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Well-Posed and Ill-Posed Boundary Value Problems for PDE

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Volume 2012 |Article ID 846582 | https://doi.org/10.1155/2012/846582

Allaberen Ashyralyev, Ozgur Yildirim, "A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem", Abstract and Applied Analysis, vol. 2012, Article ID 846582, 29 pages, 2012. https://doi.org/10.1155/2012/846582

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

Academic Editor: Sergey Piskarev
Received23 Mar 2012
Accepted11 Jun 2012
Published06 Sep 2012

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 [1–5] 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., [6–16] and the references given therein).

In the present work, we consider the nonlocal boundary value problem 𝑑2𝑒(𝑑)𝑑𝑑2𝑒+𝐴𝑒(𝑑)=𝑓(𝑑)(0≀𝑑≀1),(0)=𝑛𝑗=1π›Όπ‘—π‘’ξ€·πœ†π‘—ξ€Έ+πœ‘,𝑒𝑑(0)=𝑛𝑗=1π›½π‘—π‘’π‘‘ξ€·πœ†π‘—ξ€Έ+πœ“,0<πœ†1<πœ†2<β‹―<πœ†π‘›β‰€1,(1.1) where 𝐴 is a self-adjoint positive definite operator in a Hilbert space 𝐻.

A function 𝑒(𝑑) is called a solution of the problem (1.1), if the following conditions are satisfied:(i)𝑒(𝑑) 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 𝑒(𝑑) belongs to 𝐷(𝐴), independent of 𝑑, and dense in 𝐻 for all π‘‘βˆˆ[0,1] and the function 𝐴𝑒(𝑑) is continuous on the segment [0,1].(iii)𝑒(𝑑) 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 πœ‘βˆˆπ·(𝐴), πœ“βˆˆπ·(𝐴1/2), and 𝑓(𝑑) is a continuously differentiable function on [0,1] and the assumption π‘›ξ“π‘˜=1||π›Όπ‘˜+π›½π‘˜||+π‘›ξ“π‘š=1||π›Όπ‘š||π‘›ξ“π‘˜=1π‘˜β‰ π‘š||π›½π‘˜||<|||||1+π‘›ξ“π‘˜=1π›Όπ‘˜π›½π‘˜|||||(1.2) holds. Then, there is a unique solution of problem (1.1) and the stability inequalities max0≀𝑑≀1‖𝑒(𝑑)β€–π»ξ‚Έβ‰€π‘€β€–πœ‘β€–π»+β€–β€–π΄βˆ’1/2πœ“β€–β€–π»+max0≀𝑑≀1β€–β€–π΄βˆ’1/2‖‖𝑓(𝑑)𝐻,max0≀𝑑≀1‖‖𝐴1/2‖‖𝑒(𝑑)𝐻‖‖𝐴≀𝑀1/2πœ‘β€–β€–π»+β€–πœ“β€–π»+max0≀𝑑≀1‖𝑓(𝑑)‖𝐻,max0≀𝑑≀1‖‖‖𝑑2𝑒(𝑑)𝑑𝑑2‖‖‖𝐻+max0≀𝑑≀1‖𝐴𝑒(𝑑)β€–π»ξ‚Έβ‰€π‘€β€–π΄πœ‘β€–π»+‖‖𝐴1/2πœ“β€–β€–π»+‖𝑓(0)‖𝐻+ξ€œ10β€–β€–π‘“ξ…žβ€–β€–(𝑑)𝐻𝑑𝑑(1.3) hold, where 𝑀 does not depend on πœ‘,β€‰β€‰πœ“, and 𝑓(𝑑),β€‰β€‰π‘‘βˆˆ[0,1].

Moreover, the first order of accuracy difference scheme πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1ξ€Έ+π΄π‘’π‘˜+1=π‘“π‘˜,π‘“π‘˜ξ€·π‘‘=π‘“π‘˜ξ€Έ,π‘‘π‘˜π‘’=π‘˜πœ,1β‰€π‘˜β‰€π‘βˆ’1,π‘πœ=1,0=π‘›ξ“π‘Ÿ=1π›Όπ‘Ÿπ‘’[πœ†π‘Ÿ/𝜏]𝜏+πœ‘,βˆ’1𝑒1βˆ’π‘’0ξ€Έ=π‘›ξ“π‘Ÿ=1π›½π‘Ÿξ€·π‘’[πœ†π‘Ÿ/𝜏]+1βˆ’π‘’[πœ†π‘Ÿ/𝜏]ξ€Έ1𝜏+πœ“,(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 π‘›ξ“π‘˜=1||π›Όπ‘˜||+π‘›ξ“π‘˜=1||π›½π‘˜||+π‘›ξ“π‘˜=1||π›Όπ‘˜||π‘›ξ“π‘˜=1||π›½π‘˜||<1,(1.5) were established.

In the development of numerical techniques for solving PDEs, the stability has been an important research topic (see [6–31]). 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 β„Ž with respect to the time and space variables, are connected. In abstract terms, this particularly means that πœβ€–π΄β„Žβ€–β†’0 when πœβ†’0.

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 β„Ž. 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., [11–16, 20–24, 32–38] and the references therein).

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

Throughout this paper for simplicity πœ†1>2𝜏 and πœ†π‘›<1 will be considered. Let us associate boundary value problem (1.1) with the second order of accuracy difference scheme πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1ξ€Έ+π΄π‘’π‘˜+𝜏24𝐴2π‘’π‘˜+1=π‘“π‘˜,π‘“π‘˜ξ€·π‘‘=π‘“π‘˜ξ€Έ,π‘‘π‘˜π‘’=π‘˜πœ,1β‰€π‘˜β‰€π‘βˆ’1,π‘πœ=1,0=π‘›ξ“π‘š=1π›Όπ‘šξ‚»π‘’[πœ†π‘š/𝜏]+πœβˆ’1𝑒[πœ†π‘š/𝜏]βˆ’π‘’[πœ†π‘š/𝜏]βˆ’1ξ€ΈΓ—ξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚΅πœξ‚Άξ‚Ό+πœ‘,𝐼+2𝐴2ξ‚Άπœβˆ’1𝑒1βˆ’π‘’0ξ€Έβˆ’πœ2𝑓0βˆ’π΄π‘’0ξ€Έ=π‘›ξ“π‘˜=1π›½π‘˜ξ‚»πœβˆ’1𝑒[πœ†π‘˜/𝜏]βˆ’π‘’[πœ†π‘˜/𝜏]βˆ’1ξ€Έ+ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚ΉπœΓ—ξ€·π‘“ξ‚Άξ‚Ά[πœ†π‘˜/𝜏]βˆ’π΄π‘’[πœ†π‘˜/𝜏]𝑓+πœ“,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 π΄β„Ž that act in the Hilbert space and are uniformly self-adjoint positive definite in β„Ž for 0<β„Žβ‰€β„Ž0.

