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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 846582, 29 pages
http://dx.doi.org/10.1155/2012/846582
Research Article

A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem

1Department of Mathematics, Fatih University, Buyukcekmece 34500, Istanbul, Turkey
2Department of Mathematics, Yildiz Technical University, Esenler 34210, Istanbul, Turkey

Received 23 March 2012; Accepted 11 June 2012

Academic Editor: Sergey Piskarev

Copyright © 2012 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.

Abstract

The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert space H with the self-adjoint positive definite operator A. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.

1. Introduction

Hyperbolic partial differential equations play an important role in many branches of science and engineering and can be used to describe a wide variety of phenomena such as acoustics, electromagnetics, hydrodynamics, elasticity, fluid mechanics, and other areas of physics (see [15] and the references given therein).

While applying mathematical modelling to several phenomena of physics, biology, and ecology, there often arise problems with nonclassical boundary conditions, which the values of unknown function on the boundary are connected with inside of the given domain. Such type of boundary conditions are called nonlocal boundary conditions. Over the last decades, boundary value problems with nonlocal boundary conditions have become a rapidly growing area of research (see, e.g., [616] and the references given therein).

In the present work, we consider the nonlocal boundary value problem 𝑑2𝑢(𝑡)𝑑𝑡2𝑢+𝐴𝑢(𝑡)=𝑓(𝑡)(0𝑡1),(0)=𝑛𝑗=1𝛼𝑗𝑢𝜆𝑗+𝜑,𝑢𝑡(0)=𝑛𝑗=1𝛽𝑗𝑢𝑡𝜆𝑗+𝜓,0<𝜆1<𝜆2<<𝜆𝑛1,(1.1) where 𝐴 is a self-adjoint positive definite operator in a Hilbert space 𝐻.

A function 𝑢(𝑡) is called a solution of the problem (1.1), if the following conditions are satisfied:(i)𝑢(𝑡) is twice continuously differentiable on the segment [0,1]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii) The element 𝑢(𝑡) belongs to 𝐷(𝐴), independent of 𝑡, and dense in 𝐻 for all 𝑡[0,1] and the function 𝐴𝑢(𝑡) is continuous on the segment [0,1].(iii)𝑢(𝑡) satisfies the equation and nonlocal boundary conditions (1.1).

In the paper of [8], the following theorem on the stability estimates for the solution of the nonlocal boundary value problem (1.1) was proved.

Theorem 1.1. Suppose that 𝜑𝐷(𝐴), 𝜓𝐷(𝐴1/2), and 𝑓(𝑡) is a continuously differentiable function on [0,1] and the assumption 𝑛𝑘=1||𝛼𝑘+𝛽𝑘||+𝑛𝑚=1||𝛼𝑚||𝑛𝑘=1𝑘𝑚||𝛽𝑘||<|||||1+𝑛𝑘=1𝛼𝑘𝛽𝑘|||||(1.2) holds. Then, there is a unique solution of problem (1.1) and the stability inequalities max0𝑡1𝑢(𝑡)𝐻𝑀𝜑𝐻+𝐴1/2𝜓𝐻+max0𝑡1𝐴1/2𝑓(𝑡)𝐻,max0𝑡1𝐴1/2𝑢(𝑡)𝐻𝐴𝑀1/2𝜑𝐻+𝜓𝐻+max0𝑡1𝑓(𝑡)𝐻,max0𝑡1𝑑2𝑢(𝑡)𝑑𝑡2𝐻+max0𝑡1𝐴𝑢(𝑡)𝐻𝑀𝐴𝜑𝐻+𝐴1/2𝜓𝐻+𝑓(0)𝐻+10𝑓(𝑡)𝐻𝑑𝑡(1.3) hold, where 𝑀 does not depend on 𝜑,  𝜓, and 𝑓(𝑡),  𝑡[0,1].

Moreover, the first order of accuracy difference scheme 𝜏2𝑢𝑘+12𝑢𝑘+𝑢𝑘1+𝐴𝑢𝑘+1=𝑓𝑘,𝑓𝑘𝑡=𝑓𝑘,𝑡𝑘𝑢=𝑘𝜏,1𝑘𝑁1,𝑁𝜏=1,0=𝑛𝑟=1𝛼𝑟𝑢[𝜆𝑟/𝜏]𝜏+𝜑,1𝑢1𝑢0=𝑛𝑟=1𝛽𝑟𝑢[𝜆𝑟/𝜏]+1𝑢[𝜆𝑟/𝜏]1𝜏+𝜓,(1.4) for the approximate solution of problem (1.1) was presented. The stability estimates for the solution of this difference scheme, under the assumption 𝑛𝑘=1||𝛼𝑘||+𝑛𝑘=1||𝛽𝑘||+𝑛𝑘=1||𝛼𝑘||𝑛𝑘=1||𝛽𝑘||<1,(1.5) were established.

In the development of numerical techniques for solving PDEs, the stability has been an important research topic (see [631]). A large cycle of works on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitude of the grid steps 𝜏 and with respect to the time and space variables, are connected. In abstract terms, this particularly means that 𝜏𝐴0 when 𝜏0.

We are interested in studying the high order of accuracy difference schemes for hyperbolic PDEs, in which stability is established without any assumption with respect to the grid steps 𝜏 and . Particularly, a convenient model for analyzing the stability is provided by a proper unconditionally absolutely stable difference scheme with an unbounded operator.

In the present paper, the second order of accuracy unconditionally stable difference schemes for approximately solving boundary value problem (1.1) is presented. The stability estimates for the solutions of these difference schemes and their first and second order difference derivatives are established. This operator approach permits one to obtain the stability estimates for the solutions of difference schemes of nonlocal boundary value problems, for one-dimensional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional hyperbolic equation with Dirichlet condition in space variables.

