Abstract and Applied Analysis

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

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Volume 2012 |Article ID 120192 | https://doi.org/10.1155/2012/120192

Burkhan T. Kalimbetov, Marat A. Temirbekov, Zhanibek O. Khabibullayev, "Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator", Abstract and Applied Analysis, vol. 2012, Article ID 120192, 16 pages, 2012. https://doi.org/10.1155/2012/120192

Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator

Academic Editor: Allaberen Ashyralyev
Received02 Mar 2012
Accepted03 Apr 2012
Published24 Jul 2012

Abstract

The regularization method is applied for the construction of algorithm for an asymptotical solution for linear singular perturbed systems with the irreversible limit operator. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to study of systems of first-order partial differential equations with incomplete initial data.

1. Introduction

The investigation of singular perturbed systems for ordinary and partial differential equations occurring in systems with slow and fast variables, chemical kinetics, the mathematical theory of boundary layer, control with application of geoinformational technologies, quantum mechanics, and plasma physics (the Samarsky-Ionkin problem) has been studied by many researchers (see, e.g., [1โ€“19]).

In this work, the algorithm for construction of an asymptotical solution for linear singular perturbed systems with the irreversible limit operator is givenโ€”the regularization method [1]. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to the study of systems of first-order partial differential equations with incomplete (more exactly, point) initial data.

In this paper, we consider linear singular perturbed systems in the form๐œ€ฬ‡๐‘ฆ=๐ด(๐‘ก)๐‘ฆ+โ„Ž(๐‘ก),๐‘ฆ(0,๐œ€)=๐‘ฆ0[],,๐‘กโˆˆ0,๐‘‡(1.1) where ๐‘ฆ={๐‘ฆ1,โ€ฆ,๐‘ฆ๐‘›},๐ด(๐‘ก) is a matrix of order (๐‘›ร—๐‘›),โ„Ž(๐‘ก)={โ„Ž1,โ€ฆ,โ„Ž๐‘›} is a known function, ๐‘ฆ0โˆˆ๐ถ๐‘› is a constant vector, and ๐œ€>0 is a small parameter, in the case of violation of stability of a spectrum {๐œ†๐‘—(๐‘ก)} of the limiting operator ๐ด(๐‘ก).

Difference of such type problems from similar problems with a stable spectrum (i.e., in the case of ๐œ†๐‘–(๐‘ก)โ‰ 0,๐œ†๐‘–(๐‘ก)โ‰ ๐œ†๐‘—(๐‘ก),๐‘–โ‰ ๐‘—,๐‘–,๐‘—=1,๐‘›forall๐‘กโˆˆ[0,๐‘‡]) is that the limiting system 0=๐ด(๐‘ก)๐‘ฆ+โ„Ž(๐‘ก)at violation of stability of the spectrum can have either no solutions or uncountable set of them. In the last case, presence of discontinuous on the segment [0,๐‘‡] solutions ๐‘ฆ(๐‘ก)of the limiting system is not excluded. Under conditions, one can prove (see, e.g., [1, 6]) that the exact solution ๐‘ฆ(๐‘ก,๐œ€) of problem (1.1) tends (at ๐œ€โ†’+0) to a smooth solution of the limiting system. However, there is a problematic problem about construction of an asymptotic solution of problem (1.1). When the spectrum is instable, essentially special singularities are arising in the solution of system (1.1). These singularities are not selected by the spectrum {๐œ†๐‘—(๐‘ก)} of the limiting operator ๐ด(๐‘ก). As it was shown in [3โ€“7], they were induced by instability points ๐‘ก๐‘— of the spectrum.

In the present work, the algorithm of regularization method [1] is generalized on singular perturbed systems of the form (1.1), the limiting operator of which has some instable points of the spectrum. In order to construct the spectrum, we use the new algorithm requiring more constructive theory of solvability of iterative problems. These problems arose in application of the algorithm.

We will consider the problem (1.1) at the following conditions. Assume that(i)๐ด(๐‘ก)โˆˆ๐ถโˆž([0,๐‘‡],๐ถ๐‘›),โ„Ž(๐‘ก)โˆˆ๐ถโˆž[0,๐‘‡]; for any ๐‘กโˆˆ[0,๐‘‡], the spectrum {๐œ†๐‘—(๐‘ก)} of the operator ๐ด(๐‘ก) satisfies the conditions:(ii)๐œ†๐‘–(๐‘ก)=โˆ’(๐‘กโˆ’๐‘ก๐‘–)๐‘ ๐‘–๐‘˜๐‘–(๐‘ก),๐‘˜๐‘–(๐‘ก)โ‰ 0,๐‘ก๐‘–โˆˆ[0,๐‘‡],๐‘–=1,๐‘š,๐‘š<๐‘› (here ๐‘ ๐‘–- are even natural numbers),(iii)๐œ†๐‘–(๐‘ก)โ‰ 0,๐‘—=๐‘š+1,๐‘›,(iv)๐œ†๐‘–(๐‘ก)โ‰ ๐œ†๐‘—(๐‘ก),๐‘–โ‰ ๐‘—,๐‘–,๐‘—=1,๐‘›,(v)Re๐œ†๐‘—(๐‘ก)โฉฝ0,๐‘—=1,๐‘›.

2. Regularization of the Problem

We introduce basic regularized variables by the spectrum of the limiting operator๐œ๐‘—=๐œ€โˆ’1๎€œ๐‘ก0๐œ†๐‘—๐œ‘(๐‘ )๐‘‘๐‘ โ‰ก๐‘—(๐‘ก)๐œ€,๐‘—=1,๐‘›.(2.1) Instable points ๐‘ก๐‘–โˆˆ[0,๐‘‡] of the spectrum {๐œ†๐‘—(๐‘ก)} induce additional regularized variables described by the formulas๐œŽ๐‘–๐‘ž๐‘–=๐‘’๐œ‘๐‘–(๐‘ก)/๐œ€๎€œ๐‘ก0๐‘’โˆ’๐œ‘๐‘–(๐‘ )/๐œ€๎€ท๐‘ โˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–๐‘‘๐‘ โ‰ก๐œ“๐‘–๐‘ž๐‘–(๐‘ก,๐œ€),๐‘–=1,๐‘š,๐‘ž๐‘–=0,๐‘ ๐‘–โˆ’1.(2.2) We consider a vector function ฬƒ๐‘ฆ(๐‘ก,๐œ,๐œŽ,๐œ€) instead of the solution ๐‘ฆ(๐‘ก,๐œ€) to be found for problem (1.1). This vector function is such thatฬƒ๐‘ฆ(๐‘ก,๐œ,๐œŽ,๐œ€)โˆฃ๐œ=๐œ‘,๐œŽ=๐œ“โ‰ก๐‘ฆ(๐‘ก,๐œ€).(2.3)

For ฬƒ๐‘ฆ(๐‘ก,๐œ,๐œŽ,๐œ€), it is natural to set the following problem:๐ฟ๐œ€ฬƒ๐‘ฆ(๐‘ก,๐œ,๐œŽ,๐œ€)โ‰ก๐œ€๐œ•ฬƒ๐‘ฆ+๐œ•๐‘ก๐‘›๎“๐‘—=1๐œ†๐‘—(๐‘ก)๐œ•ฬƒ๐‘ฆ๐œ•๐œ๐‘—+๐‘š๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๎ƒฌ๐œ†๐‘–(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–๎€ท+๐œ€๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๎ƒญ๐œ•ฬƒ๐‘ฆ๐œ•๐œŽ๐‘–๐‘ž๐‘–โˆ’๐ด(๐‘ก)ฬƒ๐‘ฆ=โ„Ž(๐‘ก),ฬƒ๐‘ฆ(0,0,0,๐œ€)=๐‘ฆ0.(2.4) We determine the solution of problem (2.4) in the form of a seriesฬƒ๐‘ฆ(๐‘ก,๐œ,๐œŽ,๐œ€)=โˆž๎“๐‘˜=โˆ’1๐œ€๐‘˜๐‘ฆ๐‘˜(๐‘ก,๐œ,๐œŽ),(2.5) with coefficients ๐‘ฆ๐‘˜(๐‘ก,๐œ,๐œŽ)โˆˆ๐ถโˆž[0,๐‘‡].