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 π‘›ξ“π‘˜=1||π›Όπ‘˜||||||πœ†1+π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||ξƒͺ+π‘›ξ“π‘˜=1||π›½π‘˜||||||πœ†1+π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||ξƒͺ+π‘›ξ“π‘˜=1||π›Όπ‘˜||π‘›ξ“π‘˜=1||π›½π‘˜||||||πœ†1+π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||ξƒͺ+π‘›ξ“π‘˜=1||π›Όπ‘˜||||||πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||π‘›ξ“π‘˜=1||π›½π‘˜||||||πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||<1.(2.2) Now, let us give some lemmas that will be needed in the sequel.

Lemma 2.1. The following estimates hold: ‖𝑅‖𝐻↦𝐻‖‖𝑅‖‖≀1,𝐻↦𝐻‖‖𝑅≀1,βˆ’1𝑅‖‖𝐻↦𝐻‖‖𝑅≀1,βˆ’1𝑅‖‖𝐻↦𝐻‖‖≀1,𝜏𝐴1/2𝑅‖‖𝐻↦𝐻‖‖≀1,𝜏𝐴1/2𝑅‖‖𝐻↦𝐻≀1.(2.3) Here, 𝐻 is the Hilbert space, 𝑅=(𝐼+π‘–πœπ΄1/2βˆ’(𝜏2/2)𝐴)βˆ’1, and 𝑅=(πΌβˆ’π‘–πœπ΄1/2βˆ’(𝜏2/2)𝐴)βˆ’1.

Lemma 2.2. Suppose that assumption (2.2) holds. Denote 𝐡𝜏=π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1βŽ‘βŽ’βŽ’βŽ£π‘…[πœ†π‘š/𝜏]βˆ’1ξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1𝐼+π‘–πœπ΄1/2ξ€Έ2⎀βŽ₯βŽ₯⎦+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1+𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒ­βˆ’12π‘›ξ“π‘›π‘š=1ξ“π‘˜=1π›Όπ‘šπ›½π‘˜ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑅[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1+𝑅[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚„+π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚Άξ€·π‘–π΄1/2ξ€Έξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ‘βŽ’βŽ’βŽ£ξ‚π‘…[πœ†π‘š/𝜏]βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2𝐼+π‘–πœπ΄1/2ξ€Έβˆ’π‘…[πœ†π‘š/𝜏]βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ2⎀βŽ₯βŽ₯⎦+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœξ‚Άξ€·π‘–π΄1/2ξ€Έξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒ­+14π‘›ξ“π‘›π‘š=1ξ“π‘˜=1π›Όπ‘šπ›½π‘˜ξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’2×𝑖𝐴1/2𝑅[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1𝐼+π‘–πœ2𝐴1/2ξ‚βˆ’ξ‚π‘…[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚€πΌβˆ’π‘–πœ2𝐴1/2βˆ’12π‘›ξ“π‘›π‘š=1ξ“π‘˜=1π›Όπ‘šπ›½π‘˜ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœξ‚Άξ€·π‘–π΄1/2ξ€Έξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1×𝑅[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1βˆ’ξ‚π‘…[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚„βˆ’12π‘›ξ“π‘›π‘š=1ξ“π‘˜=1π›Όπ‘šπ›½π‘˜ξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœ†ξ‚Άξ‚΅π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœξ‚Άπ΄ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1×𝑅[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚€πΌβˆ’π‘–πœ2𝐴1/2+𝑅[πœ†π‘š/𝜏]βˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1𝐼+π‘–πœ2𝐴1/2.(2.4) Then, the operator πΌβˆ’π΅πœ has an inverse π‘‡πœ=ξ€·πΌβˆ’π΅πœξ€Έβˆ’1,(2.5) and the following estimate holds: β€–β€–π‘‡πœβ€–β€–π»β†¦π»β‰€π‘€.(2.6)

