€ π β€ π β 1, π₯ β Ξ©β , π’0β (π₯) = πβ (π₯) ,
Μβ, π₯βΞ©
respectively.
Μβ, π = [ π ] π₯βΞ© π
ππβ (π₯) = πβ (π‘π , π₯) ,
π π β β π) Vπ+1 (π₯) + ( β π β 1) Vπβ (π₯) π π (27)
and second order of accuracy difference scheme
(25)
Μβ, π₯βΞ©
+
π β β π) (π’π+1 (π₯) β π’πβ (π₯)) = πβ (π₯) , π
Μβ, π = [ π ] . π₯βΞ© π
σ΅©σ΅© β πβ1 σ΅©σ΅© 1 σ΅©σ΅©{π } σ΅©σ΅© ] σ΅© σ΅© πΌ (1 β πΌ) σ΅©σ΅© π 1 σ΅©σ΅©Cπ (πΏ 2β ) (28)
Μβ, π₯βΞ© β π’π (π₯) = πβ (π₯) ,
1 β€ π β€ π,
= πβ (π₯) β πβ (π₯) ,
ππβ (π₯) = πβ (π‘π , π₯) ,
π’πβ (π₯) + (
(26)
β β Vπ+1 (π₯) β 2Vπβ (π₯) + Vπβ1 (π₯) + π΄π₯β Vπβ (π₯) = ππβ (π₯) , π2
V0β (π₯) β (
β
β π’π (π₯) = πβ (π₯) ,
1 β€ π β€ π β 1,
Μβ, π₯βΞ©
for π’ (π, π₯) = π (π₯), problem (19) is replaced by first order of accuracy difference scheme β β π’π+1 (π₯) β 2π’πβ (π₯) + π’πβ1 (π₯) + π΄π₯β π’πβ (π₯) π2
π‘π = ππ,
V0β (π₯) β Vπβ (π₯) = πβ (π₯) β πβ (π₯) ,
β
β
ππβ (π₯) = πβ (π‘π , π₯) ,
and second order of accuracy difference scheme β
π₯ β Ξ©β
β β Vπ+1 (π₯) β 2Vπβ (π₯) + Vπβ1 (π₯) + π΄π₯β Vπβ (π₯) = ππβ (π₯) , 2 π
hold, where π is independent of π, πΌ, β, πβ (π₯), πβ (π₯), πβ (π₯), πβ1
and {ππβ (π₯)}1
.
Abstract and Applied Analysis
5
Theorem 7. Let π and |β| = ββ12 + β
β
β
+ βπ2 be sufficiently small positive numbers. Then, for the solutions of difference schemes (24) and (25) the following almost coercive stability estimate
V0β (π₯) = βπΊ (π
πβπ β π
π+π ) Γ (πΌ + ππΆ) (2πΌ + ππΆ)β1 πΆβ1 πβ1
Γ β (π
πβπ β π
π+π ) ππβ (π₯) π π=1
+ πΊ (πΌ β π
2π) (πΌ + ππΆ)
σ΅©σ΅© β πβ1 σ΅© σ΅©σ΅© π σ΅©σ΅© π’π+1 β 2π’πβ + π’πβ1 σ΅©σ΅© σ΅©σ΅©{ σ΅©σ΅© )} σ΅©σ΅© 2 σ΅©σ΅© π σ΅©σ΅© σ΅©σ΅©Cπ (πΏ 2β ) 1 σ΅©
πβ1
Γ (2πΌ + ππΆ)β1 πΆβ1 β (π
|πβπ| β π
π+π ) ππβ (π₯) π π=1
σ΅©σ΅© πβ1 σ΅© σ΅© σ΅© σ΅© + σ΅©σ΅©σ΅©σ΅©{π’πβ }1 σ΅©σ΅©σ΅©σ΅© + σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅© σ΅©Cπ (π2 ) σ΅© σ΅©πΏ 2β σ΅© 2β
(29)
σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© β€ π [σ΅©σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅©σ΅©π2 + σ΅©σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅©σ΅©π2 + σ΅©σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅©σ΅©π2 2β 2β 2β
+ πΊ (πΌ β π
2π) (πβ (π₯) β πβ (π₯)) + πΊ (π
πβπ β π
π+π ) (πβ (π₯) β πβ (π₯)) ,
σ΅©σ΅© πβ1 σ΅© 1 σ΅© + ln ( ] ) σ΅©σ΅©σ΅©σ΅©{ππβ }1 σ΅©σ΅©σ΅©σ΅© π+β σ΅© σ΅©Cπ (πΏ 2β )
(30) for difference scheme (24), V0β (π₯) = (
holds, where π is independent of π, πΌ, β, πβ (π₯),πβ (π₯), πβ (π₯), πβ1
and {ππβ (π₯)}1
.