If we substitute (2.5) in (2.4) and equate coefficients at identical degrees of ๐œ€, we obtain the systems for coefficients ๐‘ฆ๐‘˜(๐‘ก,๐œ,๐œŽ):๐ฟ๐‘ฆโˆ’1(๐‘ก,๐œ,๐œŽ)โ‰ก๐‘›๎“๐‘—=1๐œ†๐‘—(๐‘ก)๐œ•๐‘ฆโˆ’1๐œ•๐œ๐‘—+๐‘š๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๐œ†๐‘–(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–๐œ•๐‘ฆโˆ’1๐œ•๐œŽ๐‘–๐‘ž๐‘–โˆ’๐ด(๐‘ก)๐‘ฆโˆ’1=0,๐‘ฆโˆ’1(0,0,0)=0,(๐œ€โˆ’1)๐ฟ๐‘ฆ0(๐‘ก,๐œ,๐œŽ)=โˆ’๐œ•๐‘ฆโˆ’1โˆ’๐œ•๐‘ก๐‘šโˆ‘๐‘ ๐‘–=1๐‘–โˆ’1โˆ‘๐‘ž๐‘–=0๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐œ•๐‘ฆโˆ’1๐œ•๐œŽ๐‘–๐‘ž๐‘–+โ„Ž(๐‘ก),๐‘ฆ0(0,0,0)=๐‘ฆ0,โ‹ฎ(๐œ€0)๐ฟ๐‘ฆ๐‘˜+1(๐‘ก,๐œ,๐œŽ)=โˆ’๐œ•๐‘ฆ๐‘˜โˆ’๐œ•๐‘ก๐‘šโˆ‘๐‘ ๐‘–=1๐‘–โˆ’1โˆ‘๐‘ž๐‘–=0๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐œ•๐‘ฆ๐‘˜๐œ•๐œŽ๐‘–๐‘ž๐‘–,๐‘˜โฉพ1,๐‘ฆ๐‘˜+1โ‹ฎ(0,0,0)=0,(๐œ€๐‘˜+1)

3. Resolvability of Iterative Problems

We solve each of the iterative problems (๐œ€๐‘˜) in the following space of functions:โŽงโŽชโŽจโŽชโŽฉ๐‘ˆ=๐‘ฆ(๐‘ก,๐œ,๐œŽ)โˆถ๐‘ฆ=๐‘›๎“๐‘›๐‘˜=1๎“๐‘—=1๐‘ฆ๐‘˜๐‘—(๐‘ก)๐‘๐‘˜(๐‘ก)๐‘’๐œ๐‘—+๐‘›๎“๐‘š๐‘˜=1๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๐‘ฆ๐‘˜๐‘–๐‘ž๐‘–(๐‘ก)๐‘๐‘˜(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–+๐‘›๎“๐‘˜=1๐‘ฆ๐‘˜(๐‘ก)๐‘๐‘˜(๐‘ก),๐‘ฆ๐‘˜๐‘—(๐‘ก),๐‘ฆ๐‘˜๐‘–๐‘ž๐‘–(๐‘ก),๐‘ฆ๐‘˜(๐‘ก)โˆˆ๐ถโˆž๎€ท[]0,๐‘‡,๐ถ1๎€ธ๎ƒฐ,(3.1) where ๐‘๐‘˜(๐‘ก) are eigenvectors of the operator ๐ด(๐‘ก) corresponding eigenvalues ๐œ†๐‘˜(๐‘ก),๐‘˜=1,๐‘›. We represent ๐‘ˆ in the form of ๐‘ˆ(1)โŠ•๐‘ˆ(0) where๐‘ˆ(0)=๎ƒฏ๐‘ฆ(0)(๐‘ก)โˆถ๐‘ฆ(0)=๐‘›๎“๐‘—=1๐‘ฆ๐‘—(0)(๐‘ก)๐‘๐‘—(๐‘ก),๐‘ฆ๐‘—(0)(๐‘ก)โˆˆ๐ถโˆž๎€ท[]0,๐‘‡,๐ถ1๎€ธ๎ƒฐ,๐‘ˆ(1)=๐‘ˆ๐‘ˆ(0).(3.2) It is easy to note that each of the systems (๐œ€๐‘˜+1) can be written in the form๐ฟ๐‘ฆ(๐‘ก,๐œ,๐œŽ)=โ„Ž(๐‘ก,๐œ,๐œŽ),(3.3) where โ„Ž(๐‘ก,๐œ,๐œŽ) are the corresponding right hand side. Using representations of space ๐‘ˆ, we can write system (3.3) in the equivalent form๐ฟ๐‘ฆ(1)(๐‘ก,๐œ,๐œŽ)=โ„Ž(1)(๐‘ก,๐œ,๐œŽ),(3.4)โˆ’๐ด(๐‘ก)๐‘ฆ(0)(๐‘ก)=โ„Ž(0)(๐‘ก),(3.5) where ๐‘ฆ(1)(๐‘ก,๐œ,๐œŽ),โ„Ž(1)(๐‘ก,๐œ,๐œŽ)โˆˆ๐‘ˆ(1),๐‘ฆ(0)(๐‘ก),โ„Ž(0)(๐‘ก)โˆˆ๐‘ˆ(0).

We have the following result.

Theorem 3.1. Let โ„Ž(1)(๐‘ก,๐œ,๐œŽ)โˆˆ๐‘ˆ(1) and satisfy conditions (i)โ€“(iv). Then, system (3.4) is solvable in the ๐‘ˆ(1) if and only if ๎ซโ„Ž(1)(๐‘ก,๐œ,๐œŽ),๐œˆ๐‘—๎ฌ[](๐‘ก,๐œ,๐œŽ)โ‰ก0โˆ€๐‘กโˆˆ0,๐‘‡,๐‘—=๎ซโ„Ž1,๐‘›,(1)(๐‘ก,๐œ,๐œŽ),๐œˆ๐‘–๐‘ž๐‘–๎ฌ(๐‘ก,๐œ,๐œŽ)โ‰ก0,๐‘–=1,๐‘š,๐‘ž๐‘–=0,๐‘ ๐‘–โˆ’1,(3.6) where ๐œˆ๐‘—(๐‘ก,๐œ,๐œŽ),๐œˆ๐‘–๐‘ž๐‘–(๐‘ก,๐œ,๐œŽ) are basic elements of the kernel of the operator ๐ฟโˆ—โ‰ก๐‘›๎“๐‘—=1๐œ†๐‘—๐œ•(๐‘ก)๐œ•๐œ๐‘—+๐‘š๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๐œ†๐‘–(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–๐œ•๐œ•๐œŽ๐‘–๐‘ž๐‘–โˆ’๐ดโˆ—(๐‘ก).(3.7)

Proof. Let โ„Ž(1)โˆ‘(๐‘ก,๐œ,๐œŽ)=๐‘›๐‘˜=1โˆ‘๐‘›๐‘—=1โ„Ž๐‘˜๐‘—(๐‘ก)๐‘๐‘—(๐‘ก)๐‘’๐œ๐‘˜+โˆ‘๐‘›๐‘˜=1โˆ‘๐‘š๐‘–=1โˆ‘๐‘ ๐‘–๐‘žโˆ’1๐‘–=0โ„Ž๐‘˜๐‘–๐‘ž๐‘–(๐‘ก)๐‘๐‘˜(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–.
Determine solutions of system (3.4) in the form ๐‘ฆ(1)(๐‘ก,๐œ,๐œŽ)=๐‘›๎“๐‘›๐‘˜=1๎“๐‘—=1๐‘ฆ๐‘˜๐‘—(๐‘ก)๐‘๐‘˜(๐‘ก)๐‘’๐œ๐‘—+๐‘›๎“๐‘š๐‘˜=1๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๐‘ฆ๐‘˜๐‘–๐‘ž๐‘–(๐‘ก)๐‘๐‘˜(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–.(3.8) Substituting (3.8) in (3.4) and equating separately coefficients at ๐‘’๐œ๐‘— and ๐œŽ๐‘–๐‘ž๐‘–, we obtain the equations ๎€บ๐œ†๐‘˜(๐‘ก)โˆ’๐œ†๐‘—๎€ป๐‘ฆ(๐‘ก)๐‘˜๐‘—(๐‘ก)=โ„Ž๐‘˜๐‘—(๐‘ก),๐‘˜,๐‘—=๎€บ๐œ†1,๐‘›,๐‘–(๐‘ก)โˆ’๐œ†๐‘˜๎€ป๐‘ฆ(๐‘ก)๐‘–๐‘ž๐‘–๐‘˜(๐‘ก)=โ„Ž๐‘˜๐‘–๐‘ž๐‘–(๐‘ก),๐‘–=1,๐‘š,๐‘ž๐‘–=0,๐‘ ๐‘–โˆ’1,๐‘˜=1,๐‘›.(3.9)
One can see from this that obtained equations are solvable if and only if โ„Ž๐‘˜๐‘˜(๐‘ก)โ‰ก0,๐‘˜=1,๐‘›,โ„Ž๐‘–๐‘–๐‘ž๐‘–(๐‘ก)โ‰ก0,๐‘–=1,๐‘š,๐‘ž๐‘–=0,๐‘ ๐‘–โˆ’1,(3.10) and these conditions coincide with conditions (3.6). Theorem 3.1 is proved.

