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., [1ā€“19]).

In this work, the algorithm for construction of an asymptotical solution for linear singular perturbed systems with the irreversible limit operator is givenā€”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 formšœ€Ģ‡š‘¦=š“(š‘”)š‘¦+ā„Ž(š‘”),š‘¦(0,šœ€)=š‘¦0[],,š‘”āˆˆ0,š‘‡(1.1) where š‘¦={š‘¦1,ā€¦,š‘¦š‘›},š“(š‘”) is a matrix of order (š‘›Ć—š‘›),ā„Ž(š‘”)={ā„Ž1,ā€¦,ā„Žš‘›} is a known function, š‘¦0āˆˆš¶š‘› is a constant vector, and šœ€>0 is a small parameter, in the case of violation of stability of a spectrum {šœ†š‘—(š‘”)} of the limiting operator š“(š‘”).

Difference of such type problems from similar problems with a stable spectrum (i.e., in the case of šœ†š‘–(š‘”)ā‰ 0,šœ†š‘–(š‘”)ā‰ šœ†š‘—(š‘”),š‘–ā‰ š‘—,š‘–,š‘—=1,š‘›forallš‘”āˆˆ[0,š‘‡]) is that the limiting system 0=š“(š‘”)š‘¦+ā„Ž(š‘”)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,š‘‡] solutions š‘¦(š‘”)of the limiting system is not excluded. Under conditions, one can prove (see, e.g., [1, 6]) that the exact solution š‘¦(š‘”,šœ€) of problem (1.1) tends (at šœ€ā†’+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 {šœ†š‘—(š‘”)} of the limiting operator š“(š‘”). As it was shown in [3ā€“7], they were induced by instability points š‘”š‘— 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)š“(š‘”)āˆˆš¶āˆž([0,š‘‡],š¶š‘›),ā„Ž(š‘”)āˆˆš¶āˆž[0,š‘‡]; for any š‘”āˆˆ[0,š‘‡], the spectrum {šœ†š‘—(š‘”)} of the operator š“(š‘”) satisfies the conditions:(ii)šœ†š‘–(š‘”)=āˆ’(š‘”āˆ’š‘”š‘–)š‘ š‘–š‘˜š‘–(š‘”),š‘˜š‘–(š‘”)ā‰ 0,š‘”š‘–āˆˆ[0,š‘‡],š‘–=1,š‘š,š‘š<š‘› (here š‘ š‘–- are even natural numbers),(iii)šœ†š‘–(š‘”)ā‰ 0,š‘—=š‘š+1,š‘›,(iv)šœ†š‘–(š‘”)ā‰ šœ†š‘—(š‘”),š‘–ā‰ š‘—,š‘–,š‘—=1,š‘›,(v)Rešœ†š‘—(š‘”)ā©½0,š‘—=1,š‘›.

2. Regularization of the Problem

We introduce basic regularized variables by the spectrum of the limiting operatoršœš‘—=šœ€āˆ’1ī€œš‘”0šœ†š‘—šœ‘(š‘ )š‘‘š‘ ā‰”š‘—(š‘”)šœ€,š‘—=1,š‘›.(2.1) Instable points š‘”š‘–āˆˆ[0,š‘‡] of the spectrum {šœ†š‘—(š‘”)} induce additional regularized variables described by the formulasšœŽš‘–š‘žš‘–=š‘’šœ‘š‘–(š‘”)/šœ€ī€œš‘”0š‘’āˆ’šœ‘š‘–(š‘ )/šœ€ī€·š‘ āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–š‘‘š‘ ā‰”šœ“š‘–š‘žš‘–(š‘”,šœ€),š‘–=1,š‘š,š‘žš‘–=0,š‘ š‘–āˆ’1.(2.2) We consider a vector function Ģƒš‘¦(š‘”,šœ,šœŽ,šœ€) instead of the solution š‘¦(š‘”,šœ€) to be found for problem (1.1). This vector function is such thatĢƒš‘¦(š‘”,šœ,šœŽ,šœ€)āˆ£šœ=šœ‘,šœŽ=šœ“ā‰”š‘¦(š‘”,šœ€).(2.3)