Proofs of Theorems 6 and 7 are based on the symmetry property of operator π΄π₯ , on Lemmas 3β5, the formulas
π’πβ (π₯) = (πΌ β π
2π)
β1
β1
πβ1
πβπ
Γ πΆ β (π
π β π β 1) πΊ1 (πΌ β π
2π) π
πβ1
π+π
βπ
π=1
) ππβ
π=1
πβ1 π=1
(π₯) π + π (π₯) β π΄π₯β πβ
(π₯) β
Γ (πΌ + ππΆ) (2πΌ + ππΆ)β1 πΆβ1 πβ1
+(
V0β
π΄π₯β V0β
π β π) πΊ1 (π
πβπβ1 β π
π+π+1 ) π
π=1
Γ β (π
|πβπ| β π
π+π ) β
(π₯) π]
β(
Γ β (π
πβπ β π
π+π ) ππβ (π₯) π
+ (πΌ + ππΆ) (2πΌ + ππΆ)β1 πΆβ1
π (π₯) =
π=1
Γ β (π
|πβπ| β π
π+π ) ππβ (π₯) π
β (π
πβπ β π
π+π ) (πΌ + ππΆ) (2πΌ + ππΆ)β1
β
πβ1
Γ β (π
πβπ β π
π+π ) ππβ (π₯) π
Γ (πΌ + ππΆ) (2πΌ + ππΆ)β1 πΆβ1
β + (π
πβπ β π
π+π ) Vπ (π₯))
Γ
Γ (πΌ + ππΆ) (2πΌ + ππΆ)β1 πΆβ1
β(
Γ [ ((π
π β π
2πβπ ) V0β (π₯)
ππβ
π β π β 1) πΊ1 (π
πβπ β π
π+π ) π
(π₯) ,
π β π) πΊ1 (πΌ β π
2π) π
Γ (πΌ + ππΆ) (2πΌ + ππΆ)β1 πΆβ1 πβ1
(π₯) ,
β Vπ (π₯) = V0β (π₯) + πβ (π₯) β πβ (π₯) ,
Γ β (π
|π+1βπ| β π
π+1+π ) ππβ (π₯) π π=1
+ πΊ1 (πΌ β π
2π) (πβ (π₯) β πβ (π₯))
6
Abstract and Applied Analysis + ((
π β π β 1) πΊ1 (π
πβπ β π
π+π ) π
+(
We can obtain π’(π‘, π₯) by formula π’(π‘, π₯) = V(π‘, π₯)+π€(π‘, π₯), where V(π‘, π₯) is the solution of the nonlocal boundary value problem
π β π) πΊ1 (π
πβπβ1 β π
π+π+1 )) π
Γ (πβ (π₯) β πβ (π₯)) , (31)
β
for difference scheme (25), and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in πΏ 2β .