Remark 3.2. Equations (1.1) imply that under conditions (3.6), system (3.4) has a solution in ๐‘ˆ(1) representable in the form ๐‘ฆ(1)(๐‘ก,๐œ,๐œŽ)=๐‘›๎“๐‘›๐‘˜=1๎“๐‘—=1,๐‘—โ‰ ๐‘˜โ„Ž๐‘˜๐‘—(๐‘ก)๎€บ๐œ†๐‘˜(๐‘ก)โˆ’๐œ†๐‘—๎€ป๐‘(๐‘ก)๐‘—(๐‘ก)๐‘’๐œ๐‘—+๐‘›๎“๐‘˜=1๐›ผ๐‘˜(๐‘ก)๐‘๐‘˜(๐‘ก)๐‘’๐œ๐‘˜+๐‘š๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๐›พ๐‘–๐‘ž๐‘–(๐‘ก)๐‘๐‘–(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–+๐‘›๎“๐‘š๐‘˜=1๎“๐‘ ๐‘–=1,๐‘–โ‰ ๐‘˜๐‘–โˆ’1๎“๐‘ž๐‘–=0โ„Ž๐‘˜๐‘–๐‘ž๐‘–(๐‘ก)๎€บ๐œ†๐‘–(๐‘ก)โˆ’๐œ†๐‘˜๎€ป๐‘(๐‘ก)๐‘˜(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–,(3.11) where ๐›ผ๐‘˜(๐‘ก),๐›พ๐‘–๐‘ž๐‘–(๐‘ก)โˆˆ๐ถโˆž([0,๐‘‡],๐ถ1) are arbitrary functions.
Consider now system (3.5). As det๐ด(๐‘ก)โ‰ก0 in points ๐‘ก=๐‘ก๐‘–, ๐‘–=1,๐‘š, this system does not always have a solution in ๐‘ˆ(0). Introduce the space ๐‘‰(0)โŠ‚๐‘ˆ(0) consisting of vector functions ๐‘ง(0)(๐‘ก)=๐‘›๎“๐‘—=1๐‘ง๐‘—(๐‘ก)๐‘๐‘—(๐‘ก),๐‘ง๐‘—(๐‘ก)โˆˆ๐ถโˆž๎€ท[]0,๐‘‡,๐ถ1๎€ธ,๐‘—=1,๐‘›,(3.12) having the properties ๎€บ๐ท๐‘™๐‘–๎€ท๐‘ง(0)(๐‘ก),๐‘‘๐‘–(๐‘ก)๎€ธ๎€ป๐‘ก=๐‘ก๐‘–=๎€ท๐ท๐‘™๐‘–๐‘ง๐‘–๐‘ก๎€ธ๎€ท๐‘–๎€ธ=0,โˆ€๐‘™๐‘–=0,๐‘ ๐‘–โˆ’1,๐‘–=1,๐‘š,(3.13) where ๐‘‘๐‘–(๐‘ก) are eigenvectors of the operator ๐ดโˆ—(๐‘ก) with regard to eigenvalues ๐œ†๐‘–(๐‘ก),๐‘–=1,๐‘š. Let โ„Ž(0)โˆ‘(๐‘ก)=๐‘›๐‘—=1โ„Ž๐‘—(๐‘ก)๐‘๐‘—(๐‘ก)โˆˆ๐‘‰(0), that is, ๎€ท๐ท๐‘™๐‘–โ„Ž๐‘–๐‘ก๎€ธ๎€ท๐‘–๎€ธ=0โˆ€๐‘™๐‘–=0,๐‘ ๐‘–โˆ’1,๐‘–=1,๐‘š.(3.14) Determine a solution of system (3.5) in the ๐‘ฆ(0)(๐‘ก)=๐‘›๎“๐‘—=1๐‘ฆ๐‘—(๐‘ก)๐‘๐‘—(๐‘ก).(3.15) Substituting this function in (3.5), we obtain โˆ’๐‘›๎“๐‘—=1๐‘ฆ๐‘—(๐‘ก)๐œ†๐‘—(๐‘ก)๐‘๐‘—(๐‘ก)=๐‘›๎“๐‘—=1โ„Ž๐‘—(๐‘ก)๐‘๐‘—(๐‘ก).(3.16) Since {๐‘๐‘—(๐‘ก)} is a basis in ๐ถ๐‘›, we get โˆ’๐œ†๐‘–(๐‘ก)๐‘ฆ๐‘–(๐‘ก)=โ„Ž๐‘–(๐‘ก),๐‘–=1,๐‘š,(3.17)โˆ’๐œ†๐‘—(๐‘ก)๐‘ฆ๐‘—(๐‘ก)=โ„Ž๐‘—(๐‘ก),๐‘—=๐‘š+1,๐‘›.(3.18) It is easy to see that (3.18) has the unique solution ๐‘ฆ๐‘—(๐‘ก)=โˆ’โ„Ž๐‘—(๐‘ก)๐œ†๐‘—(๐‘ก),๐‘—=๐‘š+1,๐‘›.(3.19) By virtue of conditions (3.14), the function โ„Ž๐‘–(๐‘ก) can be represented in the form โ„Ž๐‘–๎€ท(๐‘ก)=๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ ๐‘–๎โ„Ž๐‘–(๐‘ก),๐‘–=1,๐‘š,(3.20) where ๎โ„Ž๐‘–(๐‘ก)โˆˆ๐ถโˆž([0,๐‘‡],๐ถ1 is the certain scalar function, โˆ’(๐‘กโˆ’๐‘ก๐‘–)๐‘ ๐‘–๐‘˜๐‘–(๐‘ก)๐‘ฆ๐‘–(๐‘ก)=(๐‘กโˆ’๐‘ก๐‘–)๐‘ ๐‘–๎โ„Ž๐‘–(๐‘ก), and we see that ๐‘ฆ๐‘–โŽงโŽชโŽจโŽชโŽฉโˆ’๎โ„Ž(๐‘ก)=๐‘–(๐‘ก)๐‘˜๐‘–(๐‘ก),๐‘กโ‰ ๐‘ก๐‘–,๐›พ๐‘–,๐‘ก=๐‘ก๐‘–,(3.21) where ๐›พ๐‘– are arbitrary constants, ๐‘–=1,๐‘š. However, the solution of system (3.5) should belong to the space ๐‘ˆ(0), and it means that ๐‘ฆ๐‘–(๐‘ก)โˆˆ๐ถโˆž([0,๐‘‡],๐ถ1). Therefore, constants in (3.21) ๐›พ๐‘–๎โ„Ž=(๐‘–(๐‘ก)/๐‘˜๐‘–(๐‘ก))โˆฃ๐‘ก=๐‘ก๐‘– and functions are determined uniquely in the form ๐‘ฆ๐‘–โˆ’๎โ„Ž(๐‘ก)=๐‘–(๐‘ก)๐‘˜๐‘–[](๐‘ก),โˆ€๐‘กโˆˆ0,๐‘‡,๐‘–=1,๐‘š.(3.22) Thus, under conditions (3.14), system (3.5) has the solution ๐‘ฆ(0)(๐‘ก) in ๐‘ˆ(0) of ๐‘ฆ(0)(๐‘ก)=โˆ’๐‘š๎“๐‘–=1๎โ„Ž๐‘–(๐‘ก)๐‘˜๐‘–๐‘(๐‘ก)๐‘–(๐‘ก)โˆ’๐‘›๎“๐‘—=๐‘š+1โ„Ž๐‘–(๐‘ก)๐œ†๐‘–๐‘(๐‘ก)๐‘–(๐‘ก),(3.23) where โ„Ž๐‘–๎โ„Ž(๐‘ก)=๐‘–(๐‘ก)/(๐‘กโˆ’๐‘ก๐‘–)๐‘ ๐‘– (in points ๐‘ก=๐‘ก๐‘–,๐‘–=1,๐‘š, this equality is understood in the limiting sense). We summarize received outcome in the form of the following assertion.

Theorem 3.3. Let the operator ๐ด(๐‘ก) satisfy condition (i), and let its spectrum satisfy conditions (ii)โ€“(iv). Then, for any vector function โ„Ž(0)(๐‘ก)โˆˆ๐‘‰(0), system (3.5) has the unique solution ๐‘ฆ(0)(๐‘ก) in space ๐‘ˆ(0).