Proof. The proof of estimate (2.6) is based on the estimate β€–β€–πΌβˆ’π΅πœβ€–β€–π»β†¦π»β‰₯1βˆ’π‘›ξ“π‘˜=1||π›Όπ‘˜||||||πœ†1+π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||ξƒͺβˆ’π‘›ξ“π‘˜=1||π›½π‘˜||||||πœ†1+π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||ξƒͺβˆ’π‘›ξ“π‘˜=1||π›Όπ‘˜||π‘›ξ“π‘˜=1||π›½π‘˜||||||πœ†1+π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||ξƒͺβˆ’π‘›ξ“π‘˜=1||π›Όπ‘˜||||||πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||π‘›ξ“π‘˜=1||π›½π‘˜||||||πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœ||||.(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 πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1ξ€Έ+π΄π‘’π‘˜+𝜏24𝐴2π‘’π‘˜+1=π‘“π‘˜,π‘“π‘˜ξ€·π‘‘=π‘“π‘˜ξ€Έ,π‘‘π‘˜ξ‚΅πœ=π‘˜πœ,1β‰€π‘˜β‰€π‘βˆ’1,π‘πœ=1,𝐼+2𝐴2ξ‚Άπœβˆ’1𝑒1βˆ’π‘’0ξ€Έβˆ’πœ2𝑓0βˆ’π΄π‘’0ξ€Έ=πœ”,𝑓0=𝑓(0),𝑒0=πœ‡,(2.8) and the following formula holds: 𝑒0=πœ‡,𝑒1=ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1ξ‚΅πœπœ‡+πœπœ”+22𝑓0ξ‚Ή,π‘’π‘˜=ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1βŽ›βŽœβŽœβŽπ‘…π‘˜βˆ’1ξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ+ξ‚π‘…π‘˜βˆ’1𝐼+π‘–πœπ΄1/2ξ€Έ2βŽžβŽŸβŽŸβŽ πœ‡+ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1ξƒ©ξ‚π‘…π‘˜βˆ’1π‘…βˆ’1βˆ’π‘…π‘˜βˆ’1ξ‚π‘…βˆ’12ξƒͺξ‚€πœπœ”+2𝑓0ξ‚βˆ’π‘˜βˆ’1𝑠=1𝜏𝐴2π‘–βˆ’1/2ξ‚€π‘…π‘˜βˆ’π‘ βˆ’ξ‚π‘…π‘˜βˆ’π‘ ξ‚π‘“π‘ ,2β‰€π‘˜β‰€π‘.(2.9) Applying formula (2.9) and the nonlocal boundary conditions in problem (2.1), we get πœ‡=π‘‡πœξƒ―ξƒ¬π‘›ξ“π‘š=1π›Όπ‘šπœ2𝑓0Γ—ξƒ©ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1×𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’12+π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽπ‘›ξ“π‘š=1π›Όπ‘š[πœ†π‘š/𝜏]βˆ’1𝑠=1𝜏𝐴2π‘–βˆ’1/2𝑅[πœ†π‘š/𝜏]βˆ’sβˆ’ξ‚π‘…[πœ†π‘š/𝜏]βˆ’sξ‚π‘“π‘ βˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚ΆΓ—[πœ†π‘š/𝜏]βˆ’2𝑠=1𝜏𝐴2𝑖1/2ξ€·π‘–π΄βˆ’1/2𝑅[πœ†π‘š/𝜏]βˆ’sξ‚€πΌβˆ’π‘–πœ2𝐴1/2+𝑅[πœ†π‘š/𝜏]βˆ’sξ‚€πΌβˆ’π‘–πœ2𝐴1/2𝑓𝑠+π‘›ξ“π‘š=1π›Όπ‘šπœξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚Άπ‘…ξ‚π‘…π‘“[πœ†π‘š/𝜏]βˆ’1Γ—ξƒ¬βˆ’πœ‘ξƒͺ𝐼+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ΄ξ‚΅πœξ‚Άξ‚ΆπΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1×𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒͺβˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦+ξƒ¬π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’12+π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2⎞⎟⎟⎠⎀βŽ₯βŽ₯βŽ¦Γ—ξƒ¬πœ2𝑓0ξƒ©π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ2+𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βˆ’π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœξ‚΅πœξ‚Άξ‚Άπ΄Γ—πΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1×𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒͺ+π‘›ξ“π‘˜=1π›½π‘˜[πœ†π‘˜/𝜏]βˆ’2𝑠=1𝜏2π‘–Γ—π΄βˆ’1/2𝑖𝐴1/2𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚€πΌ+π‘–πœ2𝐴1/2+𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚€πΌβˆ’π‘–πœ2𝐴1/2𝑓𝑠+π‘›ξ“π‘˜=1π›½π‘˜ξ‚πœπ‘…π‘…π‘“[πœ†π‘˜/𝜏]βˆ’1+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ΄ξ‚Άξ‚Ά[πœ†π‘˜/𝜏]βˆ’1𝑠=1𝜏𝐴2π‘–βˆ’1/2×𝑅[πœ†π‘˜/𝜏]βˆ’π‘ βˆ’ξ‚π‘…[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚π‘“π‘ +π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ‘“ξ‚Άξ‚Ά[πœ†π‘˜/𝜏]βˆ’1,+πœ“ξƒ­ξƒ°πœ”=π‘‡πœξƒ―ξƒ¬πΌβˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽπ‘…[πœ†π‘š/𝜏]βˆ’1ξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ2+𝑅[πœ†π‘š/𝜏]βˆ’1𝐼+π‘–πœπ΄1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2×𝑅[πœ†π‘š/𝜏]βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2𝐼+π‘–πœπ΄1/2ξ€Έβˆ’π‘…[πœ†π‘š/𝜏]βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ2⎀βŽ₯βŽ₯βŽ¦Γ—ξƒ¬πœ2𝑓0ξƒ©π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ2+𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βˆ’π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœξ‚΅πœξ‚Άξ‚Άπ΄Γ—πΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1×𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒͺ+π‘›ξ“π‘˜=1π›½π‘˜[πœ†π‘˜/𝜏]βˆ’2𝑠=1𝜏2π‘–Γ—π΄βˆ’1/2𝑖𝐴1/2𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚€πΌ+π‘–πœ2𝐴1/2+𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚€πΌβˆ’π‘–πœ2𝐴1/2𝑓𝑠+π‘›ξ“π‘˜=1π›½π‘˜π‘…ξ‚π‘…π‘“[πœ†π‘˜/𝜏]βˆ’1+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ΄ξ‚Άξ‚Ά[πœ†π‘˜/𝜏]βˆ’1𝑠=1𝜏2π‘–Γ—π΄βˆ’1/2𝑅[πœ†π‘˜/𝜏]βˆ’π‘ βˆ’ξ‚π‘…[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚π‘“π‘ +ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ‘“ξ‚Άξ‚Ά[πœ†π‘˜/𝜏]βˆ’1ξƒ­βˆ’βŽ‘βŽ’βŽ’βŽ£+πœ“π‘›ξ“π‘˜=1π›½π‘˜πœ2π΄ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1ξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ+𝑅[πœ†π‘˜/𝜏]βˆ’1𝐼+π‘–πœπ΄1/2ξ€Έ2βˆ’12π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚€πΌβˆ’π‘–πœ2𝐴1/2𝐼+π‘–πœπ΄1/2ξ€Έβˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1𝐼+π‘–πœ2𝐴1/2ξ‚ξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έξ‚ξƒ­Γ—ξƒ¬πœ2𝑓0ξƒ©π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’12+π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽπ‘›ξ“π‘š=1π›Όπ‘š[πœ†π‘š/𝜏]βˆ’1𝑠=1𝜏𝐴2π‘–βˆ’1/2𝑅[πœ†π‘š/𝜏]βˆ’π‘ βˆ’ξ‚π‘…[πœ†π‘š/𝜏]βˆ’π‘ ξ‚π‘“π‘ +π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚ΆΓ—[πœ†π‘š/𝜏]βˆ’2𝑠=1𝜏𝐴2π‘–βˆ’1/2𝑖𝐴1/2𝑅[πœ†π‘š/𝜏]βˆ’π‘ ξ‚€πΌ+π‘–πœ2𝐴1/2+𝑅[πœ†π‘š/𝜏]βˆ’π‘ ξ‚€πΌβˆ’π‘–πœ2𝐴1/2𝑓𝑠+π‘›ξ“π‘š=1π›Όπ‘šπœξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚Άπ‘…ξ‚π‘…π‘“[πœ†π‘š/𝜏]βˆ’1.βˆ’πœ‘ξƒͺξƒ­ξƒ°(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 πœ‘βˆˆπ·(𝐴),β€‰β€‰πœ“βˆˆπ·(𝐴1/2). Then, for the solution of difference scheme (2.1) the stability inequalities max0β‰€π‘˜β‰€π‘β€–β€–π‘’π‘˜β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1ξ“π‘˜=0β€–β€–π΄βˆ’1/2π‘“π‘˜β€–β€–π»β€–β€–π΄πœ+βˆ’1/2πœ“β€–β€–π»+β€–πœ‘β€–π»ξƒ°,(2.11)max0β‰€π‘˜β‰€π‘β€–β€–π΄1/2π‘’π‘˜β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1ξ“π‘˜=0β€–β€–π‘“π‘˜β€–β€–π»β€–β€–π΄πœ+1/2πœ‘β€–β€–π»+β€–πœ“β€–π»ξƒ°,(2.12)max1β‰€π‘˜β‰€π‘βˆ’1β€–β€–πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1‖‖𝐻+max0β‰€π‘˜β‰€π‘βˆ’1β€–β€–β€–π΄π‘’π‘˜+𝜏24𝐴2π‘’π‘˜+1β€–β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1ξ“π‘˜=1β€–β€–π‘“π‘˜βˆ’π‘“π‘˜βˆ’1‖‖𝐻+‖‖𝑓0‖‖𝐻+‖‖𝐴1/2πœ“β€–β€–π»+β€–π΄πœ‘β€–π»ξƒ°(2.13) hold, where 𝑀 does not depend on 𝜏,β€‰β€‰πœ‘,β€‰β€‰πœ“, and π‘“π‘˜,  0β‰€π‘˜β‰€π‘βˆ’1.