= π (π‘, π₯) ,
π·β π’β (π₯) = 0,
Μβ, π₯βΞ©
0β€π₯β€π V (π, π₯) β V (π, π₯) = (exp (βπ) β exp (βπ) + π β π) cos (π₯) ,
(32)
π₯ β πβ ,
0 β€ π₯ β€ π, Vπ₯ (π‘, 0) = Vπ₯ (π‘, π) = 0,
the following coercivity inequality holds:
2β
0 β€ π‘ β€ π, (35)
π σ΅© σ΅© σ΅© σ΅© βσ΅©σ΅©σ΅©σ΅©(π’πβ )π₯π π₯π ,ππ σ΅©σ΅©σ΅©σ΅©πΏ β€ πσ΅©σ΅©σ΅©σ΅©πβ σ΅©σ΅©σ΅©σ΅©πΏ ,
π=1
0 < π₯ < π, 0 < π‘ < π,
V (0, π₯) β V (π, π₯) = (1 β exp (βπ) β π) cos (π₯) ,
Theorem 8 (see [28]). For the solution of the elliptic difference problem π΄π₯β π’β (π₯) = πβ (π₯) ,
πV (π‘, π₯) π π2 V (π‘, π₯) β ((2 + cos π₯) ) + V (π‘, π₯) 2 ππ‘ ππ₯ ππ₯
(33)
2β
and π€(π‘, π₯) is the solution of the boundary value problem
where π does not depend on β and πβ . β
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 β
ππ’ (π‘, π₯) π π2 π’ (π‘, π₯) β ((2 + cos π₯) ) + π’ (π‘, π₯) ππ‘2 ππ₯ ππ₯ = π (π‘, π₯) + π (π₯) ,
π2 π€ (π‘, π₯) ππ€ (π‘, π₯) π β ((2 + cos π₯) ) + π€ (π‘, π₯) ππ‘2 ππ₯ ππ₯ = π (π₯) ,
0 < π₯ < π, 0 < π‘ < π,
π€ (0, π₯) = (exp (βπ) + π + 1) cos (π₯) β V (π, π₯) , 0 β€ π₯ β€ π, π€ (π, π₯) = (exp (βπ) + π + 1) cos (π₯) β V (π, π₯) , 0 β€ π₯ β€ π, π€π₯ (π‘, 0) = π€π₯ (π‘, π) = 0,
0 < π₯ < π, 0 < π‘ < π,
0 β€ π‘ β€ π. (36)
π (π‘, π₯) = β exp (βπ‘) cos (π₯) + (exp (βπ‘) + π‘) (3 cos (π₯) + cos (2π₯)) , π’ (0, π₯) = 2 cos (π₯) ,
π’ (π, π₯) = (exp (βπ) + π + 1) cos (π₯) ,
0 β€ π₯ β€ π,
π’ (π, π₯) = (exp (βπ) + π + 1) cos (π₯) ,
0 β€ π₯ β€ π,
π’π₯ (π‘, 0) = π’π₯ (π‘, π) = 0,
Introduce small parameters π and β such that ππ = π, πβ = π. For approximate solution of nonlocal boundary value problem (35) we consider the set [0, π]π Γ [0, π]β of a family of grid points
0 β€ π₯ β€ π,
0 β€ π‘ β€ π, π =
3π . 5
(34)
It is clear that π’(π‘, π₯) = (exp(βπ‘) + π‘ + 1) cos(π₯) and π(π₯) = sin(π₯) + (π₯ + 2) cos(π₯) are the exact solutions of (34).
[0, π]π Γ [0, π]β = {(π‘π , π₯π ) : π‘π = ππ, π = 0, . . . , π, π₯π = πβ, π = 0, . . . , π} .
(37)
Abstract and Applied Analysis
7
Applying (21), we obtain difference schemes of the first order of accuracy in π‘ and the second order of accuracy in π₯ π π Vπ+1 β 2Vππ + Vπβ1 Vππ+1 β 2Vππ + Vππβ1 + (2 + cos (π₯ )) π π2 β2
β sin (π₯π )
π Vπ+1
πππ = βπ (π‘π , π₯π ) ,
β 2β
π Vπβ1
π΄ π Vπ+1 + π΅π Vπ + πΆπ Vπβ1 = πΌπππ ,
β Vππ = πππ ,
V0 = V 1 ,
π = 0, . . . , π,
Vπ0 β Vππ = (1 β exp (βπ) β π) cos (π₯π ) ,
π = 0, . . . , π,
Vππ β Vππ = (exp (βπ‘π) β exp (βπ) + π‘π β π) cos (π₯π ) , π π = 0, . . . , π, π = [ ] , π
(38)
for the approximate solutions of the nonlocal boundary value problem (35), and π π π€π+1 β 2π€ππ + π€πβ1 π€ππ+1 β 2π€ππ + π€ππβ1 +(2 + cos (π₯ )) π π2 β2
= βππ ,
π π π€π+1 β π€πβ1 β π€ππ 2β
π = 1, . . . , π β 1,
π π β π€πβ1 = 0, π€0π β π€1π = π€π
π = 0, . . . , π,
= (exp (βπ) + π + 1) cos (π₯π ) β
Vππ ,
π π = 0, . . . , π, π = [ ] , π π€ππ = (exp (βπ) + π + 1) cos (π₯π ) β Vππ , π = 0, . . . , π, (39) for the approximate solutions of the boundary value problem (36). By using (22) and second order of accuracy in π₯ approximation of π΄, we get the following values of π in grid points: ππ = β
0 [0 [ [0 [ [0 [ [ π΄ π = [ ... [ [0 [ [0 [ [0 [0
0 ππ 0 0 .. .