For uniquely determination of functions ๐›ผ๐‘—(๐‘ก),๐›พ๐‘–๐‘ž๐‘–(๐‘ก), consider system (3.4) with additional conditions:๐‘ฆ(1)(0,0,0)=๐‘ฆโˆ—,๎ƒกโˆ’(3.24)๐œ•๐‘ฆ(1)๐œ•๐‘ก,๐œˆ๐‘—๎ƒข[](๐‘ก,๐œ,๐œŽ)โ‰ก0โˆ€๐‘กโˆˆ0,๐‘‡,๐‘—=๎ƒกโˆ’1,๐‘›,(3.25)๐œ•๐‘ฆ(1)๐œ•๐‘ก,๐œˆ๐‘–๐‘ž๐‘–๎ƒข(๐‘ก,๐œ,๐œŽ)โ‰ก0,๐‘–=1,๐‘š,๐‘ž๐‘–=0,๐‘ ๐‘–โˆ’1,(3.26) where ๐‘ฆโˆ—โˆˆ๐ถ๐‘› is a constant vector.

We have the following result.

Theorem 3.4. Let conditions of Theorem 3.1 hold. Then, the system (3.4) with additional conditions (3.24)-(3.25) has solutions of the form (3.11) in which all summands are uniquely determinate except for ๐›พ๐‘–๐‘ž๐‘–(๐‘ก)๐‘๐‘–(๐‘ก)๐œŽ๐‘–๐‘ž๐‘–(๐‘–=1,๐‘š,๐‘ž๐‘–=0,๐‘ ๐‘–โˆ’1). Functions ๐›พ๐‘–๐‘ž๐‘–(๐‘ก) in the last summand are determined by the formula ๐›พ๐‘–๐‘ž๐‘–(๐‘ก)=๐›พ0๐‘–๐‘ž๐‘–โ‹…๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)+๐‘“๐‘–๐‘ž๐‘–(๐‘ก),(3.27) where ๐‘ƒ๐‘–๐‘ž๐‘–(๐‘ก),๐‘“๐‘–๐‘ž๐‘–(๐‘ก) are known functions, and ๐›พ0๐‘–๐‘ž๐‘–arbitrary constants.

Proof. Denote in (3.11) that ๐‘”๐‘˜๐‘—โ„Ž(๐‘ก)=๐‘˜๐‘—(๐‘ก)๐œ†๐‘—(๐‘ก)โˆ’๐œ†๐‘˜(๐‘ก),๐‘”๐‘˜๐‘–๐‘ž๐‘–โ„Ž(๐‘ก)=๐‘˜๐‘–๐‘ž๐‘–(๐‘ก)๐œ†๐‘–(๐‘ก)โˆ’๐œ†๐‘˜.(๐‘ก)(3.28) Using (3.11) and condition (3.24), we obtain the equality ๐‘›๎“๐‘›๐‘˜=1๎“๐‘—=1๐‘”๐‘˜๐‘—(0)๐‘๐‘—(0)+๐‘›๎“๐‘˜=1๐›ผ๐‘˜(0)๐‘๐‘˜(0)=๐‘ฆโˆ—.(3.29) Multiplying this equality scalarly by ๐‘‘๐‘ (0), we get ๐›ผ๐‘ ๎€ท๐‘ฆ(0)=โˆ—,๐‘‘๐‘ ๎€ธโˆ’(0)๐‘›๎“๐‘˜=1,๐‘˜โ‰ ๐‘ ๐‘”๐‘˜๐‘ (0)โ‰ก๐›ผ0๐‘ ,๐‘ =1,๐‘›.(3.30) By (3.11) and conditions (3.25), we have โˆ’ฬ‡๐›ผ๐‘ ๎€ท(๐‘ก)โˆ’ฬ‡๐‘๐‘ (๐‘ก),๐‘‘๐‘ ๎€ธ๐›ผ(๐‘ก)๐‘ (๐‘ก)โˆ’๐‘›๎“๐‘—=1,๐‘—โ‰ ๐‘ ๐‘”๐‘ ๐‘—๎€ท(๐‘ก)ฬ‡๐‘๐‘—(๐‘ก),๐‘‘๐‘ ๎€ธ(๐‘ก)=0,๐‘ =1,๐‘›.(3.31) Considering these equations with initial conditions (3.30), we can uniquely obtain functions ๐›ผ๐‘ (๐‘ก),๐‘ =1,๐‘›.
Now, using (3.11) and conditions (3.26), we get โˆ’ฬ‡๐›พ๐‘–๐‘ž๐‘–๎€ท(๐‘ก)โˆ’ฬ‡๐‘๐‘–(๐‘ก),๐‘‘๐‘–๎€ธ๐›พ(๐‘ก)๐‘–๐‘ž๐‘–(๐‘ก)โˆ’๐‘›๎“๐‘˜=1,๐‘˜โ‰ ๐‘–๐‘”๐‘˜๐‘–๐‘ž๐‘–๎€ท(๐‘ก)ฬ‡๐‘๐‘˜(๐‘ก),๐‘‘๐‘–๎€ธ(๐‘ก)=0,๐‘–=1,๐‘š,๐‘ž๐‘–=0,๐‘ ๐‘–โˆ’1.(3.32) This implies that ๐›พ๐‘–๐‘ž๐‘–(๐‘ก) have the form (3.27) where ๐‘ƒ๐‘–๐‘ž๐‘–(๎€œ๐‘ก)=โˆ’๐‘ก๐‘ก๐‘–๎€ทฬ‡๐‘๐‘–(๐‘ ),๐‘‘๐‘–(๎€ธ๐‘“๐‘ )๐‘‘๐‘ ,๐‘–๐‘ž๐‘–(๐‘ก)=๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎€œ๐‘ก๐‘ก๐‘–๐‘’โˆ’๐‘ƒ๐‘–๐‘–๐‘ž๐‘›(๐‘ )๎“๐‘˜=1,๐‘˜โ‰ ๐‘–๐‘”๐‘˜๐‘–๐‘ž๐‘–๎€ท(๐‘ )ฬ‡๐‘๐‘˜(๐‘ ),๐‘‘๐‘–๎€ธ(๐‘ )๐‘‘๐‘ .(3.33) Theorem 3.4 is proved.

Remark 3.5. If conditions (3.6) hold for โ„Ž(1)(๐‘ก,๐œ,๐œŽ)โˆˆ๐‘ˆ(1)and โ„Ž(0)(๐‘ก)โˆˆ๐‘ˆ(0), then system (3.3) has a solution in the space ๐‘ˆ, representable in the form of ๐‘ฆ(๐‘ก,๐œ,๐œŽ)=๐‘ฆ(1)(๐‘ก,๐œ,๐œŽ)+๐‘ฆ(0)(๐‘ก),(3.34) where ๐‘ฆ(1)(๐‘ก,๐œ,๐œŽ)is a function in the form of (3.11), and ๐‘ฆ(0)(๐‘ก) is a function in the form of (3.23); moreover, functions ๐›ผ๐‘˜(๐‘ก)โˆˆ๐ถโˆž([0,๐‘‡],๐ถ1) are found uniquely in (3.11), and functions ๐›พ๐‘–๐‘ž๐‘–(๐‘ก) are determined up to arbitrary constants ๐›พ0๐‘–๐‘ž๐‘– in the form of (3.27).

Let us give the following result.

Theorem 3.6. Let โ„Ž(0)(๐‘ก)โˆˆ๐‘ˆ(0),โ„Ž(1)(๐‘ก,๐œ,๐œŽ)โˆˆ๐‘ˆ(1),and conditions (i)โ€“(iv), (3.6), (3.24)โ€“(3.26) hold. Then, there exist unique numbers ๐›พ0๐‘–๐‘ž๐‘– involved in (3.27), such that the function (3.34) satisfies the condition ๐‘ƒ๐‘ฆโ‰กโˆ’๐œ•๐‘ฆ(0)โˆ’๐œ•๐‘ก๐‘š๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐œ•๐‘ฆ(1)๐œ•๐œŽ๐‘–๐‘ž๐‘–+๐ป(0)(๐‘ก)โˆˆ๐‘‰(0),(3.35) where ๐ป(0)(๐‘ก)โˆˆ๐‘‰(0)is a fixed vector function.