Proof. By [18], the following estimates max0β‰€π‘˜β‰€π‘β€–β€–π‘’π‘˜β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1ξ“π‘˜=0β€–β€–π΄βˆ’1/2π‘“π‘˜β€–β€–π»β€–β€–π΄πœ+βˆ’1/2πœ”β€–β€–π»+β€–πœ‡β€–π»ξƒ°,(2.14)max0β‰€π‘˜β‰€π‘β€–β€–π΄1/2π‘’π‘˜β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1ξ“π‘˜=0β€–β€–π‘“π‘˜β€–β€–π»β€–β€–π΄πœ+1/2πœ‡β€–β€–π»+β€–πœ”β€–π»ξƒ°,(2.15)max1β‰€π‘˜β‰€π‘βˆ’1β€–β€–πœβˆ’2ξ€·π‘’π‘˜+1βˆ’2π‘’π‘˜+π‘’π‘˜βˆ’1‖‖𝐻+max0β‰€π‘˜β‰€π‘βˆ’1β€–β€–β€–π΄π‘’π‘˜+𝜏24𝐴2π‘’π‘˜+1β€–β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1ξ“π‘˜=1β€–β€–π‘“π‘˜βˆ’π‘“π‘˜βˆ’1‖‖𝐻+‖‖𝑓0‖‖𝐻+‖‖𝐴1/2πœ”β€–β€–π»+β€–π΄πœ‡β€–π»ξƒ°(2.16) hold for the solution of (2.8). Using formulas of πœ‡, πœ”, and (2.3) and (2.6) the following estimates obtained β€–πœ‡β€–π»ξƒ―β‰€π‘€π‘βˆ’1𝑠=0β€–β€–π΄βˆ’1/2π‘“π‘ β€–β€–π»β€–β€–π΄πœ+βˆ’1/2πœ“β€–β€–π»+β€–πœ‘β€–π»ξƒ°,‖‖𝐴(2.17)βˆ’1/2πœ”β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1𝑠=0β€–β€–π΄βˆ’1/2π‘“π‘ β€–β€–π»β€–β€–π΄πœ+βˆ’1/2πœ“β€–β€–π»+β€–πœ‘β€–π»ξƒ°.(2.18) Estimate (2.11) follows from (2.14), (2.17), and (2.18). In a similar manner, we obtain max0β‰€π‘˜β‰€π‘β€–β€–π΄1/2π‘’π‘˜β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1ξ“π‘˜=0β€–β€–π‘“π‘˜β€–β€–π»β€–β€–π΄πœ+1/2πœ‘β€–β€–π»+β€–πœ“β€–π»ξƒ°.(2.19) Now, we obtain the estimates for β€–π΄πœ‡β€–π» and ‖𝐴1/2πœ”β€–π». First, applying 𝐴 to the formula of πœ‡ and using Abel’s formula, we can write π΄πœ‡=π‘‡πœβŽ§βŽͺ⎨βŽͺβŽ©ξƒ¬πœ2𝑓0ξƒ©π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝐴1/2𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1+2π‘–π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚Άπ΄ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1×𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽ12π‘›ξ“π‘š=1π›Όπ‘š[πœ†π‘š/𝜏]βˆ’1𝑠=2𝑅[πœ†π‘š/𝜏]βˆ’s𝐼+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘š/𝜏]βˆ’sξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1ξ‚ΆΓ—ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έ+π‘›ξ“π‘š=1π›Όπ‘š12𝑅[πœ†π‘š/𝜏]βˆ’1𝐼+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1𝑓1βˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴4ξ‚Άβˆ’1𝑓[πœ†π‘š/𝜏]βˆ’1+π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚ΆΓ—βŽ‘βŽ’βŽ’βŽ£[πœ†π‘š/𝜏]βˆ’2𝑠=21𝐴2𝑖1/2×𝑅[πœ†π‘š/𝜏]βˆ’π‘ +𝑅[πœ†π‘š/𝜏]βˆ’π‘ ξ‚ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έ+1𝐴2𝑖1/2𝑅[πœ†π‘š/𝜏]βˆ’1+𝑅[πœ†π‘š/𝜏]βˆ’1𝑓1+𝑖𝐴1/2𝑓[πœ†π‘š/𝜏]βˆ’2⎀βŽ₯βŽ₯⎦+π‘›ξ“π‘š=1π›Όπ‘šπœξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚Άξ‚π΄π‘…π‘…π‘“[πœ†π‘š/𝜏]βˆ’1Γ—ξƒ¬βˆ’π΄πœ‘ξƒͺ𝐼+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ΄ξ‚΅πœξ‚Άξ‚ΆπΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2ξ€Έβˆ’1×𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒͺβˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1×𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2⎀βŽ₯βŽ₯⎦+ξƒ¬π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1+2π‘–π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚ΆΓ—π΄1/2ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1×𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2⎀βŽ₯βŽ₯βŽ¦Γ—βŽ‘βŽ’βŽ’βŽ£πœ2𝑓0βŽ›βŽœβŽœβŽπ‘›ξ“π‘˜=1π›½π‘˜π΄1/2ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ2+𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2⎞⎟⎟⎠+π‘›ξ“π‘˜=1π›½π‘˜π‘–ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœξ‚΅πœξ‚Άξ‚Άπ΄Γ—πΌ+2𝐴2ξ‚Άβˆ’1×𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒͺβˆ’π‘›ξ“π‘˜=1π›½π‘˜[πœ†π‘˜/𝜏]βˆ’2𝑠=21𝑅2𝑖[πœ†π‘˜/𝜏]βˆ’π‘ +𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έβˆ’π‘›ξ“π‘˜=1π›½π‘˜1𝑅2𝑖[πœ†π‘˜/𝜏]βˆ’1+𝑅[πœ†π‘˜/𝜏]βˆ’1𝑓1βˆ’π‘›ξ“π‘˜=1π›½π‘˜π‘–π‘“[πœ†π‘š/𝜏]βˆ’2+π‘›ξ“π‘˜=1π›½π‘˜π΄1/2ξ‚πœπ‘…π‘…π‘“[πœ†π‘˜/𝜏]βˆ’1+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚ΉπœΓ—βŽ‘βŽ’βŽ’βŽ£1ξ‚Άξ‚Ά2π΄βˆ’1/2[πœ†π‘˜/𝜏]βˆ’1𝑠=2𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚€πΌ+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1ξ‚ΆΓ—ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έ+12π΄βˆ’1/2𝑅[πœ†π‘˜/𝜏]βˆ’1𝐼+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1𝑓1βˆ’π΄βˆ’1/2ξ‚΅πœπΌ+2𝐴4ξ‚Άβˆ’1𝑓[πœ†π‘˜/𝜏]βˆ’1ξƒ­+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ΄ξ‚Άξ‚Ά1/2𝑓[πœ†π‘˜/𝜏]βˆ’1+𝐴1/2πœ“βŽ€βŽ₯βŽ₯⎦⎫βŽͺ⎬βŽͺ⎭.(2.20) Second, applying 𝐴1/2 to the formula of πœ” and using Abel’s formula, we can write 𝐴1/2πœ”=π‘‡πœξƒ―ξƒ¬πΌβˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽπ‘…[πœ†π‘š/𝜏]βˆ’1ξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ2+𝑅[πœ†π‘š/𝜏]βˆ’1𝐼+π‘–πœπ΄1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœπœξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1𝑖𝐴1/2×𝑅[πœ†π‘š/𝜏]βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2𝐼+π‘–πœπ΄1/2ξ€Έβˆ’π‘…[πœ†π‘š/𝜏]βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ2×12𝑓0ξƒ©π‘›ξ“π‘˜=1π›½π‘˜π΄1/2πœξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ2+𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2⎞⎟⎟⎠+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπœξ‚Άξ‚Άξ‚΅πΌ+2𝐴2ξ‚Άβˆ’1×𝐴1/2𝑖𝑅[πœ†π‘˜/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1ξ‚π‘…βˆ’12ξƒͺβˆ’π‘›ξ“π‘˜=1π›½π‘˜[πœ†π‘˜/𝜏]βˆ’2𝑠=21𝑅2𝑖[πœ†π‘˜/𝜏]βˆ’s+𝑅[πœ†π‘˜/𝜏]βˆ’sξ‚Γ—ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έβˆ’π‘›ξ“π‘˜=1π›½π‘˜1𝑅2𝑖[πœ†π‘˜/𝜏]βˆ’1+𝑅[πœ†π‘˜/𝜏]βˆ’1𝑓1βˆ’π‘›ξ“π‘˜=1π›½π‘˜π‘–π‘“[πœ†π‘˜/𝜏]βˆ’2+π‘›ξ“π‘˜=1π›½π‘˜π΄1/2ξ‚πœπ‘…π‘…π‘“[πœ†π‘˜/𝜏]βˆ’1+π‘›ξ“π‘˜=1π›½π‘˜ξ‚΅πœ2+ξ‚΅πœ†π‘˜βˆ’ξ‚Έπœ†π‘˜πœξ‚Ήπœπ΄ξ‚Άξ‚Άβˆ’1/2Γ—βŽ‘βŽ’βŽ’βŽ£12[πœ†π‘˜/𝜏]βˆ’1𝑠=2𝑅[πœ†π‘˜/𝜏]βˆ’s𝐼+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘˜/𝜏]βˆ’π‘ ξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1ξ‚ΆΓ—ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έ+12𝑅[πœ†π‘˜/𝜏]βˆ’1𝐼+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘˜/𝜏]βˆ’1ξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1𝑓1βˆ’ξ‚΅πœπΌ+2𝐴4ξ‚Άβˆ’1𝑓[πœ†π‘˜/𝜏]βˆ’1ξƒ­+𝐴1/2πœ“ξƒ­βˆ’βŽ‘βŽ’βŽ’βŽ£π‘›ξ“π‘˜=1π›½π‘˜π΄1/2𝜏2ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽπ‘…[πœ†π‘˜/𝜏]βˆ’1ξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ+𝑅[πœ†π‘˜/𝜏]βˆ’1𝐼+π‘–πœπ΄1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βˆ’π‘›ξ“π‘˜=1π›½π‘˜π΄βˆ’1/2ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘˜/𝜏]βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2𝐼+π‘–πœπ΄1/2ξ€Έ2βˆ’π‘…[πœ†π‘˜/𝜏]βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έξ€·πΌβˆ’π‘–πœπ΄1/2ξ€Έ2×1ξƒͺξƒ­2𝑓0ξƒ©π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—πœπ΄1/2𝑅[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1βˆ’π‘…[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1+2π‘–π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚Άπ΄ξ‚΅πœπΌ+2𝐴2ξ‚Άβˆ’1Γ—βŽ›βŽœβŽœβŽξ‚π‘…[πœ†π‘š/𝜏]βˆ’1π‘…βˆ’1ξ€·πΌβˆ’(π‘–πœ/2)𝐴1/2ξ€Έ+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚π‘…βˆ’1𝐼+(π‘–πœ/2)𝐴1/2ξ€Έ2βŽžβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽ βˆ’ξƒ©12π‘›ξ“π‘š=1π›Όπ‘šπ΄βˆ’1/2Γ—[πœ†π‘š/𝜏]βˆ’1𝑠=2𝑅[πœ†π‘š/𝜏]βˆ’π‘ ξ‚€πΌ+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘š/𝜏]βˆ’π‘ ξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1ξ‚ΆΓ—ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έ+12π‘›ξ“π‘š=1π›Όπ‘šπ΄βˆ’1/2×𝑅[πœ†π‘š/𝜏]βˆ’1𝐼+π‘–πœ2𝐴1/2ξ‚βˆ’1+𝑅[πœ†π‘š/𝜏]βˆ’1ξ‚€πΌβˆ’π‘–πœ2𝐴1/2ξ‚βˆ’1𝑓1βˆ’π‘›ξ“π‘š=1π›Όπ‘šπ΄βˆ’1/2ξ‚΅πœπΌ+2𝐴4ξ‚Άβˆ’1𝑓[πœ†π‘š/𝜏]βˆ’1βˆ’π‘›ξ“π‘š=1π›Όπ‘šξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚ΆΓ—βŽ‘βŽ’βŽ’βŽ£[πœ†π‘š/𝜏]βˆ’2𝑠=21𝐴2𝑖1/2×𝑅[πœ†π‘š/𝜏]βˆ’π‘ +𝑅[πœ†π‘š/𝜏]βˆ’π‘ ξ‚ξ€·π‘“π‘ βˆ’π‘“π‘ βˆ’1ξ€Έ+1𝐴2𝑖1/2𝑅[πœ†π‘š/𝜏]βˆ’1+𝑅[πœ†π‘š/𝜏]βˆ’1𝑓1+𝑖𝐴1/2𝑓[πœ†π‘š/𝜏]βˆ’2⎀βŽ₯βŽ₯⎦+π‘›ξ“π‘š=1π›Όπ‘šπœξ‚΅πœ†π‘šβˆ’ξ‚Έπœ†π‘šπœξ‚Ήπœξ‚Άξ‚π΄π‘…π‘…π‘“[πœ†π‘š/𝜏]βˆ’1.βˆ’π΄πœ‘ξƒͺξƒ­ξƒ°(2.21) The following estimates β€–π΄πœ‡β€–π»ξƒ―β‰€π‘€π‘βˆ’1𝑠=1β€–β€–π‘“π‘ βˆ’π‘“π‘ βˆ’1‖‖𝐻+‖‖𝑓0‖‖𝐻+‖‖𝐴1/2πœ“β€–β€–π»+β€–π΄πœ‘β€–π»ξƒ°,‖‖𝐴1/2πœ”β€–β€–π»ξƒ―β‰€π‘€π‘βˆ’1𝑠=1β€–β€–π‘“π‘ βˆ’π‘“π‘ βˆ’1‖‖𝐻+‖‖𝑓0‖‖𝐻+‖‖𝐴1/2πœ“β€–β€–π»+β€–π΄πœ‘β€–π»ξƒ°(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 π‘’π‘‘π‘‘βˆ’ξ€·π‘Ž(π‘₯)𝑒π‘₯ξ€Έπ‘₯+𝛿𝑒=𝑓(𝑑,π‘₯),0<𝑑<1,0<π‘₯<1,𝑒(0,π‘₯)=π‘›ξ“π‘š=1π›Όπ‘šπ‘’ξ€·πœ†π‘šξ€Έπ‘’,π‘₯+πœ‘(π‘₯),0≀π‘₯≀1,𝑑(0,π‘₯)=π‘›ξ“π‘˜=1π›½π‘˜π‘’π‘‘ξ€·πœ†π‘˜ξ€Έ,π‘₯+πœ“(π‘₯),0≀π‘₯≀1,𝑒(𝑑,0)=𝑒(𝑑,1),𝑒π‘₯(𝑑,0)=𝑒π‘₯(𝑑,1),0≀𝑑≀1,(2.23) under assumption (2.2), is considered. Here π‘Žπ‘Ÿ(π‘₯),  (π‘₯∈(0,1)),β€‰β€‰πœ‘(π‘₯),πœ“(π‘₯)(π‘₯∈[0,1]) and 𝑓(𝑑,π‘₯)(π‘‘βˆˆ(0,1),π‘₯∈(0,1)) are given smooth functions and π‘Žπ‘Ÿ(π‘₯)β‰₯π‘Ž>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: []0,1β„Ž=ξ€½π‘₯∢π‘₯π‘Ÿξ€Ύ=π‘Ÿβ„Ž,0β‰€π‘Ÿβ‰€πΎ,πΎβ„Ž=1.(2.24) We introduce the Hilbert space 𝐿2β„Ž=𝐿2([0,1]β„Ž),β€‰β€‰π‘Š12β„Ž=π‘Š12β„Ž([0,1]β„Ž) and π‘Š22β„Ž=π‘Š22β„Ž([0,1]β„Ž) of the grid functions πœ‘β„Ž(π‘₯)={πœ‘π‘Ÿ}1πΎβˆ’1 defined on [0,1]β„Ž, equipped with the norms β€–β€–πœ‘β„Žβ€–β€–πΏ2β„Ž=ξƒ©πΎβˆ’1ξ“π‘Ÿ=1||πœ‘β„Ž||(π‘₯)2β„Žξƒͺ1/2,β€–β€–πœ‘β„Žβ€–β€–π‘Š12β„Ž=β€–β€–πœ‘β„Žβ€–β€–πΏ2β„Ž+ξƒ©πΎβˆ’1ξ“π‘Ÿ=1|||ξ€·πœ‘β„Žξ€Έπ‘₯,𝑗|||2β„Žξƒͺ1/2,β€–β€–πœ‘β„Žβ€–β€–π‘Š22β„Ž=β€–β€–πœ‘β„Žβ€–β€–πΏ2β„Ž+ξƒ©πΎβˆ’1ξ“π‘Ÿ=1|||ξ€·πœ‘β„Žξ€Έπ‘₯,𝑗|||2β„Žξƒͺ1/2+ξƒ©πΎβˆ’1ξ“π‘Ÿ=1|||ξ€·πœ‘β„Žξ€Έπ‘₯π‘₯,𝑗|||2β„Žξƒͺ1/2,(2.25) respectively. To the differential operator 𝐴 generated by problem (2.23), we assign the difference operator 𝐴π‘₯β„Ž by the formula 𝐴π‘₯β„Žπœ‘β„Žξ€½(π‘₯)=βˆ’(π‘Ž(π‘₯)πœ‘π‘₯)π‘₯,π‘Ÿ+π›Ώπœ‘π‘Ÿξ€Ύ1πΎβˆ’1,(2.26) acting in the space of grid functions πœ‘β„Ž(π‘₯)={πœ‘π‘Ÿ}𝐾0 satisfying the conditions πœ‘0=πœ‘πΎ,β€‰β€‰πœ‘1βˆ’πœ‘0=πœ‘πΎβˆ’πœ‘πΎβˆ’1. With the help of 𝐴π‘₯β„Ž, we arrive at the nonlocal boundary value problem 𝑑2π‘£β„Ž(𝑑,π‘₯)𝑑𝑑2+𝐴π‘₯β„Žπ‘£β„Ž(𝑑,π‘₯)=π‘“β„Ž[](𝑑,π‘₯),0≀𝑑≀1,π‘₯∈0,1β„Ž,π‘£β„Ž(0,π‘₯)=𝑛𝑗=1π›Όπ‘—π‘£β„Žξ€·πœ†π‘—ξ€Έ,π‘₯+πœ‘β„Ž[](π‘₯),π‘₯∈0,1β„Ž,π‘£β„Žπ‘‘(0,π‘₯)=𝑛𝑗=1π›½π‘—π‘£β„Žπ‘‘ξ€·πœ†π‘—ξ€Έ,π‘₯+πœ“β„Ž[](π‘₯),π‘₯∈0,1β„Ž,(2.27) for an infinite system of ordinary differential equations.
In the second step, we replace problem (2.27) by difference scheme (2.28) π‘’β„Žπ‘˜+1(π‘₯)βˆ’2π‘’β„Žπ‘˜(π‘₯)+π‘’β„Žπ‘˜βˆ’1(π‘₯)𝜏2+𝐴π‘₯β„Žπ‘’β„Žπ‘˜πœ(π‘₯)+24𝐴π‘₯β„Žξ€Έ2π‘’β„Žπ‘˜+1(π‘₯)=π‘“β„Žπ‘˜[](π‘₯),π‘₯∈0,1β„Ž,π‘“β„Žπ‘˜(π‘₯)=π‘“β„Žξ€·π‘‘π‘˜ξ€Έ,π‘₯,π‘‘π‘˜π‘’=π‘˜πœ,1β‰€π‘˜β‰€π‘βˆ’1,π‘πœ=1,β„Ž0(π‘₯)=𝑛𝑗=1π›Όπ‘—ξ‚»π‘’β„Žξ€Ίπœ†π‘—ξ€»/𝜏(π‘₯)+πœβˆ’1ξ‚€π‘’β„Ž[πœ†π‘—/𝜏](π‘₯)βˆ’π‘’β„Ž[πœ†π‘—/𝜏]βˆ’1ξ‚ξ‚΅πœ†(π‘₯)π‘—βˆ’ξ‚Έπœ†π‘—πœξ‚Ήπœξ‚Άξ‚Ό+πœ‘β„Ž[](π‘₯),π‘₯∈0,1β„Ž,ξƒ©πœπΌ+2𝐴π‘₯β„Ž2ξƒͺπ‘’β„Ž1(π‘₯)βˆ’π‘’β„Ž0(π‘₯)πœβˆ’πœ2ξ€·π‘“β„Ž0(π‘₯)βˆ’π΄π‘₯β„Žπ‘’β„Ž0ξ€Έ=(π‘₯)𝑛𝑗=1π›½π‘—βŽ§βŽͺ⎨βŽͺβŽ©π‘’β„Žξ€Ίπœ†π‘—ξ€»/𝜏(π‘₯)βˆ’π‘’β„Žξ€Ίπœ†π‘—ξ€»/πœβˆ’1(π‘₯)𝜏+ξ‚΅πœ2+ξ‚΅πœ†π‘—βˆ’ξ‚Έπœ†π‘—πœξ‚ΉπœΓ—ξ‚€π‘“ξ‚Άξ‚Άβ„Ž[πœ†π‘—/𝜏](π‘₯)βˆ’π΄π‘₯β„Žπ‘’β„Ž[πœ†π‘—/𝜏]ξ‚βŽ«βŽͺ⎬βŽͺ⎭(π‘₯)+πœ“β„Žπ‘“(π‘₯),β„Ž0(π‘₯)=π‘“β„Ž[](0,π‘₯),π‘₯∈0,1β„Ž.(2.28)