Some results of this paper without proof were presented in [7].

Note that nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [1116, 2024, 3238] and the references therein).

2. The Second Order of Accuracy Difference Scheme Generated by 𝐴2

Throughout this paper for simplicity 𝜆1>2𝜏 and 𝜆𝑛<1 will be considered. Let us associate boundary value problem (1.1) with the second order of accuracy difference scheme 𝜏2𝑢𝑘+12𝑢𝑘+𝑢𝑘1+𝐴𝑢𝑘+𝜏24𝐴2𝑢𝑘+1=𝑓𝑘,𝑓𝑘𝑡=𝑓𝑘,𝑡𝑘𝑢=𝑘𝜏,1𝑘𝑁1,𝑁𝜏=1,0=𝑛𝑚=1𝛼𝑚𝑢[𝜆𝑚/𝜏]+𝜏1𝑢[𝜆𝑚/𝜏]𝑢[𝜆𝑚/𝜏]1×𝜆𝑚𝜆𝑚𝜏𝜏𝜏+𝜑,𝐼+2𝐴2𝜏1𝑢1𝑢0𝜏2𝑓0𝐴𝑢0=𝑛𝑘=1𝛽𝑘𝜏1𝑢[𝜆𝑘/𝜏]𝑢[𝜆𝑘/𝜏]1+𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏×𝑓[𝜆𝑘/𝜏]𝐴𝑢[𝜆𝑘/𝜏]𝑓+𝜓,0=𝑓(0).(2.1) A study of discretization, over time only, of the nonlocal boundary value problem also permits one to include general difference schemes in applications, if the differential operator in space variables 𝐴 is replaced by the difference operator 𝐴 that act in the Hilbert space and are uniformly self-adjoint positive definite in for 0<0.

In general, we have not been able to obtain the stability estimates for the solution of difference scheme (2.1) under assumption (1.5). Note that the stability of solution of difference scheme (2.1) will be obtained under the strong assumption 𝑛𝑘=1||𝛼𝑘||||||𝜆1+𝑘𝜆𝑘𝜏𝜏||||+𝑛𝑘=1||𝛽𝑘||||||𝜆1+𝑘𝜆𝑘𝜏𝜏||||+𝑛𝑘=1||𝛼𝑘||𝑛𝑘=1||𝛽𝑘||||||𝜆1+𝑘𝜆𝑘𝜏𝜏||||+𝑛𝑘=1||𝛼𝑘||||||𝜆𝑘𝜆𝑘𝜏𝜏||||𝑛𝑘=1||𝛽𝑘||||||𝜆𝑘𝜆𝑘𝜏𝜏||||<1.(2.2) Now, let us give some lemmas that will be needed in the sequel.

Lemma 2.1. The following estimates hold: 𝑅𝐻𝐻𝑅1,𝐻𝐻𝑅1,1𝑅𝐻𝐻𝑅1,1𝑅𝐻𝐻1,𝜏𝐴1/2𝑅𝐻𝐻1,𝜏𝐴1/2𝑅𝐻𝐻1.(2.3) Here, 𝐻 is the Hilbert space, 𝑅=(𝐼+𝑖𝜏𝐴1/2(𝜏2/2)𝐴)1, and 𝑅=(𝐼𝑖𝜏𝐴1/2(𝜏2/2)𝐴)1.

Lemma 2.2. Suppose that assumption (2.2) holds. Denote 𝐵𝜏=𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21𝑅[𝜆𝑚/𝜏]1𝐼𝑖𝜏𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝐼+𝑖𝜏𝐴1/22+𝑛𝑘=1𝛽𝑘𝜏𝐼+2𝐴21𝑅[𝜆𝑘/𝜏]1𝑅1+𝑅[𝜆𝑘/𝜏]1𝑅1212𝑛𝑛𝑚=1𝑘=1𝛼𝑚𝛽𝑘𝜏𝐼+2𝐴21𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]1+𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]1+𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝑖𝐴1/2𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝐼(𝑖𝜏/2)𝐴1/2𝐼+𝑖𝜏𝐴1/2𝑅[𝜆𝑚/𝜏]1𝐼+(𝑖𝜏/2)𝐴1/2𝐼𝑖𝜏𝐴1/22+𝑛𝑘=1𝛽𝑘𝜆𝑘𝜆𝑘𝜏𝜏𝑖𝐴1/2𝜏𝐼+2𝐴21𝑅[𝜆𝑘/𝜏]1𝑅1𝑅[𝜆𝑘/𝜏]1𝑅12+14𝑛𝑛𝑚=1𝑘=1𝛼𝑚𝛽𝑘𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴22×𝑖𝐴1/2𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]1𝐼+𝑖𝜏2𝐴1/2𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]1𝐼𝑖𝜏2𝐴1/212𝑛𝑛𝑚=1𝑘=1𝛼𝑚𝛽𝑘𝜆𝑘𝜆𝑘𝜏𝜏𝑖𝐴1/2𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]1𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]112𝑛𝑛𝑚=1𝑘=1𝛼𝑚𝛽𝑘𝜆𝑚𝜆𝑚𝜏𝜏𝜆𝑘𝜆𝑘𝜏𝜏𝐴𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]1𝐼𝑖𝜏2𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅[𝜆𝑘/𝜏]1𝐼+𝑖𝜏2𝐴1/2.(2.4) Then, the operator 𝐼𝐵𝜏 has an inverse 𝑇𝜏=𝐼𝐵𝜏1,(2.5) and the following estimate holds: 𝑇𝜏𝐻𝐻𝑀.(2.6)