Proof. To determine functions uniquely, calculate ๐‘ƒ๐‘ฆโ‰กโˆ’๐‘š๎“๐‘–=1๎‚ธโ„Ž๐‘–(๐‘ก)๐‘˜๐‘–๐‘(๐‘ก)๐‘–๎‚น(๐‘ก)๎…žโˆ’๐‘›๎“๐‘—=๐‘š+1๎‚ธโ„Ž๐‘—(๐‘ก)๐œ†๐‘—๐‘(๐‘ก)๐‘—๎‚น(๐‘ก)๎…žโˆ’๐‘š๎“๐‘ ๐‘–=1๐‘–โˆ’1๎“๐‘ž๐‘–=0๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐›พ๐‘–๐‘ž๐‘–(๐‘ก)๐‘๐‘–+(๐‘ก)๐‘›๎“๐‘š๐‘˜=1๎“๐‘ ๐‘–=1๐‘–โ‰ ๐‘˜๐‘–โˆ’1๎“๐‘ž๐‘–=0๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!โ‹…โ„Ž๐‘–๐‘ž๐‘–(๐‘ก)๐œ†๐‘–(๐‘ก)โˆ’๐œ†๐‘˜๐‘(๐‘ก)๐‘˜(๐‘ก)+๐ป(0)๎€ท(๐‘ก),๐‘ƒ๐‘ฆ,๐‘‘๐‘–๎€ธ๎‚ธโ„Ž(๐‘ก)โ‰กโˆ’๐‘–(๐‘ก)๐‘˜๐‘–๎‚น(๐‘ก)๎…žโˆ’๐‘š๎“๐‘–=1โ„Ž๐‘–(๐‘ก)๐‘˜๐‘–๎€บ(๐‘ก)ฬ‡๐‘๐‘–(๐‘ก),๐‘‘๐‘–๎€ปโˆ’(๐‘ก)๐‘›๎“๐‘—=๐‘š+1๎‚ธโ„Ž๐‘—(๐‘ก)๐‘˜๐‘—๎‚น๎€ท(๐‘ก)ฬ‡๐‘๐‘—(๐‘ก),๐‘‘๐‘–๎€ธโˆ’(๐‘ก)๐‘ ๐‘–โˆ’1๎“๐‘ž๐‘–๐‘š=0๎“๐‘–=1๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐›พ๐‘–๐‘ž๐‘–๎€บ๐ป(๐‘ก)+(0)(๐‘ก),๐‘‘๐‘–๎€ป(๐‘ก),๐‘–=1,๐‘š.(3.36) Denote by ๐‘Ÿ๐‘–(๐‘ก)the known function ๐‘Ÿ๐‘–๎‚ธโ„Ž(๐‘ก)โ‰กโˆ’๐‘–(๐‘ก)๐‘˜๐‘–๎‚น(๐‘ก)๎…žโˆ’๐‘š๎“๐‘–=1โ„Ž๐‘–(๐‘ก)๐‘˜๐‘–๎€บ(๐‘ก)ฬ‡๐‘๐‘–(๐‘ก),๐‘‘๐‘–๎€ปโˆ’(๐‘ก)๐‘›๎“๐‘—=๐‘š+1๎‚ธโ„Ž๐‘—(๐‘ก)๐‘˜๐‘—๎‚น๎€ท(๐‘ก)ฬ‡๐‘๐‘—(๐‘ก),๐‘‘๐‘–๎€ธ+๎€ท๐ป(๐‘ก)(0)(๐‘ก),๐‘‘๐‘–๎€ธ,(๐‘ก)(3.37) and write the conditions (3.13) for (๐‘ƒ๐‘ฆ,๐‘‘๐‘–(๐‘ก)).Taking into account expression (3.27) for ๐›พ๐‘–๐‘ž๐‘–(๐‘ก), we get ๐‘ ๐‘–โˆ’1๎“๐‘ž๐‘–=0๐›พ0๐‘–๐‘ž๐‘–๎ƒฌ๐ท๐‘™๐‘–๎ƒฉ๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ฅ)๎ƒช๎ƒญ๐‘ก=๐‘ก๐‘–+๐‘ ๐‘–โˆ’1๎“๐‘ž๐‘–=0๎€บ๐ท๐‘™๐‘–๐‘“๐‘–๐‘ž๐‘–๎€ป(๐‘ก)๐‘ก=๐‘ก๐‘–=๎€บ๐ท๐‘™๐‘–๐‘Ÿ๐‘–๎€ป(๐‘ก)๐‘ก=๐‘ก๐‘–,๐‘–=1,๐‘š,๐‘™๐‘–=0,๐‘ ๐‘–โˆ’1.(3.38) Using the Leibnitz formula, we obtain that ๎ƒฌ๐ท๐‘™๐‘–๎ƒฉ๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎ƒช๎ƒญ๐‘ก=๐‘ก๐‘–=โŽกโŽขโŽขโŽฃ๐‘™๐‘–๎“๐œˆ=0๐ถ๐œˆ๐‘™๐‘–๎ƒฉ๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๎ƒช(๐œˆ)๎€ท๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎€ธ(๐‘™๐‘–โˆ’๐œˆ)โŽคโŽฅโŽฅโŽฆ๐‘ก=๐‘ก๐‘–=โŽกโŽขโŽขโŽฃ๐‘ž๐‘–๎“๐œˆ=0๐ถ๐œˆ๐‘™๐‘–๎ƒฉ๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๎ƒช(๐œˆ)๎€ท๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎€ธ(๐‘™๐‘–โˆ’๐œˆ)โŽคโŽฅโŽฅโŽฆ๐‘ก=๐‘ก๐‘–=๐ถ๐‘ž๐‘–๐‘™๐‘–๎€ท๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎€ธ(๐‘™๐‘–โˆ’๐‘ž๐‘–)๐‘ก=๐‘ก๐‘–,(3.39) for ๐‘™๐‘–โ‰ฅ๐‘ž๐‘–, ๎ƒฌ๐ท๐‘™๐‘–๎ƒฉ๎€ท๐‘กโˆ’๐‘ก๐‘–๎€ธ๐‘ž๐‘–๐‘ž๐‘–!๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎ƒช๎ƒญ๐‘ก=๐‘ก๐‘–=0,(3.40) for 0โ‰ค๐‘™๐‘–โ‰ค๐‘ž๐‘–.
Therefore, previous equalities are written in the form of ๐‘ ๐‘–โˆ’1๎“๐‘ž๐‘–=0๐›พ0๐‘–๐‘ž๐‘–๐ถ๐‘ž๐‘–๐‘™๐‘–๎€ท๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎€ธ(๐‘™๐‘–โˆ’๐‘ž๐‘–)๐‘ก=๐‘ก๐‘–=๐‘Ÿ0๐‘–๐‘™๐‘–๎‚€๐‘–=1,๐‘š,๐‘™๐‘–=0,๐‘ ๐‘–๎‚โˆ’1,(3.41) where ๐‘Ÿ0๐‘–๐‘™๐‘–=โˆ’๐‘ ๐‘–โˆ’1๎“๐‘ž๐‘–=0๎€บ๐ท๐‘™๐‘–๐‘“๐‘–๐‘ž๐‘–๎€ป(๐‘ก)๐‘ก=๐‘ก๐‘–โˆ’๎€บ๐ท๐‘™๐‘–๐‘Ÿ๐‘–๎€ป(๐‘ก)๐‘ก=๐‘ก๐‘–,for๐‘™๐‘–=0,weget๐›พ0๐‘–0๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก๐‘–)=๐‘Ÿ0๐‘–0;for๐‘™๐‘–=1,weget๐›พ0๐‘–0๐‘01๎€บ๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎€ป๎…ž๐‘ก=๐‘ก๐‘–+๐›พ0๐‘–1๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก๐‘–)=๐‘Ÿ0๐‘–1;โ‹ฎfor๐‘™๐‘–=๐‘ ๐‘–โˆ’1,weget๐›พ0๐‘–0๐‘0๐‘ ๐‘–โˆ’1๎€บ๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก)๎€ป๐‘ ๐‘–โˆ’1๐‘ก=๐‘ก๐‘–+โ‹ฏ+๐›พ0๐‘–๐‘ ๐‘–โˆ’1๐‘’๐‘ƒ๐‘–๐‘–๐‘ž(๐‘ก๐‘–)=๐‘Ÿ0๐‘–๐‘ ๐‘–โˆ’1.(3.42)
We obtain from here sequentially the numbers ๐›พ0๐‘–0,โ€ฆ,๐›พ0๐‘–๐‘ ๐‘–โˆ’1.Theorem 3.6 is proved.

Thus, if conditions (3.24)โ€“(3.26), (3.35) hold, all summands of solution (3.11) are defined uniquely.