0 0 ππ 0 .. .
0 0 0 ππ .. .
β
β
β
β
β
β
β
β
β
β
β
β
0 π ππ π .. .
0 0 π ππ .. .
β
β
β
β
β
β
β
β
β
β
β
β
.. .
β1 0 0 0 .. .
0 [0 [ [0 [ [0 [ [ πΆπ = [ ... [ [0 [ [0 [ [0 [0
0 ππ 0 0 .. .
0 0 ππ 0 .. .
(2 + cos (π₯π )) 0 ((ππ+1 β Vπ+1 ) β 2 (ππ β Vπ0 ) β2
0 + (ππβ1 β Vπβ1 )) +
Γ ((ππ+1 β
(41)
Vπ = Vπβ1 .
Here, πΌ is the (π + 1) Γ (π + 1) identity matrix, π΄ π , π΅π , πΆπ are (π + 1) Γ (π + 1) square matrices, and ππ is a (π + 1) Γ 1 column matrix which are defined by
1 [π [ [0 [ [0 [ [ π΅π = [ ... [ [0 [ [0 [ [0 [0
ππ = π (π₯π ) , π = 1, . . . , π β 1,
π€π0
π = 1, . . . , π β 1,
π = 1, . . . , π β 1, π = 1, . . . , π β 1,
π π β Vπβ1 = 0, V0π β V1π = Vπ
β sin (π₯π )
We can rewrite difference scheme (38) in the matrix form
0 Vπ+1 )
+ (ππ β Vπ0 ) ,
sin (π₯π ) 2β
β (ππβ1 β
0 0 0 0
0 0 0 0
ππ = (40)
ππ = β
0 Vπβ1 ))
π = 1, . . . , π β 1.
0 ππ π 0 .. .
0 0 0 0
ππ =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
β
β
β
β
β
β
β
β
β
β
β
β
0 0 0 0
0 0 0 0 . β
β
β
.. β
β
β
ππ β
β
β
0 β
β
β
0 β
β
β
0
0 0 0 β1
0 0 0 ππ .. . 0 0 0 0
0 0 0 0 .. .
0 ππ 0 0
0 0 0 0 .. .
0 0 ππ 0
0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] 0] 0]
0 0 0 0 . β
β
β
.. β
β
β
ππ β
β
β
π β
β
β
0 β
β
β
0
0 0 0 0 .. .
0 0 0 0 .. .
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
0 0 0 0 . β
β
β
.. β
β
β
ππ β
β
β
0 β
β
β
0 β
β
β
0
0 0 0 0 .. .
0 ππ 0 0
π ππ π 0
0 π ππ 0
0 0 0 0 .. .
0 0 ππ 0
(42)
0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] π] 1]
0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] 0] 0]
(43)
2 + cos (π₯π ) sin (π₯π ) β , β2 2β 2 2 (2 + cos (π₯π )) β β 1, π2 β2
2 + cos (π₯π ) sin (π₯π ) + , β2 2β
π=
(44) 1 , π2
8
Abstract and Applied Analysis ππ0 [ ] ππ = [ ... ] ,
π΄ π and πΆπ are defined by (42) and (43) and (π+1)Γ1 column matrix ππ is defined by
π
[ππ ] ππ0 [ .. ] ππ = [ . ] , π [ππ ]
ππ0 = (1 β exp (βπ) β π) cos (π₯π ) , πππ = (exp (βπ‘π) β exp (βπ) + π‘π β π) cos (π₯π ) , π = 1, . . . , π β 1, πππ = βπ (π‘π , π₯π ) ,
ππ0 = (exp (βπ) + π + 1) cos (π₯π ) β Vππ ,
π = 1, . . . , π β 1, π = 1, . . . , π β 1,
πππ = (exp (βπ) + π + 1) cos (π₯π ) β Vππ ,
Vπ 0 [ .. ] Vπ = [ . ]
,
π = π β 1, π, π + 1.