Proof. The proof of estimate (2.6) is based on the estimate 𝐼𝐵𝜏𝐻𝐻1𝑛𝑘=1||𝛼𝑘||||||𝜆1+𝑘𝜆𝑘𝜏𝜏||||𝑛𝑘=1||𝛽𝑘||||||𝜆1+𝑘𝜆𝑘𝜏𝜏||||𝑛𝑘=1||𝛼𝑘||𝑛𝑘=1||𝛽𝑘||||||𝜆1+𝑘𝜆𝑘𝜏𝜏||||𝑛𝑘=1||𝛼𝑘||||||𝜆𝑘𝜆𝑘𝜏𝜏||||𝑛𝑘=1||𝛽𝑘||||||𝜆𝑘𝜆𝑘𝜏𝜏||||.(2.7) Estimate (2.7) follows from the triangle inequality and estimate (2.3). Lemma 2.2 is proved.
Now, we will obtain the formula for the solution of problem (2.1). It is easy to show that (see, e.g., [18]) there is unique solution of the problem 𝜏2𝑢𝑘+12𝑢𝑘+𝑢𝑘1+𝐴𝑢𝑘+𝜏24𝐴2𝑢𝑘+1=𝑓𝑘,𝑓𝑘𝑡=𝑓𝑘,𝑡𝑘𝜏=𝑘𝜏,1𝑘𝑁1,𝑁𝜏=1,𝐼+2𝐴2𝜏1𝑢1𝑢0𝜏2𝑓0𝐴𝑢0=𝜔,𝑓0=𝑓(0),𝑢0=𝜇,(2.8) and the following formula holds: 𝑢0=𝜇,𝑢1=𝜏𝐼+2𝐴21𝜏𝜇+𝜏𝜔+22𝑓0,𝑢𝑘=𝜏𝐼+2𝐴21𝑅𝑘1𝐼𝑖𝜏𝐴1/2+𝑅𝑘1𝐼+𝑖𝜏𝐴1/22𝜇+𝜏𝐼+2𝐴21𝑖𝐴1/21𝑅𝑘1𝑅1𝑅𝑘1𝑅12𝜏𝜔+2𝑓0𝑘1𝑠=1𝜏𝐴2𝑖1/2𝑅𝑘𝑠𝑅𝑘𝑠𝑓𝑠,2𝑘𝑁.(2.9) Applying formula (2.9) and the nonlocal boundary conditions in problem (2.1), we get 𝜇=𝑇𝜏𝑛𝑚=1𝛼𝑚𝜏2𝑓0×𝜏𝐼+2𝐴21𝑖𝐴1/21×𝑅[𝜆𝑚/𝜏]1𝑅1𝑅[𝜆𝑚/𝜏]1𝑅12+𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22𝑛𝑚=1𝛼𝑚[𝜆𝑚/𝜏]1𝑠=1𝜏𝐴2𝑖1/2𝑅[𝜆𝑚/𝜏]s𝑅[𝜆𝑚/𝜏]s𝑓𝑠𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏×[𝜆𝑚/𝜏]2𝑠=1𝜏𝐴2𝑖1/2𝑖𝐴1/2𝑅[𝜆𝑚/𝜏]s𝐼𝑖𝜏2𝐴1/2+𝑅[𝜆𝑚/𝜏]s𝐼𝑖𝜏2𝐴1/2𝑓𝑠+𝑛𝑚=1𝛼𝑚𝜏𝜆𝑚𝜆𝑚𝜏𝜏𝑅𝑅𝑓[𝜆𝑚/𝜏]1×𝜑𝐼+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝐴𝜏𝐼+2𝐴21𝑖𝐴1/21×𝑅[𝜆𝑘/𝜏]1𝑅1𝑅[𝜆𝑘/𝜏]1𝑅12𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22+𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21𝑖𝐴1/21𝑅[𝜆𝑚/𝜏]1𝑅1𝑅[𝜆𝑚/𝜏]1𝑅12+𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22×𝜏2𝑓0𝑛𝑘=1𝛽𝑘𝜏𝐼+2𝐴21×𝑅[𝜆𝑘/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/22+𝑅[𝜆𝑘/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝜏𝐴×𝐼+2𝐴21𝑖𝐴1/21×𝑅[𝜆𝑘/𝜏]1𝑅1𝑅[𝜆𝑘/𝜏]1𝑅12+𝑛𝑘=1𝛽𝑘[𝜆𝑘/𝜏]2𝑠=1𝜏2𝑖×𝐴1/2𝑖𝐴1/2𝑅[𝜆𝑘/𝜏]𝑠𝐼+𝑖𝜏2𝐴1/2+𝑅[𝜆𝑘/𝜏]𝑠𝐼𝑖𝜏2𝐴1/2𝑓𝑠+𝑛𝑘=1𝛽𝑘𝜏𝑅𝑅𝑓[𝜆𝑘/𝜏]1+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝐴[𝜆𝑘/𝜏]1𝑠=1𝜏𝐴2𝑖1/2×𝑅[𝜆𝑘/𝜏]𝑠𝑅[𝜆𝑘/𝜏]𝑠𝑓𝑠+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝑓[𝜆𝑘/𝜏]1,+𝜓𝜔=𝑇𝜏𝐼𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝐼𝑖𝜏𝐴1/22+𝑅[𝜆𝑚/𝜏]1𝐼+𝑖𝜏𝐴1/22𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴21𝑖𝐴1/2×𝑅[𝜆𝑚/𝜏]1𝐼(𝑖𝜏/2)𝐴1/2𝐼+𝑖𝜏𝐴1/2𝑅[𝜆𝑚/𝜏]1𝐼+(𝑖𝜏/2)𝐴1/2𝐼𝑖𝜏𝐴1/22×𝜏2𝑓0𝑛𝑘=1𝛽𝑘𝜏𝐼+2𝐴21×𝑅[𝜆𝑘/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/22+𝑅[𝜆𝑘/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝜏𝐴×𝐼+2𝐴21𝑖𝐴1/21×𝑅[𝜆𝑘/𝜏]1𝑅1𝑅[𝜆𝑘/𝜏]1𝑅12+𝑛𝑘=1𝛽𝑘[𝜆𝑘/𝜏]2𝑠=1𝜏2𝑖×𝐴1/2𝑖𝐴1/2𝑅[𝜆𝑘/𝜏]𝑠𝐼+𝑖𝜏2𝐴1/2+𝑅[𝜆𝑘/𝜏]𝑠𝐼𝑖𝜏2𝐴1/2𝑓𝑠+𝑛𝑘=1𝛽𝑘𝑅𝑅𝑓[𝜆𝑘/𝜏]1+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝐴[𝜆𝑘/𝜏]1𝑠=1𝜏2𝑖×𝐴1/2𝑅[𝜆𝑘/𝜏]𝑠𝑅[𝜆𝑘/𝜏]𝑠𝑓𝑠+𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝑓[𝜆𝑘/𝜏]1+𝜓𝑛𝑘=1𝛽𝑘𝜏2𝐴𝜏𝐼+2𝐴21𝑅[𝜆𝑘/𝜏]1𝐼𝑖𝜏𝐴1/2+𝑅[𝜆𝑘/𝜏]1𝐼+𝑖𝜏𝐴1/2212𝑛𝑘=1𝛽𝑘𝜏𝐼+2𝐴21𝑅[𝜆𝑘/𝜏]1𝐼𝑖𝜏2𝐴1/2𝐼+𝑖𝜏𝐴1/2𝑅[𝜆𝑘/𝜏]1𝐼+𝑖𝜏2𝐴1/2𝐼𝑖𝜏𝐴1/2×𝜏2𝑓0𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21𝑖𝐴1/21𝑅[𝜆𝑚/𝜏]1𝑅1𝑅[𝜆𝑚/𝜏]1𝑅12+𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22𝑛𝑚=1𝛼𝑚[𝜆𝑚/𝜏]1𝑠=1𝜏𝐴2𝑖1/2𝑅[𝜆𝑚/𝜏]𝑠𝑅[𝜆𝑚/𝜏]𝑠𝑓𝑠+𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏×[𝜆𝑚/𝜏]2𝑠=1𝜏𝐴2𝑖1/2𝑖𝐴1/2𝑅[𝜆𝑚/𝜏]𝑠𝐼+𝑖𝜏2𝐴1/2+𝑅[𝜆𝑚/𝜏]𝑠𝐼𝑖𝜏2𝐴1/2𝑓𝑠+𝑛𝑚=1𝛼𝑚𝜏𝜆𝑚𝜆𝑚𝜏𝜏𝑅𝑅𝑓[𝜆𝑚/𝜏]1.𝜑(2.10) Thus, formulas (2.9) and (2.10) give a solution of problem (2.1).