So, if โ„Ž(0)(๐‘ก)โˆˆ๐‘ˆ(0),โ„Ž(1)(๐‘ก,๐œ,๐œŽ)โˆˆ๐‘ˆ(1),and conditions (3.6), (3.24)โ€“(3.26), and (3.35) are valid, then the systems (3.4), (3.5) (and (3.3) together with them) are solvable uniquely in the class ๐‘ˆ=๐‘ˆ(1)โŠ•๐‘ˆ(0).Two sequential problems (๐œ€๐‘˜) and (๐œ€๐‘˜+1) are connected uniquely by conditions (3.23)โ€“(3.25), (3.30); therefore, by virtue of Theorems 3.1โ€“3.6, they are solvable uniquely in the space ๐‘ˆ.

4. Asymptotical Character of Formal Solutions

Let ๐‘ฆโˆ’1(๐‘ก,๐œ,๐œŽ),โ€ฆ,๐‘ฆ๐‘˜(๐‘ก,๐œ,๐œŽ) be solutions of formal problems (๐œ€โˆ’1), โ€ฆ,(๐œ€๐‘˜) in the space ๐‘ˆ,respectively. Compose the partial sum for series (2.4):๐‘†๐‘›(๐‘ก,๐œ,๐œŽ)=๐‘›๎“๐‘˜=โˆ’1๐œ€๐‘˜๐‘ฆ๐‘˜(๐‘ก,๐œ,๐œŽ),(4.1) and take its restriction ๐‘ฆ๐œ€๐‘›(๐‘ก)=๐‘†๐‘›(๐‘ก,๐œ‘(๐‘ก)/๐œ€,๐œ“(๐‘ก,๐œ€)).

We have the following result.

Theorem 4.1. Let conditions (i)โ€“(v) hold. Then, for sufficiently small ๐œ€(0โ‰ค๐œ€โ‰ค๐œ€0), the estimates โ€–โ€–๐‘ฆ(๐‘ก,๐œ)โˆ’๐‘ฆ๐œ€๐‘›โ€–โ€–(๐‘ก)๐ถ[0,๐‘‡]โ‰ค๐ถ๐‘›๐œ€๐‘›+1,๐‘›=โˆ’1,0,1,โ€ฆ,(4.2) hold. Here, ๐‘ฆ(๐‘ก,๐œ€) is the exact solution of problem (1.1), and ๐‘ฆ๐œ€๐‘›(๐‘ก) is the states above restriction of the ๐‘›th partial sum of series (2.4).

Proof. The restriction ๐‘ฆ๐œ€๐‘›(๐‘ก) of series (2.4) satisfies the initial condition ๐‘ฆ๐œ€๐‘›(0)=๐‘ฆ0 and system (1.1) up to terms containing ๐œ€๐‘›+1, that is, ๐œ€๐‘‘๐‘ฆ๐œ€๐‘›(๐‘ก)๐‘‘๐‘ก=๐ด(๐‘ก)๐‘ฆ๐œ€๐‘›(๐‘ก)+๐œ€๐‘›+1๐‘…๐‘›(๐‘ก,๐œ€)+โ„Ž(๐‘ก),(4.3) where ๐‘…๐‘›(๐‘ก,๐‘ ) is a known function satisfying the estimate (โ€–๐‘…๐‘ก,๐œ€)โ€–๐ถ[0,๐‘‡]โ‰ค๐‘…๐‘›,๐‘…๐‘›---const.(4.4) Under conditions of Theorem 4.1 on the spectrum of the operator ๐ด(๐‘ก) for the fundamental matrix ๐‘Œ(๐‘ก,๐‘ ,๐œ€)โ‰ก๐‘Œ(๐‘ก,๐œ€)๐‘Œโˆ’1(๐‘ก,๐œ€) of the system ๐œ€ฬ‡๐‘Œ=๐ด(๐‘ก)๐‘Œ, the estimate ๎€บโ€–๐‘Œ(๐‘ก,๐‘ ,๐œ€)โ€–โ‰คconstโˆ€(๐‘ก,๐œ€)โˆˆ๐‘„โ‰ก{0โ‰ค๐‘ โ‰ค๐‘กโ‰ค๐‘‡},โˆ€๐œ€>0โˆˆ0,๐œ€0๎€ป,(4.5) is valid. Here, ๐œ€0>0โˆ’is sufficiently small. Now, write the equation ๐œ€๐‘‘ฮ”(๐‘ก,๐œ€)๐‘‘๐‘ก=๐ด(๐‘ก)ฮ”(๐‘ก,๐œ€)โˆ’๐œ€๐‘›+1๐‘…๐‘›(๐‘ก,๐œ€),ฮ”(0,๐œ€)=0,(4.6) for the remainder term ฮ”(๐‘ก,๐œ€)โ‰ก๐‘ฆ(๐‘ก,๐œ€)โˆ’๐‘ฆ๐œ€๐‘›(๐‘ก).We obtain from this equation that ฮ”(๐‘ก,๐œ€)=โˆ’๐œ€๐‘›๎€œ๐‘ก0๐‘Œ(๐‘ฅ,๐‘ ,๐œ€)๐‘…๐‘›(๐‘ ,๐œ€)๐‘‘๐‘ ,(4.7) whence we get the estimate (โ€–ฮ”๐‘ก,๐œ€)โ€–๐ถ[0,๐‘‡]โ‰คโˆ’๐œ€๐‘›๐‘…๐‘›,(4.8) where ๐‘…๐‘›=max(๐‘ก,๐‘ )โˆˆ๐‘„โ€–๐‘Œ(๐‘ก,๐‘ ,๐œ€)โ€–โ‹…โ€–๐‘…๐‘›(๐‘ก,๐‘ )โ€–โ‹…๐‘‡. So, we obtain the estimate โ€–โ€–๐‘ฆ(๐‘ก,๐œ€)โˆ’๐‘ฆ๐œ€๐‘›(โ€–โ€–๐‘ก)๐ถ[0,๐‘‡]โ‰ค๐œ€๐‘›๐‘…๐‘›,๐‘›=โˆ’1,0,1,โ€ฆ.(4.9) Taking instead of ๐‘ฆ๐œ€๐‘›(๐‘ก) the partial sum ๐‘ฆ๐œ€,๐‘›+1(๐‘ก)โ‰ก๐‘ฆ๐œ€๐‘›(๐‘ก)+๐œ€๐‘›+1๐‘ฆ๐‘›+1๎‚ต๐‘ก,๐œ‘(๐‘ก)๐œ€๎‚ถ,๐œ“(๐‘ก,๐œ€),(4.10) we get โ€–โ€–โ€–๎€ท๐‘ฆ(๐‘ก,๐œ€)โˆ’๐‘ฆ๐œ€๐‘›๎€ธ(๐‘ก)โˆ’๐œ€๐‘›+1๐‘ฆ๐‘›+1๎‚ต๐‘ก,๐œ‘(๐‘ก)๐œ€๎‚ถโ€–โ€–โ€–,๐œ“(๐‘ก,๐œ€)โ‰ค๐œ€๐‘›+1๐‘…๐‘›+1,(4.11) which implies the estimates (4.2). Theorem 4.1 is proved.

5. Example

Let it be required to construct the asymptotical solution for the Cauchy problem๐œ€โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽฬ‡๐‘ฆฬ‡๐‘งโˆ’5๐‘ก2+42๐‘ก2โˆ’2โˆ’10๐‘ก2+104๐‘ก2โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐‘ฆ๐‘งโŽžโŽŸโŽŸโŽ +โŽ›โŽœโŽœโŽ๐‘กโˆ’52โ„Ž10โŽžโŽŸโŽŸโŽ (๐‘ก),๐‘ฆ(0,๐œ€)=๐‘ฆ0,๐‘ง(0,๐œ€)=๐‘ง0,(5.1) where โ„Ž1(๐‘ก)โˆˆ๐ถโˆž[0,2],๐œ€>0is a small parameter. Eigenvalues of the matrix ๎‚€๐ด(๐‘ก)=โˆ’5๐‘ก2+42๐‘ก2โˆ’2โˆ’10๐‘ก2+104๐‘ก2โˆ’5๎‚are ๐œ†1(๐‘ก)=โˆ’๐‘ก2,๐œ†2(๐‘ก)=โˆ’1.Eigenvectors of matrices ๐ด(๐‘ก)and ๐ดโˆ—(๐‘ก), are, respectively,๐œ‘1=โŽ›โŽœโŽœโŽ12โŽžโŽŸโŽŸโŽ ,๐œ‘2=โŽ›โŽœโŽœโŽ25โŽžโŽŸโŽŸโŽ ,๐œ“1=โŽ›โŽœโŽœโŽ5โŽžโŽŸโŽŸโŽ โˆ’2,๐œ“2=โŽ›โŽœโŽœโŽ1โŽžโŽŸโŽŸโŽ โˆ’2.(5.2) We get (โ„Ž(๐‘ก),๐œ“1(๐‘ก))โ‰ก5๐‘ก2โ„Ž1(๐‘ก). Therefore,๎€ทโ„Ž(0),๐œ“1๎€ธ๐‘‘(0)=0,๎€ท๐‘‘๐‘กโ„Ž(0),๐œ“1๎€ธ(0)=0.(5.3)

Hence, we can apply to problem (5.1) the above developed algorithm of the regularization method.