πππ = βππ ,
π [Vπ ](π+1)Γ1
(45)
Vπ = πΌπ+1 Vπ+1 + π½π+1 ,
π = π β 1, . . . , 1,
(46)
where Vπ = 0,β πΌπ (π = 1, . . . , π β 1) are (π + 1) Γ (π + 1) square matrices and π½π (π = 1, . . . , π β 1) are (π + 1) Γ 1 column matrices. For πΌπ+1 , π½π+1 , we get formulas
π = 1, . . . , π β 1, (47)
where πΌ1 is the (π + 1) Γ (π + 1) identity matrix and π½1 is the (π + 1) Γ 1 zero column vector. Futher, we rewrite difference scheme (39) in the matrix form
Now we present second order of accuracy in π‘ and π₯ difference schemes for problems (35) and (36). Applying (27) and formulas for sufficiently smooth function π π (π₯π+1 ) β π (π₯πβ1 ) β πσΈ (π₯π ) = π (β2 ) , 2β π (π₯π+1 ) β 2π (π₯π ) + π (π₯πβ1 ) β πσΈ σΈ (π₯π ) = π (β2 ) , β2 10π (0) β 15π (β) + 6π (2β) β π (3β) β πσΈ σΈ σΈ (0) = π (β2 ) , β3 β3π (0) + 4π (β) β π (2β) β πσΈ (0) = π (β2 ) , 2β
π΄ π π€π+1 + πΈπ π€π + πΆπ π€πβ1 = πΌπππ , π = 1, . . . , π β 1, π€0 = π€1 ,
π = π β 1, π, π + 1. (50)
πΌπ+1 = β(π΅π + πΆπ πΌπ ) π΄ π , β1
,
π [π€π ](π+1)Γ1
β1
π½π+1 = β(π΅π + πΆπ πΌπ ) (πΌππ β πΆπ π½π ) ,
π = 1, . . . , π β 1, π = 1, . . . , π β 1,
π€π 0 [ ] π€π = [ ... ]
For solving (41) we use the modified Gauss elimination method (see [29]). Namely, we seek solution of (41) by the formula
π = 1, . . . , π β 1,
10π (π) β 15π (π β β) + 6π (π β 2β) β π (π β 3β) β3
(48)
β πσΈ σΈ σΈ (π) = π (β2 ) ,
π€π = π€πβ1 .
β3π (π) + 4π (π β β) β π (π β 2β) β πσΈ (π) = π (β2 ) , 2β
Here, 1 [π [ [0 [ [0 [ [ πΈπ = [ ... [ [0 [ [0 [ [0 [0
0 ππ π 0 .. . 0 0 0 0
0 π ππ π .. . 0 0 0 0
0 0 π ππ .. . 0 0 0 0
β
β
β
β
β
β
β
β
β
β
β
β
0 0 0 0 . d .. β
β
β
ππ β
β
β
π β
β
β
0 β
β
β
0
0 0 0 0 .. .
π ππ π 0
0 0 0 0 .. .
0 π ππ 0
0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] π] 1]
we get (49) π π Vπ+1 β 2Vππ + Vπβ1 Vππ+1 β 2Vππ + Vππβ1 + (2 + cos (π₯ )) π π2 β2 π π Vπ+1 β Vπβ1 β sin (π₯π ) β Vππ = πππ , 2β
(51)
Abstract and Applied Analysis πππ = βπ (π‘π , π₯π ) ,
9
π = 1, . . . , π β 1,
π = 1, . . . , π β 1,
π΄ π Vπ+1 + π΅π Vπ + πΆπ Vπβ1 = πΌπππ ,
β 3V0π + 4V1π β V2π π π π = β3Vπ + 4Vπβ1 β Vπβ2 = 0,
where π΄ π , πΆπ are defined by (42), (43), (44), and π΅π is defined by
π π β π β 1) Vππ β ( β π) Vππ+1 π π
= (1 β exp (βπ) β π) cos (π₯π ) ,
1 [π [ [0 [ [0 [ [ π΅π = [ ... [ [0 [ [0 [ [0 [0
π = 0, . . . , π,
π π β π β 1) Vππ β ( β π) Vππ+1 π π
= (exp (βπ‘π) β exp (βπ) + π‘π β π) cos (π₯π ) , π = 0, . . . , π,
0 ππ π 0 .. . 0 0 0 0
0 π ππ π .. . 0 0 0 0
0 0 π ππ .. . 0 0 0 0
(52) ππ = β difference scheme for nonlocal problem (35), and π€ππ+1
β
2π€ππ π2
+
π€ππβ1
+ (2 + cos (π₯π ))
π π€π+1
β
2π€ππ β2
+
π π€πβ1
ππ = π (π₯π ) ,
π=
1 , π2
β
β
β
β
β
β
β
β
β
β
β
β
.. .