Theorem 2.3. Suppose that assumption (2.2) holds and 𝜑𝐷(𝐴),  𝜓𝐷(𝐴1/2). Then, for the solution of difference scheme (2.1) the stability inequalities max0𝑘𝑁𝑢𝑘𝐻𝑀𝑁1𝑘=0𝐴1/2𝑓𝑘𝐻𝐴𝜏+1/2𝜓𝐻+𝜑𝐻,(2.11)max0𝑘𝑁𝐴1/2𝑢𝑘𝐻𝑀𝑁1𝑘=0𝑓𝑘𝐻𝐴𝜏+1/2𝜑𝐻+𝜓𝐻,(2.12)max1𝑘𝑁1𝜏2𝑢𝑘+12𝑢𝑘+𝑢𝑘1𝐻+max0𝑘𝑁1𝐴𝑢𝑘+𝜏24𝐴2𝑢𝑘+1𝐻𝑀𝑁1𝑘=1𝑓𝑘𝑓𝑘1𝐻+𝑓0𝐻+𝐴1/2𝜓𝐻+𝐴𝜑𝐻(2.13) hold, where 𝑀 does not depend on 𝜏,  𝜑,  𝜓, and 𝑓𝑘,  0𝑘𝑁1.

Proof. By [18], the following estimates max0𝑘𝑁𝑢𝑘𝐻𝑀𝑁1𝑘=0𝐴1/2𝑓𝑘𝐻𝐴𝜏+1/2𝜔𝐻+𝜇𝐻,(2.14)max0𝑘𝑁𝐴1/2𝑢𝑘𝐻𝑀𝑁1𝑘=0𝑓𝑘𝐻𝐴𝜏+1/2𝜇𝐻+𝜔𝐻,(2.15)max1𝑘𝑁1𝜏2𝑢𝑘+12𝑢𝑘+𝑢𝑘1𝐻+max0𝑘𝑁1𝐴𝑢𝑘+𝜏24𝐴2𝑢𝑘+1𝐻𝑀𝑁1𝑘=1𝑓𝑘𝑓𝑘1𝐻+𝑓0𝐻+𝐴1/2𝜔𝐻+𝐴𝜇𝐻(2.16) hold for the solution of (2.8). Using formulas of 𝜇, 𝜔, and (2.3) and (2.6) the following estimates obtained 𝜇𝐻𝑀𝑁1𝑠=0𝐴1/2𝑓𝑠𝐻𝐴𝜏+1/2𝜓𝐻+𝜑𝐻,𝐴(2.17)1/2𝜔𝐻𝑀𝑁1𝑠=0𝐴1/2𝑓𝑠𝐻𝐴𝜏+1/2𝜓𝐻+𝜑𝐻.(2.18) Estimate (2.11) follows from (2.14), (2.17), and (2.18). In a similar manner, we obtain max0𝑘𝑁𝐴1/2𝑢𝑘𝐻𝑀𝑁1𝑘=0𝑓𝑘𝐻𝐴𝜏+1/2𝜑𝐻+𝜓𝐻.(2.19) Now, we obtain the estimates for 𝐴𝜇𝐻 and 𝐴1/2𝜔𝐻. First, applying 𝐴 to the formula of 𝜇 and using Abel’s formula, we can write 𝐴𝜇=𝑇𝜏𝜏2𝑓0𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21𝐴1/2𝑅[𝜆𝑚/𝜏]1𝑅1𝑅[𝜆𝑚/𝜏]1𝑅1+2𝑖𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝐴𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/2212𝑛𝑚=1𝛼𝑚[𝜆𝑚/𝜏]1𝑠=2𝑅[𝜆𝑚/𝜏]s𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑚/𝜏]s𝐼𝑖𝜏2𝐴1/21×𝑓𝑠𝑓𝑠1+𝑛𝑚=1𝛼𝑚12𝑅[𝜆𝑚/𝜏]1𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑚/𝜏]1𝐼𝑖𝜏2𝐴1/21𝑓1𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴41𝑓[𝜆𝑚/𝜏]1+𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏×[𝜆𝑚/𝜏]2𝑠=21𝐴2𝑖1/2×𝑅[𝜆𝑚/𝜏]𝑠+𝑅[𝜆𝑚/𝜏]𝑠𝑓𝑠𝑓𝑠1+1𝐴2𝑖1/2𝑅[𝜆𝑚/𝜏]1+𝑅[𝜆𝑚/𝜏]1𝑓1+𝑖𝐴1/2𝑓[𝜆𝑚/𝜏]2+𝑛𝑚=1𝛼𝑚𝜏𝜆𝑚𝜆𝑚𝜏𝜏𝐴𝑅𝑅𝑓[𝜆𝑚/𝜏]1×𝐴𝜑𝐼+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝐴𝜏𝐼+2𝐴21𝑖𝐴1/21×𝑅[𝜆𝑘/𝜏]1𝑅1𝑅[𝜆𝑘/𝜏]1𝑅12𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22+𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21𝑅[𝜆𝑚/𝜏]1𝑅1𝑅[𝜆𝑚/𝜏]1𝑅1+2𝑖𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏×𝐴1/2𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22×𝜏2𝑓0𝑛𝑘=1𝛽𝑘𝐴1/2𝜏𝐼+2𝐴21×𝑅[𝜆𝑘/