For Ģƒš‘¦(š‘”,šœ,šœŽ,šœ€), it is natural to set the following problem:šæšœ€Ģƒš‘¦(š‘”,šœ,šœŽ,šœ€)ā‰”šœ€šœ•Ģƒš‘¦+šœ•š‘”š‘›ī“š‘—=1šœ†š‘—(š‘”)šœ•Ģƒš‘¦šœ•šœš‘—+š‘šī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0īƒ¬šœ†š‘–(š‘”)šœŽš‘–š‘žš‘–ī€·+šœ€š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!īƒ­šœ•Ģƒš‘¦šœ•šœŽš‘–š‘žš‘–āˆ’š“(š‘”)Ģƒš‘¦=ā„Ž(š‘”),Ģƒš‘¦(0,0,0,šœ€)=š‘¦0.(2.4) We determine the solution of problem (2.4) in the form of a seriesĢƒš‘¦(š‘”,šœ,šœŽ,šœ€)=āˆžī“š‘˜=āˆ’1šœ€š‘˜š‘¦š‘˜(š‘”,šœ,šœŽ),(2.5) with coefficients š‘¦š‘˜(š‘”,šœ,šœŽ)āˆˆš¶āˆž[0,š‘‡].

If we substitute (2.5) in (2.4) and equate coefficients at identical degrees of šœ€, we obtain the systems for coefficients š‘¦š‘˜(š‘”,šœ,šœŽ):šæš‘¦āˆ’1(š‘”,šœ,šœŽ)ā‰”š‘›ī“š‘—=1šœ†š‘—(š‘”)šœ•š‘¦āˆ’1šœ•šœš‘—+š‘šī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0šœ†š‘–(š‘”)šœŽš‘–š‘žš‘–šœ•š‘¦āˆ’1šœ•šœŽš‘–š‘žš‘–āˆ’š“(š‘”)š‘¦āˆ’1=0,š‘¦āˆ’1(0,0,0)=0,(šœ€āˆ’1)šæš‘¦0(š‘”,šœ,šœŽ)=āˆ’šœ•š‘¦āˆ’1āˆ’šœ•š‘”š‘šāˆ‘š‘ š‘–=1š‘–āˆ’1āˆ‘š‘žš‘–=0ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!šœ•š‘¦āˆ’1šœ•šœŽš‘–š‘žš‘–+ā„Ž(š‘”),š‘¦0(0,0,0)=š‘¦0,ā‹®(šœ€0)šæš‘¦š‘˜+1(š‘”,šœ,šœŽ)=āˆ’šœ•š‘¦š‘˜āˆ’šœ•š‘”š‘šāˆ‘š‘ š‘–=1š‘–āˆ’1āˆ‘š‘žš‘–=0ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!šœ•š‘¦š‘˜šœ•šœŽš‘–š‘žš‘–,š‘˜ā©¾1,š‘¦š‘˜+1ā‹®(0,0,0)=0,(šœ€š‘˜+1)

3. Resolvability of Iterative Problems

We solve each of the iterative problems (šœ€š‘˜) in the following space of functions:āŽ§āŽŖāŽØāŽŖāŽ©š‘ˆ=š‘¦(š‘”,šœ,šœŽ)āˆ¶š‘¦=š‘›ī“š‘›š‘˜=1ī“š‘—=1š‘¦š‘˜š‘—(š‘”)š‘š‘˜(š‘”)š‘’šœš‘—+š‘›ī“š‘šš‘˜=1ī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0š‘¦š‘˜š‘–š‘žš‘–(š‘”)š‘š‘˜(š‘”)šœŽš‘–š‘žš‘–+š‘›ī“š‘˜=1š‘¦š‘˜(š‘”)š‘š‘˜(š‘”),š‘¦š‘˜š‘—(š‘”),š‘¦š‘˜š‘–š‘žš‘–(š‘”),š‘¦š‘˜(š‘”)āˆˆš¶āˆžī€·[]0,š‘‡,š¶1ī€øīƒ°,(3.1) where š‘š‘˜(š‘”) are eigenvectors of the operator š“(š‘”) corresponding eigenvalues šœ†š‘˜(š‘”),š‘˜=1,š‘›. We represent š‘ˆ in the form of š‘ˆ(1)āŠ•š‘ˆ(0) whereš‘ˆ(0)=īƒÆš‘¦(0)(š‘”)āˆ¶š‘¦(0)=š‘›ī“š‘—=1š‘¦š‘—(0)(š‘”)š‘š‘—(š‘”),š‘¦š‘—(0)(š‘”)āˆˆš¶āˆžī€·[]0,š‘‡,š¶1ī€øīƒ°,š‘ˆ(1)=š‘ˆš‘ˆ(0).(3.2) It is easy to note that each of the systems (šœ€š‘˜+1) can be written in the formšæš‘¦(š‘”,šœ,šœŽ)=ā„Ž(š‘”,šœ,šœŽ),(3.3) where ā„Ž(š‘”,šœ,šœŽ) are the corresponding right hand side. Using representations of space š‘ˆ, we can write system (3.3) in the equivalent formšæš‘¦(1)(š‘”,šœ,šœŽ)=ā„Ž(1)(š‘”,šœ,šœŽ),(3.4)āˆ’š“(š‘”)š‘¦(0)(š‘”)=ā„Ž(0)(š‘”),(3.5) where š‘¦(1)(š‘”,šœ,šœŽ),ā„Ž(1)(š‘”,šœ,šœŽ)āˆˆš‘ˆ(1),š‘¦(0)(š‘”),ā„Ž(0)(š‘”)āˆˆš‘ˆ(0).