β
β
β
β
β
β
β
β
β
β
β
β
0 0 0 0 .. .
0 0 0 0
π¦ 0 0 0 .. .
π§ 0 0 0 .. .
0 0 0 π¦
0 0 0 π§
0 0 0 0 .. .
0 0 0 0
β
β
β
β
β
β
β
β
β
β
β
β
.. . β
β
β
β
β
β
β
β
β
β
β
β
0 0 0 0 .. .
ππ π 0 0
0 0 0 0 .. .
0 0 0 0 .. .
π ππ π 0
0 π ππ 0
0 0] ] 0] ] 0] ] .. ] , .] ] 0] ] 0] ] π] 1]
2 2 (2 + cos (π₯π )) β β 1, π2 β2
π¦=(
π β π β 1) , π
π§ = β(
π β π) . π
(55)
We seek solution of (54) by the formula
π π€π β π€πβ1 β sin (π₯π ) π+1 = βππ , 2β
π = 1, . . . , π β 1,
(54)
β3Vπ + 4Vπβ1 β Vπβ2 = 0,
π π π π = 10Vπ β 15Vπβ1 + 6Vπβ2 β Vπβ3 = 0,
Vππ + (
π = 1, . . . , π β 1,
β3V0 + 4V1 β V2 = 0,
π = 0, . . . , π,
10V0π β 15V1π + 6V2π β V3π
Vπ0 + (
By difference scheme (52), we write in matrix form
Vπ = πΌπ Vπ+1 + π½π Vπ+2 + πΎπ ,
π = π β 2, . . . , 0,
(56)
where πΌπ , π½π (π = 0, . . . , π β 2) are (π + 1) Γ (π + 1) square matrices and πΎπ (π = 0, . . . , π β 2) are (π + 1) Γ 1 column matrices. For the solution of difference equation (41) we need to use the following formulas for πΌπ , π½π :
π = 1, . . . , π β 1,
π π π + 4π€πβ1 β π€πβ2 = 0, β 3π€0π + 4π€1π β π€2π = β3π€π
β1
πΌπ = β(π΅π + πΆπ πΌπβ1 ) (π΄ π + πΆπ π½πβ1 ) ,
π = 0, . . . , π,
π½π = 0,
10π€0π β 15π€1π + 6π€2π β π€3π =
π 10π€π
β
π 15π€πβ1
+
π 6π€πβ2
β1
β
π π€πβ3
πΎπ = β(π΅π + πΆπ πΌπβ1 ) (πΌππ β πΆπ πΎπβ1 ) ,
= 0,
π = 1, . . . , π β 1, (57)
where π€π0 = (exp (βπ) + π + 1) cos (π₯π ) +(
π π β π β 1) Vππ β ( β π) Vππ+1 , π π
π = 0, . . . , π,
π€ππ = (exp (βπ) + π + 1) cos (π₯π ) π π + ( β π β 1) Vππ β ( β π) Vππ+1 , π π ππ = π (π₯π ) ,
1 π½0 = β πΌ, 3
8 πΌ1 = πΌ, 5
3 π½1 = β πΌ, 5
πΌπβ2 = 4πΌ,
π½πβ2 = β3πΌ,
8 πΌπβ3 = πΌ, 3
5 π½πβ3 = β πΌ, 3
(58)
and πΎ0 , πΎ1 , πΎπβ2 , πΎπβ3 are the (π + 1) Γ 1 zero column vector. For Vπ and Vπβ1 we have
π = 0, . . . , π, (53)
difference scheme for boundary value problem (36).