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/22+𝑅[𝜆𝑘/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22+𝑛𝑘=1𝛽𝑘𝑖𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝜏𝐴×𝐼+2𝐴21×𝑅[𝜆𝑘/𝜏]1𝑅1𝑅[𝜆𝑘/𝜏]1𝑅12𝑛𝑘=1𝛽𝑘[𝜆𝑘/𝜏]2𝑠=21𝑅2𝑖[𝜆𝑘/𝜏]𝑠+𝑅[𝜆𝑘/𝜏]𝑠𝑓𝑠𝑓𝑠1𝑛𝑘=1𝛽𝑘1𝑅2𝑖[𝜆𝑘/𝜏]1+𝑅[𝜆𝑘/𝜏]1𝑓1𝑛𝑘=1𝛽𝑘𝑖𝑓[𝜆𝑚/𝜏]2+𝑛𝑘=1𝛽𝑘𝐴1/2𝜏𝑅𝑅𝑓[𝜆𝑘/𝜏]1+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏×12𝐴1/2[𝜆𝑘/𝜏]1𝑠=2𝑅[𝜆𝑘/𝜏]𝑠𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑘/𝜏]𝑠𝐼𝑖𝜏2𝐴1/21×𝑓𝑠𝑓𝑠1+12𝐴1/2𝑅[𝜆𝑘/𝜏]1𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑘/𝜏]1𝐼𝑖𝜏2𝐴1/21𝑓1𝐴1/2𝜏𝐼+2𝐴41𝑓[𝜆𝑘/𝜏]1+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝐴1/2𝑓[𝜆𝑘/𝜏]1+𝐴1/2𝜓.(2.20) Second, applying 𝐴1/2 to the formula of 𝜔 and using Abel’s formula, we can write 𝐴1/2𝜔=𝑇𝜏𝐼𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝐼𝑖𝜏𝐴1/22+𝑅[𝜆𝑚/𝜏]1𝐼+𝑖𝜏𝐴1/22𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝜏𝐼+2𝐴21𝑖𝐴1/2×𝑅[𝜆𝑚/𝜏]1𝐼(𝑖𝜏/2)𝐴1/2𝐼+𝑖𝜏𝐴1/2𝑅[𝜆𝑚/𝜏]1𝐼+(𝑖𝜏/2)𝐴1/2𝐼𝑖𝜏𝐴1/22×12𝑓0𝑛𝑘=1𝛽𝑘𝐴1/2𝜏𝜏𝐼+2𝐴21×𝑅[𝜆𝑘/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/22+𝑅[𝜆𝑘/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/22+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝜏𝐼+2𝐴21×𝐴1/2𝑖𝑅[𝜆𝑘/𝜏]1𝑅1𝑅[𝜆𝑘/𝜏]1𝑅12𝑛𝑘=1𝛽𝑘[𝜆𝑘/𝜏]2𝑠=21𝑅2𝑖[𝜆𝑘/𝜏]s+𝑅[𝜆𝑘/𝜏]s×𝑓𝑠𝑓𝑠1𝑛𝑘=1𝛽𝑘1𝑅2𝑖[𝜆𝑘/𝜏]1+𝑅[𝜆𝑘/𝜏]1𝑓1𝑛𝑘=1𝛽𝑘𝑖𝑓[𝜆𝑘/𝜏]2+𝑛𝑘=1𝛽𝑘𝐴1/2𝜏𝑅𝑅𝑓[𝜆𝑘/𝜏]1+𝑛𝑘=1𝛽𝑘𝜏2+𝜆𝑘𝜆𝑘𝜏𝜏𝐴1/2×12[𝜆𝑘/𝜏]1𝑠=2𝑅[𝜆𝑘/𝜏]s𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑘/𝜏]𝑠𝐼𝑖𝜏2𝐴1/21×𝑓𝑠𝑓𝑠1+12𝑅[𝜆𝑘/𝜏]1𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑘/𝜏]1𝐼𝑖𝜏2𝐴1/21𝑓1𝜏𝐼+2𝐴41𝑓[𝜆𝑘/𝜏]1+𝐴1/2𝜓𝑛𝑘=1𝛽𝑘𝐴1/2𝜏2𝜏𝐼+2𝐴21×𝑅[𝜆𝑘/𝜏]1𝐼𝑖𝜏𝐴1/2+𝑅[𝜆𝑘/𝜏]1𝐼+𝑖𝜏𝐴1/22𝑛𝑘=1𝛽𝑘𝐴1/2𝜏𝐼+2𝐴21×𝑅[𝜆𝑘/𝜏]1𝐼(𝑖𝜏/2)𝐴1/2𝐼+𝑖𝜏𝐴1/22𝑅[𝜆𝑘/𝜏]1𝐼+(𝑖𝜏/2)𝐴1/2𝐼𝑖𝜏𝐴1/22×12𝑓0𝑛𝑚=1𝛼𝑚𝜏𝐼+2𝐴21×𝜏𝐴1/2𝑅[𝜆𝑚/𝜏]1𝑅1𝑅[𝜆𝑚/𝜏]1𝑅1+2𝑖𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏𝐴𝜏𝐼+2𝐴21×𝑅[𝜆𝑚/𝜏]1𝑅1𝐼(𝑖𝜏/2)𝐴1/2+𝑅[𝜆𝑚/𝜏]1𝑅1𝐼+(𝑖𝜏/2)𝐴1/2212𝑛𝑚=1𝛼𝑚𝐴1/2×[𝜆𝑚/𝜏]1𝑠=2𝑅[𝜆𝑚/𝜏]𝑠𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑚/𝜏]𝑠𝐼𝑖𝜏2𝐴1/21×𝑓𝑠𝑓𝑠1+12𝑛𝑚=1𝛼𝑚𝐴1/2×𝑅[𝜆𝑚/𝜏]1𝐼+𝑖𝜏2𝐴1/21+𝑅[𝜆𝑚/𝜏]1𝐼𝑖𝜏2𝐴1/21𝑓1𝑛𝑚=1𝛼𝑚𝐴1/2𝜏𝐼+2𝐴41𝑓[𝜆𝑚/𝜏]1𝑛𝑚=1𝛼𝑚𝜆𝑚𝜆𝑚𝜏𝜏×[𝜆𝑚/𝜏]2𝑠=21𝐴2𝑖1/2×𝑅[𝜆𝑚/𝜏]𝑠+𝑅[𝜆𝑚/𝜏]𝑠𝑓𝑠𝑓𝑠1+1𝐴2𝑖1/2𝑅[𝜆𝑚/𝜏]1+𝑅[𝜆𝑚/𝜏]1𝑓1+𝑖𝐴1/2𝑓[𝜆𝑚/𝜏]2+𝑛𝑚=1𝛼𝑚𝜏𝜆𝑚𝜆𝑚𝜏𝜏𝐴𝑅𝑅𝑓[𝜆𝑚/𝜏]1.𝐴𝜑(2.21) The following estimates 𝐴𝜇𝐻𝑀𝑁1𝑠=1𝑓𝑠𝑓𝑠1𝐻+𝑓0𝐻+𝐴1/2𝜓𝐻+𝐴𝜑𝐻,𝐴1/2𝜔𝐻𝑀𝑁1𝑠=1𝑓𝑠𝑓𝑠1𝐻+𝑓0𝐻+𝐴1/2𝜓𝐻+𝐴𝜑𝐻(2.22) are obtained by using formulas (2.20), (2.21), (2.3), and (2.6).