Theorem 2.4. Let 𝜏 and β„Ž be sufficiently small positive numbers. Suppose that assumption (2.2) holds. Then, the solution of difference scheme (2.28) satisfies the following stability estimates: max0β‰€π‘˜β‰€π‘β€–β€–π‘’β„Žπ‘˜β€–β€–πΏ2β„Ž+max0β‰€π‘˜β‰€π‘β€–β€–ξ€·π‘’β„Žπ‘˜ξ€Έπ‘₯‖‖𝐿2β„Žβ‰€π‘€1ξ‚Έmax0β‰€π‘˜β‰€π‘βˆ’1β€–β€–π‘“β„Žπ‘˜β€–β€–πΏ2β„Ž+β€–β€–πœ“β„Žβ€–β€–πΏ2β„Ž+β€–β€–πœ‘β„Žπ‘₯‖‖𝐿2β„Žξ‚Ή,max1β‰€π‘˜β‰€π‘βˆ’1β€–β€–πœβˆ’2ξ€·π‘’β„Žπ‘˜+1βˆ’2π‘’β„Žπ‘˜+π‘’β„Žπ‘˜βˆ’1‖‖𝐿2β„Ž+max0β‰€π‘˜β‰€π‘β€–β€–ξ€·π‘’β„Žπ‘˜ξ€Έπ‘₯π‘₯‖‖𝐿2β„Žβ‰€π‘€1ξ‚Έβ€–β€–π‘“β„Ž0‖‖𝐿2β„Ž+max1β‰€π‘˜β‰€π‘βˆ’1β€–β€–πœβˆ’1ξ€·π‘“β„Žπ‘˜βˆ’π‘“β„Žπ‘˜βˆ’1‖‖𝐿2β„Ž+β€–β€–πœ“β„Žπ‘₯‖‖𝐿2β„Ž+β€–β€–ξ€·πœ‘β„Žπ‘₯ξ€Έπ‘₯‖‖𝐿2β„Žξ‚Ή.(2.29) Here, 𝑀1 does not depend on 𝜏,β€‰β€‰β„Ž,β€‰β€‰πœ‘β„Ž(π‘₯),β€‰β€‰πœ“β„Ž(π‘₯) and π‘“β„Žπ‘˜,  0β‰€π‘˜<𝑁.

