On Second Order of Accuracy Difference Scheme of the Approximate Solution of Nonlocal Elliptic-Parabolic Problems

and Applied Analysis 3 exist and are bounded for a self-adjoint positive operator A. Here B 1 2 ( τA √ A 4 τ2A ) , K ( I 2τA 5 4 τA 2 )−1 . 2.2 Theorem 2.1. For any gk, 1 ≤ k ≤ N − 1, and fk,−N 1 ≤ k ≤ 0, the solution of problem 1.2 exists and the following formula holds: uk ( I − R2N )−1{[ R − R2N−k ] u0 [ RN−k − R k ][ Pu0 − τ 0 ∑ s −N 1 P N−1Gfs μ ] − [ RN−k − R k ] I τB 2I τB −1B−1 N−1 ∑ s 1 [ RN−s − R s ] gsτ } I τB 2I τB −1B−1 N−1 ∑ s 1 [ R|k−s| − R s ] gsτ, 1 ≤ k ≤N, 2.3


Introduction
The role played by coercive inequalities in the study of boundary value problems for elliptic and parabolic partial differential equations is well known see 1-4 .Nonlocal problems are widely used for mathematical modeling of various processes of physics, biology, chemistry, ecology, engineering, and industry when it is impossible to determine the boundary or initial values of the unknown function.Theory and numerical methods of solutions of the nonlocal boundary value problems for partial differential equations of variable type have been studied extensively by many researchers see, e.g., 5-34 and the references therein .for the differential equation in a Hilbert space H with the self-adjoint positive definite operator A was considered.The well posedness of problem 1.1 in H ölder spaces was established.The first order of accuracy difference scheme for approximate solutions of nonlocal boundary value problem 1.1 was presented.In applications, the coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations were obtained.
In the present paper, the second order of accuracy difference scheme generated by Crank-Nicholson difference scheme for the approximate solution of problem 1.1 is presented.The well posedness of difference scheme 1.2 in H ölder spaces is established.As an application, coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations are obtained.A numerical example is given.

The Formula for the Solution of Problem 1.2
The following operators:

2.1
Abstract and Applied Analysis 3 exist and are bounded for a self-adjoint positive operator A.Here exists and the following formula holds:

4 Abstract and Applied Analysis
Proof.For any {f k } −1 k −N and ξ, the solution of the auxiliary inverse Cauchy difference problem exists and the following formula holds 36 Putting ξ u 0 , we get 2.4 .Now, we consider the following auxiliary difference problem

2.8
It is well known that for the solution of 2.8 the following formula holds 37, 38 : 2.9 Applying 2.7 and putting ξ u 0 , ψ P N u 0 − τ 0 s −N 1 P N s−1 Gf s μ, in 2.9 , we get 2.3 .
For u 0 , using 2.3 , 2.4 , and the condition Abstract and Applied Analysis 5 we obtain the operator equation

2.11
The operator 12 has an inverse

2.13
Hence, we obtain that

2.14
This concludes the proof of Theorem 2.1.

Main Theorems
Here, we study well posedness of problem 1.2 .First, we give some necessary estimates for P k , R k , and T τ .For a self-adjoint positive operator A, the following estimates are satisfied 36, 38, 39 : where M is independent of τ.From these estimates, it follows that

3.4
Nonlocal boundary value problem 1.2 is said to be stable in F −1, 1 τ , H if we have the inequality where M is independent of not only f τ , g τ , and μ but also τ.
for the solution of boundary value problem 2.8 .By 36 , we get for the solution of an inverse Cauchy difference problem 2.6 .Then, the proof of Theorem 3.1 is based on the stability inequalities 3.6 , 3.7 , and on the estimates for the solution of the boundary value problem 1.2 .Estimates 3.8 and 3.9 follow from formula 2.5 and estimates 3.1 , 3.2 , and 3.3 which conclude the proof of Theorem 3.1.
Theorem 3.2.Assume that μ ∈ D A and f 0 , f −1 , g 1 ∈ D I τB .Then, for the solution of difference problem 1.2 , we have the following almost coercivity inequality:

3.10
where M does not dependent on not only f τ , g τ , and μ but also τ.
Proof.By 40 , we have for the solution of an inverse Cauchy difference problem 2.6 .By 38 , we get for the solution of boundary value problem 2.8 .Then, the proof of Theorem 3.2 is based on almost coercivity inequalities 3.11 , 3.12 , and on the estimates for the solution of boundary value problem 1.2 .The proof of these estimates follows the scheme of the papers 38, 40 and relies on both the formula 2.5 and the estimates 3.1 , 3.2 , and 3.3 .This concludes the proof of Theorem 3.2.
where M is independent of not only f τ , g τ , and μ but also τ and α.
Proof.By 39, 40 , Au 0 H 3.17 Abstract and Applied Analysis for the solution of an inverse Cauchy difference problem 2.6 can be written.By 37, 38 , we get for the solution of boundary value problem 2.8 .Then, the proof of Theorem 3.3 is based on coercivity inequalities 3.16 -3.18 , and the estimates 19 for the solution of boundary value problem 1.2 .Estimates 3.19 and 3.20 follow from the formulas for the solution of problem 1.2 and estimates 3.1 , 3.2 , and 3.3 .This finalizes the proof of Theorem 3.3.

Applications
In this section, we indicate applications of Theorems 3.1, 3.2, and 3.3 to obtain the stability, the almost coercive stability, and the coercive stability estimates for the solutions of these difference schemes for the approximate solution of nonlocal mixed problems.First, let Ω be the unit open cube in the n-dimensional Euclidean space R n 0 < x k < 1, 1 ≤ k ≤ n with boundary S, Ω Ω ∪ S. In −1, 1 × Ω, the boundary value problem for the multidimensional elliptic-parabolic equation 1 has a unique smooth solution u t, x for f t, x t ∈ −1, 0 , x ∈ Ω , g t, x t ∈ 0, 1 , x ∈ Ω the smooth functions, and a r x a > 0 x ∈ Ω .The discretization of problem 4.1 is carried out in two steps.In the first step, the grid sets are defined.To the differential operator A generated by problem 4.1 , we assign the difference operator A x h by the formula acting in the space of grid functions u h x , satisfying the conditions u h x 0 for all x ∈ S h .With the help of A x h , we arrive at the nonlocal boundary value problem for an infinite system of ordinary differential equations.Replacing problem 4.4 by the difference scheme 1.2 , one can obtain the second order of accuracy difference scheme • • • h 2 n be sufficiently small positive numbers.Then, solutions of difference scheme 4.5 satisfy the following stability and almost coercivity estimates:

4.6
Here, M is independent of not only τ, h, μ h x but also The proof of Theorem 4.1 is based on Theorems 3.1 and 3.2, the estimate the symmetry properties of the difference operator A x h defined by formula 4.3 in L 2h , and the following theorem.

Theorem 4.2. For the solution of the elliptic difference problem
the following coercivity inequality holds [41]: Let us give a corollary of Theorem 3.3.
Theorem 4.3.Let τ and |h| be sufficiently small positive numbers.Then, solutions of difference scheme 4.5 satisfy the following coercivity stability estimates: where M is independent of not only τ, h, and μ h x but also The proof of Theorem 4.3 is based on the abstract Theorems 3.3 and 4.2, and the symmetry properties of the difference operator A x h defined by the formula 4.3 .Note that in a similar manner one can construct the difference schemes of the second order of accuracy with respect to one variable for approximate solutions of the boundary value problem 4.11 .Abstract theorems given above permit us to obtain the stability, the almost stability and the coercive stability estimates for the solutions of these difference schemes.
The exact solution of this problem is u t, x e t − 1 sin x.Now, we give the results of the numerical analysis.The errors computed by of the numerical solutions are given in Table 1.