Estimate (2.13) follows from (2.16), and (2.22). Theorem 2.3 is proved.
Now, let us consider the applications of Theorem 2.3. First, the nonlocal the mixed boundary value problem for hyperbolic equation 𝑢𝑡𝑡𝑎(𝑥)𝑢𝑥𝑥+𝛿𝑢=𝑓(𝑡,𝑥),0<𝑡<1,0<𝑥<1,𝑢(0,𝑥)=𝑛𝑚=1𝛼𝑚𝑢𝜆𝑚𝑢,𝑥+𝜑(𝑥),0𝑥1,𝑡(0,𝑥)=𝑛𝑘=1𝛽𝑘𝑢𝑡𝜆𝑘,𝑥+𝜓(𝑥),0𝑥1,𝑢(𝑡,0)=𝑢(𝑡,1),𝑢𝑥(𝑡,0)=𝑢𝑥(𝑡,1),0𝑡1,(2.23) under assumption (2.2), is considered. Here 𝑎𝑟(𝑥),  (𝑥(0,1)),  𝜑(𝑥),𝜓(𝑥)(𝑥[0,1]) and 𝑓(𝑡,𝑥)(𝑡(0,1),𝑥(0,1)) are given smooth functions and 𝑎𝑟(𝑥)𝑎>0,  𝛿>0. The discretization of problem (2.23) is carried out in two steps.
In the first step, the grid space is defined as follows: []0,1=𝑥𝑥𝑟=𝑟,0𝑟𝐾,𝐾=1.(2.24) We introduce the Hilbert space 𝐿2=𝐿2([0,1]),  𝑊12=𝑊12([0,1]) and 𝑊22=𝑊22([0,1]) of the grid functions 𝜑(𝑥)={𝜑𝑟}1𝐾1 defined on [0,1], equipped with the norms 𝜑𝐿2=𝐾1𝑟=1||𝜑||(𝑥)21/2,𝜑𝑊12=𝜑𝐿2+𝐾1𝑟=1|||𝜑𝑥,𝑗|||21/2,𝜑𝑊22=𝜑𝐿2+𝐾1𝑟=1|||𝜑𝑥,𝑗|||21/2+𝐾1𝑟=1|||𝜑𝑥𝑥,𝑗|||21/2,(2.25) respectively. To the differential operator 𝐴 generated by problem (2.23), we assign the difference operator 𝐴𝑥 by the formula 𝐴𝑥𝜑(𝑥)=(𝑎(𝑥)𝜑𝑥)𝑥,𝑟+𝛿𝜑𝑟1𝐾1,(2.26) acting in the space of grid functions 𝜑(𝑥)={𝜑𝑟}𝐾0 satisfying the conditions 𝜑0=𝜑𝐾,  𝜑1𝜑0=𝜑𝐾𝜑𝐾1. With the help of 𝐴𝑥, we arrive at the nonlocal boundary value problem 𝑑2𝑣(𝑡,𝑥)𝑑𝑡2+𝐴𝑥𝑣(𝑡,𝑥)=𝑓[](𝑡,𝑥),0𝑡1,𝑥0,1,𝑣(0,𝑥)=𝑛𝑗=1𝛼𝑗𝑣𝜆𝑗,𝑥+𝜑[](𝑥),𝑥0,1,𝑣𝑡(0,𝑥)=𝑛𝑗=1𝛽𝑗𝑣𝑡𝜆𝑗,𝑥+𝜓[](𝑥),𝑥0,1,(2.27) for an infinite system of ordinary differential equations.
In the second step, we replace problem (2.27) by difference scheme (2.28) 𝑢𝑘+1(𝑥)2𝑢𝑘(𝑥)+𝑢𝑘1(𝑥)𝜏2+𝐴𝑥𝑢𝑘𝜏(𝑥)+24𝐴𝑥2𝑢𝑘+1(𝑥)=𝑓𝑘[](𝑥),𝑥0,1,𝑓𝑘(𝑥)=𝑓𝑡𝑘,𝑥,𝑡𝑘𝑢=𝑘𝜏,1𝑘𝑁1,𝑁𝜏=1,0(𝑥)=𝑛𝑗=1𝛼𝑗𝑢𝜆𝑗/𝜏(𝑥)+𝜏1𝑢[𝜆𝑗/𝜏](𝑥)𝑢[𝜆𝑗/𝜏]1𝜆(𝑥)𝑗𝜆𝑗𝜏𝜏+𝜑[](𝑥),𝑥0,1,𝜏𝐼+2𝐴𝑥2𝑢1(𝑥)𝑢0(𝑥)𝜏𝜏2𝑓0(𝑥)𝐴𝑥𝑢0=(𝑥)𝑛𝑗=1𝛽𝑗𝑢𝜆𝑗/𝜏(𝑥)𝑢𝜆𝑗/𝜏1(𝑥)𝜏+𝜏2+𝜆𝑗𝜆𝑗𝜏𝜏×𝑓[𝜆𝑗/𝜏](𝑥)𝐴𝑥𝑢[𝜆𝑗/𝜏](𝑥)+𝜓𝑓(𝑥),0(𝑥)=𝑓[](0,𝑥),𝑥0,1.(2.28)