The proof of Theorem 2.4 is based on abstract Theorem 2.3 and symmetry properties of the operator 𝐴π‘₯β„Ž defined by (2.26).

Second, for the π‘š-dimensional hyperbolic equation under assumption (2.2) is considered. Let Ξ© be the unit open cube in the π‘š-dimensional Euclidean space β„π‘š{π‘₯=(π‘₯1,…,π‘₯π‘š)∢0<π‘₯𝑗<1,1β‰€π‘—β‰€π‘š} with boundary 𝑆,Ξ©=Ξ©βˆͺ𝑆. In [0,1]Γ—Ξ©, the mixed boundary value problem for the multidimensional hyperbolic equation πœ•2𝑒(𝑑,π‘₯)πœ•π‘‘2βˆ’π‘šξ“π‘Ÿ=1ξ€·π‘Žπ‘Ÿ(π‘₯)𝑒π‘₯π‘Ÿξ€Έπ‘₯π‘Ÿξ€·π‘₯=𝑓(𝑑,π‘₯),π‘₯=1,…,π‘₯π‘šξ€ΈβˆˆΞ©,0<𝑑<1,𝑒(0,π‘₯)=𝑛𝑗=1π›Όπ‘—π‘’ξ€·πœ†π‘—ξ€Έ,π‘₯+πœ‘(π‘₯),π‘₯βˆˆπ‘’Ξ©;𝑑(0,π‘₯)=π‘›ξ“π‘˜=1π›½π‘˜π‘’π‘‘ξ€·πœ†π‘˜ξ€Έ,π‘₯+πœ“(π‘₯),π‘₯∈Ω;𝑒(𝑑,π‘₯)=0,π‘₯βˆˆπ‘†(2.30) is considered.