At first, obtain the basic Lagrange-Silvestre polynomials ๐พ๐‘—๐‘–(๐‘ก).Since ๐œ“(๐‘ก)โ‰ก๐œ†1(๐‘ก)=โˆ’๐‘ก2,there will be two such polynomials: ๐พ00(๐‘ก)and ๐พ01(๐‘ก).

Take the arbitrary numbers ๐‘Ž00(๐‘ก)and ๐‘Ž01(๐‘ก),and set the interpolation conditions for the polynomial ๐‘Ÿ(๐‘ก),๐‘Ÿ(๐‘ก)=๐‘Ž00,ฬ‡๐‘Ÿ(1)=๐‘Ž01.(5.4) Expand ๐‘Ÿ(๐‘ก)onto partial fractions๐‘Ÿ(๐‘ก)=๐ด๐œ“(๐‘ก)๐‘ก2+๐ต๐‘ก,(5.5) from where๐‘Ÿ(๐‘ก)โ‰ก๐ด+๐ต๐‘ก.(5.6) Use the interpolation polynomial (5.4). We get ๐ด=๐‘Ž00,๐ต=๐‘Ž01. Hence, (5.6) takes the form๐‘Ÿ(๐‘ก)โ‰ก๐‘Ž00+๐‘ก๐‘Ž01.(5.7) Since numbers ๐‘Ž00and ๐‘Ž01 are arbitrary, basic Lagrange-Silvestre polynomials will be coefficients standing before them, that is,๐พ00(๐‘ก)โ‰ก1,๐พ01(๐‘ก)โ‰ก๐‘ก.(5.8)

Introduce the regularizing variables๐œŽ00=๐‘’โˆซ(1/๐œ€)๐‘ก0๐œ†1๐‘‘๐‘ ๎€œ๐‘ก0๐‘’โˆซโˆ’(1/๐œ€)๐‘ 0๐œ†1๐‘‘๐‘ฅโ‹…๐พ00(๐‘ )๐‘‘๐‘ =๐‘’โˆ’๐‘ก3/3๐œ€๎€œ๐‘ก0๐‘’๐‘ 3/3๐œ€๐‘‘๐‘ โ‰ก๐‘00(๐œŽ๐‘ก),01=๐‘’โˆซ(1/๐œ€)๐‘ก0๐œ†1๐‘‘๐‘ ๎€œ๐‘ก0๐‘’โˆซโˆ’(1/๐œ€)๐‘ 0๐œ†1๐‘‘๐‘ฅโ‹…๐พ01(๐‘ )๐‘‘๐‘ =๐‘’โˆ’๐‘ก3/3๐œ€๎€œ๐‘ก0๐‘’๐‘ 3/3๐œ€โ‹…๐‘ ๐‘‘๐‘ โ‰ก๐‘01๐œ(๐‘ก),1=1๐œ€๎€œ๐‘ก0๐œ†1๐‘ก๐‘‘๐‘ =โˆ’33๐œ€โ‰ก๐‘ž1(๐‘ก),๐œ2=1๐œ€๎€œ๐‘ก0๐œ†2๐‘ก๐‘‘๐‘ =โˆ’๐œ€โ‰ก๐‘ž2(๐‘ก).(5.9)

Construct the extended problem corresponding to problem (5.1):๐œ€๐œ•๐‘ค๐œ•๐‘ก+๐œ†1(๐‘ก)๐œ•๐‘ค๐œ•๐œ1+๐œ†2(๐‘ก)๐œ•๐‘ค๐œ•๐œ2+๐œ†1(๐‘ก)๐œŽ00๐œ•๐‘ค๐œ•๐œŽ00+๐œ†1(๐‘ก)๐œŽ01๐œ•๐‘ค๐œ•๐œŽ01+๐œ€๐œ•๐‘ค๐œ•๐œŽ00+๐œ€๐‘ก๐œ•๐‘ค๐œ•๐œŽ01โˆ’๐ด(๐‘ก)๐‘ค=โ„Ž(๐‘ก),๐‘ค(0,0,0,๐œ€)=๐‘ค0,(5.10) where ๐œโ‰ก(๐œ1,๐œ2),๐œŽ=(๐œŽ00,๐œŽ01),๐‘ค=๐‘ค(๐‘ก,๐œ,๐œŽ,๐œ€).

Determining solutions of problem (5.10) in the form of a series๐‘ค(๐‘ก,๐œ,๐œŽ,๐œ€)=โˆž๎“๐‘˜=0๐œ€๐‘˜๐‘ค๐‘˜(๐‘ก,๐œ,๐œŽ),(5.11) we obtain the following iteration problems: ๐ฟ๐‘ค0โ‰ก๐œ†1๎‚ธ(๐‘ก)๐œ•๐‘ค0+๐œ•๐œ๐œ•๐‘ค0๐œ•๐œŽ00+๐‘กโ‹…๐œŽ01๐œ•๐‘ค0๐œ•๐œŽ01๎‚น+๐œ†2(๐‘ก)๐œ•๐‘ค0๐œ•๐œ2โˆ’๐ด(๐‘ก)๐‘ค0=โ„Ž(๐‘ก),๐‘ค0(0,0,0)=๐‘ค0,(5.12)๐ฟ๐‘ค1=โˆ’๐œ•๐‘ค0โˆ’๐œ•๐‘ก๐œ•๐‘ค0๐œ•๐œŽ00โˆ’๐‘ก๐œ•๐‘ค0๐œ•๐œŽ01,๐‘ค1โ‹ฎ(0,0,0)=0,(5.13)

We determine solutions of iteration problems (5.12), (5.13), and so on in the space ๐‘ˆ of functions in the form of๐‘ค(๐‘ก,๐œ,๐œŽ)=๐‘ค1(๐‘ก)๐‘’๐œ1+๐‘ค2(๐‘ก)๐‘’๐œ2+๐‘ค00(๐‘ก)๐œŽ00+๐‘ค01(๐‘ก)๐œŽ01+๐‘ค0๐‘ค(๐‘ก),0(๐‘ก),๐‘ค1(๐‘ก),๐‘ค2(๐‘ก),๐‘ค00(๐‘ก),๐‘ค01(๐‘ก)โˆˆ๐ถโˆž๎€ท[]0,2,๐ถ2๎€ธ.(5.14)

Directly calculating, we obtain the solution of system (5.12) in the form of๐‘ค0(๐‘ก,๐œ,๐œŽ)=๐›ผ1(๐‘ก)๐œ‘1๐‘’๐œ1+๐›ผ2(๐‘ก)๐œ‘2๐‘’๐œ2+๐›พ00(๐‘ก)๐œ‘1๐œŽ00+๐›พ01(๐‘ก)๐œ‘1๐œŽ01+5โ„Ž1(๐‘ก)๐œ‘1โˆ’2๐‘ก2โ„Ž1(๐‘ก)๐œ‘2,(5.15) where ๐›ผ๐‘—(๐‘ก),๐›พ๐‘—๐‘–(๐‘ก)โˆˆ๐ถโˆž[0,2]are for now arbitrary functions.