Theorem 2.4. Let 𝜏 and be sufficiently small positive numbers. Suppose that assumption (2.2) holds. Then, the solution of difference scheme (2.28) satisfies the following stability estimates: max0𝑘𝑁𝑢𝑘𝐿2+max0𝑘𝑁𝑢𝑘𝑥𝐿2𝑀1max0𝑘𝑁1𝑓𝑘𝐿2+𝜓𝐿2+𝜑𝑥𝐿2,max1𝑘𝑁1𝜏2𝑢𝑘+12𝑢𝑘+𝑢𝑘1𝐿2+max0𝑘𝑁𝑢𝑘𝑥𝑥𝐿2𝑀1𝑓0𝐿2+max1𝑘𝑁1𝜏1𝑓𝑘𝑓𝑘1𝐿2+𝜓𝑥𝐿2+𝜑𝑥𝑥𝐿2.(2.29) Here, 𝑀1 does not depend on 𝜏,  ,  𝜑(𝑥),  𝜓(𝑥) and 𝑓𝑘,  0𝑘<𝑁.

The proof of Theorem 2.4 is based on abstract Theorem 2.3 and symmetry properties of the operator 𝐴𝑥 defined by (2.26).

Second, for the 𝑚-dimensional hyperbolic equation under assumption (2.2) is considered. Let Ω be the unit open cube in the 𝑚-dimensional Euclidean space 𝑚{𝑥=(𝑥1,,𝑥𝑚)0<𝑥𝑗<1,1𝑗𝑚} with boundary 𝑆,Ω=Ω𝑆. In [0,1]×Ω, the mixed boundary value problem for the multidimensional hyperbolic equation 𝜕2𝑢(𝑡,𝑥)𝜕𝑡2𝑚𝑟=1𝑎𝑟(𝑥)𝑢𝑥𝑟𝑥𝑟𝑥=𝑓(𝑡,𝑥),𝑥=1,,𝑥𝑚Ω,0<𝑡<1,𝑢(0,𝑥)=𝑛𝑗=1𝛼𝑗𝑢𝜆𝑗,𝑥+𝜑(𝑥),𝑥𝑢Ω;𝑡(0,𝑥)=𝑛𝑘=1𝛽𝑘𝑢𝑡𝜆𝑘,𝑥+𝜓(𝑥),𝑥Ω;𝑢(𝑡,𝑥)=0,𝑥𝑆(2.30) is considered.