4 πΌ0 = πΌ, 3
β1
β1 β1 Vπ = (π11 β π12 π22 π21 ) (πΊ1 β π12 π22 πΊ2 ) , β1 (πΊ2 β π21 Vπ) , Vπβ1 = π22
(59)
10
Abstract and Applied Analysis
where
Table 1: Error analysis for nonlocal problem.
π11 = β3π΄ πβ2 β 8π΅πβ2 β 8πΆπβ2 πΌπβ3 β 3πΆπβ2 π½πβ3 , π12 = 4π΄ πβ2 + 9π΅πβ2 + 9πΆπβ2 πΌπβ3 + 4πΆπβ2 π½πβ3 ,
Difference scheme (38) Difference scheme (52)
π21 = β3π΅πβ1 β 8πΆπβ1 , π22 = π΄ πβ1 + 4π΅πβ1 + 9πΆπβ1 , πΊ1 = πΌππβ2 β πΆπβ2 πΎπβ3 ,
πΊ2 = πΌππβ1 .
We can rewrite difference scheme (53) in the matrix form π = 1, . . . , π β 1,
β3π€0 + 4π€1 β π€2 = 0,
(61)
β3π€π + 4π€πβ1 β π€πβ2 = 0,
Difference scheme (38), (40) Difference scheme (52), (40)
where π΄ π , πΈπ , πΆπ are defined by (42), (49), (43), and (44) and ππ is defined by ππ0 [ ] ππ = [ ... ] , ππ0 = (exp (βπ) + π + 1) cos (π₯π ) π π β π β 1) Vππ β ( β π) Vππ+1 , π π
πππ = (exp (βπ) + π + 1) cos (π₯π ) + ( β( πππ = βππ ,
π β π) Vππ+1 , π
π β π β 1) Vππ π
π = 1, . . . , π β 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 π = 2 and different values of π = π. Grid functions Vππ , π’ππ represent the numerical solutions of difference schemes for auxiliary nonlocal problem (35) and inverse problem (34) at (π‘π , π₯π ), respectively. Grid function ππ calculated by (40) represents numerical solution at π₯π for unknown function π. The errors are computed by the norms πβ1
σ΅¨ σ΅¨2 π πΈVπ = max ( β σ΅¨σ΅¨σ΅¨σ΅¨V (π‘π , π₯π ) β Vππ σ΅¨σ΅¨σ΅¨σ΅¨ β) 1β€πβ€πβ1
π πΈπ’π
1/2
σ΅¨ σ΅¨2 = max ( β σ΅¨σ΅¨σ΅¨σ΅¨π’ (π‘π , π₯π ) β π’ππ σ΅¨σ΅¨σ΅¨σ΅¨ β) 1β€πβ€πβ1 π=1
πΈππ
,
π=1
πβ1
πβ1
σ΅¨ σ΅¨2 = ( β σ΅¨σ΅¨σ΅¨π (π₯π ) β ππ σ΅¨σ΅¨σ΅¨ β) π=1
1/2
.
π=π= 160
0.30522
0.14933
0.073953
0.036814
0.024714
0.0031054
4.36 Γ 10β4 7.52 Γ 10β5
π=π= 20
π=π= 40
π=π= 80
π=π= 160
0.57878
0.33755
0.20387
0.12905
0.058201
0.010646
0.0020228
4.03 Γ 10β4
π=π= 20
π=π= 40
π=π= 80
π=π= 160
0.088225
0.038586
0.01815
0.008818
0.017034
0.0020225
2.47 Γ 10β4 3.08 Γ 10β5
(62)
π = 0, . . . , π,
π = 1, . . . , π β 1,
π=π= 80
Table 3: Error analysis for u.
Difference scheme (38), (40), (39) Difference scheme (52), (40), (53)
π [ππ ]
+(
π=π= 40
Table 2: Error analysis for p.