Here, π‘Žπ‘Ÿ(π‘₯),  (π‘₯∈Ω),β€‰β€‰πœ‘(π‘₯),β€‰β€‰πœ“(π‘₯)  (π‘₯∈Ω) and 𝑓(𝑑,π‘₯)  (π‘‘βˆˆ(0,1),π‘₯∈Ω) are given smooth functions and π‘Žπ‘Ÿ(π‘₯)β‰₯π‘Ž>0. The discretization of problem (2.30) is carried out in two steps. In the first step, let us define the grid sets ξ‚Ξ©β„Ž=ξ€½π‘₯=π‘₯π‘Ÿ=ξ€·β„Ž1π‘Ÿ1,…,β„Žπ‘šπ‘Ÿπ‘šξ€Έξ€·π‘Ÿ,π‘Ÿ=1,…,π‘Ÿπ‘šξ€Έ,0β‰€π‘Ÿπ‘—β‰€π‘π‘—,β„Žπ‘—π‘π‘—ξ€Ύ,Ξ©=1,𝑗=1,…,π‘šβ„Ž=ξ‚Ξ©β„Žβˆ©Ξ©,π‘†β„Ž=ξ‚Ξ©β„Žβˆ©π‘†.(2.31) We introduce the Banach space 𝐿2β„Ž=𝐿2(ξ‚Ξ©β„Ž),β€‰β€‰π‘Š12β„Ž=π‘Š12β„Ž(ξ‚Ξ©β„Ž) and π‘Š22β„Ž=π‘Š22β„Ž(ξ‚Ξ©β„Ž) of the grid functions πœ‘β„Ž(π‘₯)={πœ‘(β„Ž1π‘Ÿ1,…,β„Žπ‘šπ‘Ÿπ‘š)} defined on ξ‚Ξ©β„Ž, equipped with the norms β€–β€–