Here, 𝑎𝑟(𝑥),  (𝑥Ω),  𝜑(𝑥),  𝜓(𝑥)  (𝑥Ω) and 𝑓(𝑡,𝑥)  (𝑡(0,1),𝑥Ω) are given smooth functions and 𝑎𝑟(𝑥)𝑎>0. The discretization of problem (2.30) is carried out in two steps. In the first step, let us define the grid sets Ω=𝑥=𝑥𝑟=1𝑟1,,𝑚𝑟𝑚𝑟,𝑟=1,,𝑟𝑚,0𝑟𝑗𝑁𝑗,𝑗𝑁𝑗,Ω=1,𝑗=1,,𝑚=ΩΩ,𝑆=Ω𝑆.(2.31) We introduce the Banach space 𝐿2=𝐿2(Ω),  𝑊12=𝑊12(Ω) and 𝑊22=𝑊22(Ω) of the grid functions 𝜑(𝑥)={𝜑(1𝑟1,,𝑚𝑟𝑚)} defined on Ω, equipped with the norms 𝜑𝐿2(Ω)=𝑥Ω||𝜑||(𝑥)21𝑚1/2,𝜑𝑊12=𝜑𝐿2+𝑥Ω𝑚𝑟=1|||𝜑𝑥𝑟,𝑗𝑟|||21𝑚1/2,𝜑𝑊22=𝜑𝐿2+𝑥Ω𝑚𝑟=1|||𝜑𝑥𝑟|||21𝑚1/2+𝑥Ω𝑚𝑟=1|||𝜑𝑥𝑟𝑥𝑟,𝑗𝑟|||21𝑚1/2,(2.32) respectively. To the differential operator 𝐴 generated by problem (2.27), we assign the difference operator 𝐴𝑥 by the formula 𝐴𝑥𝑢=𝑚𝑟=1𝑎𝑟(𝑥)𝑢𝑥𝑟𝑥𝑟,𝑗𝑟,(2.33) acting in the space of grid functions 𝑢(𝑥), satisfying the conditions 𝑢(𝑥)=0 for all 𝑥𝑆. Note that 𝐴𝑥 is a self-adjoint positive definite operator in 𝐿2(Ω). With the help of 𝐴𝑥 we arrive at the nonlocal boundary value problem 𝑑2𝑣(𝑡,𝑥)𝑑𝑡2+𝐴𝑥𝑣(𝑡,𝑥)=𝑓(𝑡,𝑥),0<𝑡<1,𝑥Ω,𝑣(0,𝑥)=𝑛𝑙=1𝛼𝑙𝑣𝜆𝑙,𝑥+𝜑Ω(𝑥),𝑥,𝑑𝑣(0,𝑥)=𝑑𝑡𝑛𝑙=1𝛽𝑙𝑣𝑡𝜆𝑙,𝑥+𝜓(𝑥),𝑥Ω,(2.34) for an infinite system of ordinary differential equations.

In the second step, we replace problem (2.34) by difference scheme (2.35) 𝑢𝑘+1(𝑥)2𝑢𝑘(𝑥)+𝑢𝑘1(𝑥)𝜏2+𝐴𝑥𝑢𝑘𝜏(𝑥)+24𝐴𝑥2𝑢𝑘+1(𝑥)=𝑓𝑘(𝑥),𝑥Ω,𝑓𝑘(𝑥)=𝑓𝑡𝑘,𝑥,𝑡𝑘𝑢=𝑘𝜏,1𝑘𝑁1,𝑁𝜏=1,0(𝑥)=𝑛𝑙=1𝛼𝑙𝑢[𝜆𝑙/𝜏](𝑥)+𝜏1𝑢[𝜆𝑙/𝜏](𝑥)𝑢[𝜆𝑙/𝜏]1𝜆(𝑥)𝑙𝜆𝑙𝜏/𝜏+𝜑Ω(𝑥),𝑥,𝜏𝐼+2𝐴𝑥2𝑢1(𝑥)𝑢0(𝑥)𝜏𝜏2𝑓0(𝑥)𝐴𝑥𝑢0=(𝑥)𝑛𝑙=1𝛽𝑙𝑢[𝜆𝑙/𝜏](𝑥)𝑢[𝜆𝑙/𝜏]1(𝑥)𝜏+𝜏2+𝜆𝑙𝜆𝑙𝜏𝜏×𝑓[𝜆𝑙/𝜏](𝑥)𝐴𝑥𝑢[𝜆𝑙/𝜏](𝑥)+𝜓𝑓(𝑥),0(𝑥)=𝑓Ω(0,𝑥),𝑥.(2.35)

Theorem 2.5. Let 𝜏 and be sufficiently small positive numbers. Suppose that assumption (2.2) holds. Then, the solution of difference scheme (2.35) satisfies the following stability estimates: max0𝑘𝑁𝑢𝑘𝐿2+max𝑚0𝑘𝑁𝑟=1𝑢𝑘𝑥𝑟,𝑗𝑟𝐿2𝑀1max0𝑘𝑁1𝑓𝑘𝐿2+𝜓𝐿2+𝑚𝑟=1𝜑𝑥𝑟,𝑗𝑟𝐿2,max1𝑘𝑁1𝜏2𝑢𝑘+12𝑢𝑘+𝑢𝑘1𝐿2+max𝑚0𝑘𝑁𝑟=1𝑢𝑘𝑥𝑟𝑥𝑟,𝑗𝑟𝐿2𝑀1𝑓0𝐿2+max1𝑘𝑁1𝜏1𝑓𝑘𝑓𝑘1