To calculate the functions ๐›ผ๐‘—(๐‘ก)and ๐›พ๐‘–๐‘—(๐‘ก),we pass to the following iteration problem (5.13). Taking into account (5.15), it will be written in the form of๐ฟ๐‘ค1=โˆ’ฬ‡๐›ผ1(๐‘ก)๐œ‘1๐‘’๐œ1โˆ’ฬ‡๐›ผ2(๐‘ก)๐œ‘2๐‘’๐œ2โˆ’ฬ‡๐›พ00(๐‘ก)๐œ‘1๐œŽ00โˆ’ฬ‡๐›พ01(๐‘ก)๐œ‘1๐œŽ01ฬ‡โ„Žโˆ’51(๐‘ก)๐œ‘1โˆ’๎€ท2๐‘ก2โ„Ž1๎€ธ(๐‘ก)๎…ž๐œ‘2โˆ’๐›พ00(๐‘ก)๐œ‘1โˆ’๐‘ก๐›พ01(๐‘ก)๐œ‘1.(5.16)

For solvability of problem (5.13) in the space ๐‘ˆ, it is necessary and sufficient to fulfill the conditionsโˆ’ฬ‡๐›ผ1(๐‘ก)=0,โˆ’ฬ‡๐›ผ2(๐‘ก)=0,โˆ’ฬ‡๐›พ00(๐‘ก)=0,โˆ’ฬ‡๐›พ01ฬ‡โ„Ž(๐‘ก)=0,โˆ’51(0)โˆ’๐›พ00ฬˆโ„Ž(0)=0,โˆ’51(0)โˆ’ฬ‡๐›พ00(0)โˆ’๐›พ01(0)=0.(5.17)

Using solution (5.15) and the initial condition ๐‘ค0(0,0,0)=๐‘ค0,we obtain the equation๐›ผ1(0)๐œ‘1+๐›ผ2(0)๐œ‘2+5โ„Ž1(0)๐œ‘1=๐‘ค0.(5.18)

Multiplying it (scalar) on ๐œ“1and ๐œ“2, we obtain the values๐›ผ1๎€ท๐‘ค(0)=0,๐œ“1๎€ธโˆ’5โ„Ž1(0)โ‰ก5๐‘ฆ0โˆ’2๐‘ง0โˆ’5โ„Ž1๐›ผ(0),2๎€ท๐‘ค(0)=0,๐œ“2๎€ธ=๐‘ง0โˆ’2๐‘ฆ0.(5.19)

Using equalities (5.17), and also the initial data (5.19), we obtain uniquely the functions ๐›ผ๐‘—(๐‘ก)and ๐›พ๐‘—๐‘–(๐‘ก):๐›ผ1(๐‘ก)=5๐‘ฆ0โˆ’2๐‘ง0โˆ’5โ„Ž1(0),๐›ผ2(๐‘ก)=๐‘ง0โˆ’2๐‘ฆ0.๐›พ00ฬ‡โ„Ž(๐‘ก)=โˆ’51(0),๐›พ01ฬˆโ„Ž(๐‘ก)=โˆ’51(0).(5.20)

Substituting these functions into (5.15), we obtain uniquely the solution of problem (5.12) in the space ๐‘ˆ,๐‘ค0๎€ท(๐‘ก,๐œ,๐œŽ)=5๐‘ฆ0โˆ’2๐‘ง0โˆ’5โ„Ž1๎€ธ๐œ‘(0)1๐‘’๐œ1+๎€ท๐‘ง0โˆ’2๐‘ฆ0๎€ธ๐œ‘2๐‘’๐œ2ฬ‡โ„Žโˆ’51(0)๐œ‘1๐œŽ00ฬˆโ„Žโˆ’51(0)๐œ‘1๐œŽ01+5โ„Ž1(๐‘ก)๐œ‘1โˆ’2๐‘ก2โ„Ž1(๐‘ก)๐œ‘2.(5.21)

Producing here restriction on the functions ๐œ=๐‘ž(๐‘ก),๐œŽ=๐‘(๐‘ก),we obtain the principal term of the asymptotics for the solution of problem (5.1):๐‘ค0๐œ€๎€ท(๐‘ก)=5๐‘ฆ0โˆ’2๐‘ง0โˆ’5โ„Ž1๎€ธ๐œ‘(0)1๐‘’โˆ’๐‘ก3/3๐œ€+๎€ท๐‘ง0โˆ’2๐‘ฆ0๎€ธ๐œ‘2๐‘’โˆ’๐‘ก/๐œ€ฬ‡โ„Žโˆ’51(0)๐œ‘1๐‘’โˆ’๐‘ก3/3๐œ€๎€œ๐‘ก0๐‘’๐‘ 3/3๐œ€ฬˆโ„Ž๐‘‘๐‘ โˆ’51(0)๐œ‘1๐‘’โˆ’๐‘ก3/3๐œ€๎€œ๐‘ก0๐‘’๐‘ 3/3๐œ€๐‘ ๐‘‘๐‘ +5โ„Ž1(๐‘ก)๐œ‘1โˆ’2๐‘ก2โ„Ž1(๐‘ก)๐œ‘2.(5.22)

The zero-order asymptotical solution is obtained: it satisfies the estimateโ€–โ€–๐‘ค(๐‘ก,๐œ€)โˆ’๐‘ค0๐œ€โ€–โ€–(๐‘ก)๐ถ[0,2]โ‰ค๐ถ1โ‹…๐œ€,(5.23) where ๐‘ค(๐‘ก,๐œ€)is an exact solution of problem (1.1), and ๐ถ1>0is a constant independent of ๐œ€at sufficiently small ๐œ€(0<๐œ€โ‰ค๐œ€0).

References

  1. S. A. Lomov, Introduction to the General Theory of Singular Perturbations, vol. 112 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1992. View at: Zentralblatt MATH
  2. L. A. Skinner, โ€œMatched expansion solutions of the first-order turning point problem,โ€ SIAM Journal on Mathematical Analysis, vol. 25, no. 5, pp. 1402โ€“1411, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. A. B. Vasil'eva, โ€œOn contrast structures of step type for a system of singularly perturbed equations,โ€ Computational Mathematics and Mathematical Physics, vol. 34, no. 10, pp. 1215โ€“1223, 1994. View at: Google Scholar
  4. A. B. Vasil'eva, โ€œContrast structures of step-like type for a second-order singularly perturbed quasilinear differential equation,โ€ Computational Mathematics and Mathematical Physics, vol. 35, no. 4, pp. 411โ€“419, 1995. View at: Google Scholar
  5. A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, vol. 14, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1995.
  6. A. G. Eliseev and S. A. Lomov, โ€œThe theory of singular perturbations in the case of spectral singularities of a limit operator,โ€ Matematicheskiĭ Sbornik, vol. 131, no. 4, pp. 544โ€“557, 1986. View at: Google Scholar
  7. A. Ashyralyev, โ€œOn uniform difference schemes of a high order of approximation for evolution equations with a small parameter,โ€ Numerical Functional Analysis and Optimization, vol. 10, no. 5-6, pp. 593โ€“606, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. G. I. Shishkin, J. J. H. Miller, and E. O'Riordan, Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, River Edge, NJ, USA, 1996.
  9. A. Ashyralyev and H. O. Fattorini, โ€œOn uniform difference schemes for second-order singular perturbation problems in Banach spaces,โ€ SIAM Journal on Mathematical Analysis, vol. 23, no. 1, pp. 29โ€“54, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004.
  11. A. Ashyralyev and Y. Sözen, โ€œA note on the parabolic equation with an arbitrary parameter at the derivative,โ€ Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2565โ€“2572, 2011. View at: Publisher Site | Google Scholar
  12. A. Ashyralyev, โ€œOn uniform difference schemes of a higher order of approximation for elliptical equations with a small parameter,โ€ Applicable Analysis, vol. 36, no. 3-4, pp. 211โ€“220, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. A. Ashyralyev and H. O. Fattorini, โ€œOn difference schemes of the high order of accuracy for singular perturbation elliptic equations,โ€ in Investigation of Theory and Approximation Methods for Differential Equations, pp. 80โ€“83, Ashgabat, Turkmenistan, 1991. View at: Google Scholar
  14. A. Xu and Z. Cen, โ€œAsymptotic behaviors of intermediate points in the remainder of the Euler-Maclaurin formula,โ€ Abstract and Applied Analysis, vol. 2010, Article ID 134392, 8 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. M. De la Sen, โ€œAsymptotic comparison of the solutions of linear time-delay systems with point and distributed lags with those of their limiting equations,โ€ Abstract and Applied Analysis, vol. 2009, Article ID 216746, 37 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. F. Wang and A. Yukun, โ€œPositive solutions for singular complementary Lid-stone boundary value problems,โ€ Abstract and Applied Analysis, vol. 2011, Article ID 714728, 13 pages, 2011. View at: Publisher Site | Google Scholar
  17. A. B. Vasile'va and L. V. Kalachev, โ€œSingularly perturbed periodic parabolic equations with alternating boundary layer type solutions,โ€ Abstract and Applied Analysis, vol. 2010, Article ID 52856, 21 pages, 2006. View at: Publisher Site | Google Scholar
  18. H. Šamajová and E. Špániková, โ€œOn asymptotic behaviour of solutions to n-dimensional systems of neutral differential equations,โ€ Abstract and Applied Analysis, vol. 2011, Article ID 791323, 19 pages, 2011. View at: Publisher Site | Google Scholar
  19. I. Gavrea and M. Ivan, โ€œAsymptotic behaviour of the iterates of positive linear operators,โ€ Abstract and Applied Analysis, vol. 2011, Article ID 670509, 11 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2012 Burkhan T. Kalimbetov et al. 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.


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