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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 959216, 15 pages
On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator
1Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey
2Department of Mathematics, Yildiz Technical University, Esenler, 34210 Istanbul, Turkey
Received 13 May 2013; Accepted 20 July 2013
Academic Editor: Abdullah Said Erdogan
Copyright © 2013 Allaberen Ashyralyev and Ozgur Yildirim. 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 third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert space with self-adjoint positive definite operator . Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered.
In modeling several phenomena of physics, biology, and ecology mathematically, there often arise problems with nonlocal boundary conditions (see [1–5] and the references given therein). Nonlocal boundary value problems have been a major research area in the case when it is impossible to determine the boundary conditions of the unknown function. Over the last few decades, the study of nonlocal boundary value problems is of substantial contemporary interest (see, e.g., [6–14] and the references given therein).
We consider the nonlocal boundary value problem for hyperbolic equations in a Hilbert space with self-adjoint positive definite linear operator with domain .
A function is called a solution of problem (1) if the following conditions are satisfied.(i) is twice continuously differentiable on the segment . The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii)The element belongs to for all and the function is continuous on the segment .(iii) satisfies the equations and the nonlocal boundary conditions (1).
Here, , and are smooth functions.
In the study of numerical methods for solving PDEs, stability is an important research area (see [6–27]). Many scientists work on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitudes of the grid steps and with respect to the time and space variables are connected. This particularly means that when .
We are interested in studying high order of accuracy unconditionally stable difference schemes for hyperbolic PDEs.
In the present paper, third order of accuracy difference scheme generated by integer power of for approximately solving nonlocal boundary value problem (1) is presented. The stability estimates for solution of the difference scheme are established.
In , some results of this paper, without proof, were presented.
The well posedness of nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11–14, 19–32] and the references therein).
2. Third Order of Accuracy Difference Scheme Subject to Nonlocal Conditions
In this section, we obtain stability estimates for the solution of third order of accuracy difference scheme for numerical solution of nonlocal boundary value problem (1). Here, We study the stability of solutions of difference scheme (2) under the following assumption: We give a lemma that will be needed in the sequel which was presented in . First, let us present the following operators: and its conjugate , and its conjugate , and its conjugate , and and its conjugate .
We consider the following operators: and its conjugate , and its conjugate .
Lemma 1. The following estimates hold:
Now let us give, without proof, the second lemma.
Lemma 2. The following estimates hold:
Throughout the section, for simplicity, we denote
Lemma 3. Suppose that assumption (4) holds. Then, the operator has an inverse . From symmetry and positivity properties of operator , the following estimate is satisfied:
Now, let us obtain formula for the solution of problem (2). Using the results of , one can obtain the following formula: for the solution of difference scheme Applying formula (19) and nonlocal boundary conditions one can write Using formulas in (22), we obtain
Unfortunately, the estimates for , , and cannot be obtained under the conditions Nevertheless, we have the following theorem.
Proof. Using formulas in (23) and estimates (11), (12), and (14), we obtain
Applying to formulas in (23), we get Now, applying Abel’s formula to (23), we obtain the following formulas:
Next, let us obtain the estimates for and . First, applying to formula (28) and using estimates (11), (12), and (14) and the triangle inequality, one can obtain
Second, applying to formula (29) and using estimates (11), (12), and (14) and the triangle inequality, we get