(60)
π΄ π π€π+1 + πΈπ π€π + πΆπ π€πβ1 = πΌπππ ,
π=π= 20
1/2
,
(63)
Tables 1β3 present the error between the exact solution and numerical solutions derived by corresponding difference schemes. The results are recorded for π = π = 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 π of inverse problem (34). Table 3 represents error between the exact solution π’ 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 1β3, 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, βInverse problems of source determination for the Poisson equation on the plane,β Zhurnal Vychislitelβnoi Matematiki i Matematicheskoi Fiziki, vol. 44, no. 5, pp. 862β871, 2004 (Russian). [4] D. G. OrlovskiΛΔ±, βInverse Dirichlet problem for an equation of elliptic type,β Differential Equations, vol. 44, no. 1, pp. 124β134, 2008. [5] D. Orlovsky and S. Piskarev, βOn approximation of inverse problems for abstract elliptic problems,β Journal of Inverse and Ill-Posed Problems, vol. 17, no. 8, pp. 765β782, 2009. [6] A. Ashyralyev and A. S. Erdogan, βWell-posedness of the inverse problem of a multidimensional parabolic equation,β Vestnik of Odessa National University, Mathematics and Mechanics, vol. 15, no. 18, pp. 129β135, 2010. [7] A. Ashyralyev, βOn the problem of determining the parameter of a parabolic equation,β Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1397β1408, 2011. [8] A. Ashyralyev and A. S. ErdoΛgan, βOn the numerical solution of a parabolic inverse problem with the Dirichlet condition,β International Journal of Mathematics and Computation, vol. 11, no. J11, pp. 73β81, 2011. [9] C. Ashyralyyev, A. Dural, and Y. Sozen, βFinite difference method for the reverse parabolic problem,β Abstract and Applied Analysis, vol. 2012, Article ID 294154, 17 pages, 2012. [10] C. Ashyralyyev and O. Demirdag, βThe difference problem of obtaining the parameter of a parabolic equation,β Abstract and Applied Analysis, vol. 2012, Article ID 603018, 14 pages, 2012. [11] Ch. Ashyralyyev and M. Dedeturk, βA finite difference method for the inverse elliptic problem with the Dirichlet condition,β Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 132β155, 2013. [12] A. Ashyralyev and M. Urun, βDetermination of a control parameter for the Schrodinger equation,β Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 156β166, 2013. [13] M. Dehghan, βDetermination of a control parameter in the twodimensional diffusion equation,β Applied Numerical Mathematics, vol. 37, no. 4, pp. 489β502, 2001.
11 [14] V. G. Romanov, βA three-dimensional inverse problem of viscoelasticity,β Doklady Mathematics, vol. 84, no. 3, pp. 833β 836, 2011. [15] S. I. Kabanikhin, βDefinitions and examples of inverse and illposed problems,β Journal of Inverse and Ill-Posed Problems, vol. 16, no. 4, pp. 317β357, 2008. Β΄ [16] Y. S. Eidelβman, βAn inverse problem for an evolution equation,β Mathematical Notes, vol. 49, no. 5, pp. 535β540, 1991. [17] K. Sakamoto and M. Yamamoto, βInverse heat source problem from time distributing overdetermination,β Applicable Analysis, vol. 88, no. 5, pp. 735β748, 2009. [18] A. V. Gulin, N. I. Ionkin, and V. A. Morozova, βOn the stability of a nonlocal two-dimensional finite-difference problem,β Differential Equations, vol. 37, no. 7, pp. 970β978, 2001. [19] G. Berikelashvili, βOn a nonlocal boundary-value problem for two-dimensional elliptic equation,β Computational Methods in Applied Mathematics, vol. 3, no. 1, pp. 35β44, 2003. [20] M. P. Sapagovas, βDifference method of increased order of accuracy for the Poisson equation with nonlocal conditions,β Differential Equations, vol. 44, no. 7, pp. 1018β1028, 2008. [21] A. Ashyralyev, βA note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space,β Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 557β573, 2008. [22] D. Orlovsky and S. Piskarev, βThe approximation of BitzadzeSamarsky type inverse problem for elliptic equations with Neumann conditions,β Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 118β131, 2013. [23] A. Ashyralyev and F. S. O. Tetikoglu, βFDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions,β Abstract and Applied Analysis, vol. 2012, Article ID 454831, 22 pages, 2012. [24] A. Ashyralyev and E. Ozturk, βThe numerical solution of the Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition,β Abstract and Applied Analysis, vol. 2013, Article ID 730804, 13 pages, 2012. [25] A. Ashyralyev, βWell-posedness of the difference schemes for elliptic equations in πΆππ½,πΎ (πΈ) spaces,β Applied Mathematics Letters, vol. 22, no. 3, pp. 390β395, 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, BirkhΒ¨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, BirkhΒ¨auser, Basel, Switzerland, 1989.
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