We have the following result.

Theorem 3.1. Let ā„Ž(1)(š‘”,šœ,šœŽ)āˆˆš‘ˆ(1) and satisfy conditions (i)ā€“(iv). Then, system (3.4) is solvable in the š‘ˆ(1) if and only if ī«ā„Ž(1)(š‘”,šœ,šœŽ),šœˆš‘—ī¬[](š‘”,šœ,šœŽ)ā‰”0āˆ€š‘”āˆˆ0,š‘‡,š‘—=ī«ā„Ž1,š‘›,(1)(š‘”,šœ,šœŽ),šœˆš‘–š‘žš‘–ī¬(š‘”,šœ,šœŽ)ā‰”0,š‘–=1,š‘š,š‘žš‘–=0,š‘ š‘–āˆ’1,(3.6) where šœˆš‘—(š‘”,šœ,šœŽ),šœˆš‘–š‘žš‘–(š‘”,šœ,šœŽ) are basic elements of the kernel of the operator šæāˆ—ā‰”š‘›ī“š‘—=1šœ†š‘—šœ•(š‘”)šœ•šœš‘—+š‘šī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0šœ†š‘–(š‘”)šœŽš‘–š‘žš‘–šœ•šœ•šœŽš‘–š‘žš‘–āˆ’š“āˆ—(š‘”).(3.7)

Proof. Let ā„Ž(1)āˆ‘(š‘”,šœ,šœŽ)=š‘›š‘˜=1āˆ‘š‘›š‘—=1ā„Žš‘˜š‘—(š‘”)š‘š‘—(š‘”)š‘’šœš‘˜+āˆ‘š‘›š‘˜=1āˆ‘š‘šš‘–=1āˆ‘š‘ š‘–š‘žāˆ’1š‘–=0ā„Žš‘˜š‘–š‘žš‘–(š‘”)š‘š‘˜(š‘”)šœŽš‘–š‘žš‘–.
Determine solutions of system (3.4) in the form š‘¦(1)(š‘”,šœ,šœŽ)=š‘›ī“š‘›š‘˜=1ī“š‘—=1š‘¦š‘˜š‘—(š‘”)š‘š‘˜(š‘”)š‘’šœš‘—+š‘›ī“š‘šš‘˜=1ī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0š‘¦š‘˜š‘–š‘žš‘–(š‘”)š‘š‘˜(š‘”)šœŽš‘–š‘žš‘–.(3.8) Substituting (3.8) in (3.4) and equating separately coefficients at š‘’šœš‘— and šœŽš‘–š‘žš‘–, we obtain the equations ī€ŗšœ†š‘˜(š‘”)āˆ’šœ†š‘—ī€»š‘¦(š‘”)š‘˜š‘—(š‘”)=ā„Žš‘˜š‘—(š‘”),š‘˜,š‘—=ī€ŗšœ†1,š‘›,š‘–(š‘”)āˆ’šœ†š‘˜ī€»š‘¦(š‘”)š‘–š‘žš‘–š‘˜(š‘”)=ā„Žš‘˜š‘–š‘žš‘–(š‘”),š‘–=1,š‘š,š‘žš‘–=0,š‘ š‘–āˆ’1,š‘˜=1,š‘›.(3.9)
One can see from this that obtained equations are solvable if and only if ā„Žš‘˜š‘˜(š‘”)ā‰”0,š‘˜=1,š‘›,ā„Žš‘–š‘–š‘žš‘–(š‘”)ā‰”0,š‘–=1,š‘š,š‘žš‘–=0,š‘ š‘–āˆ’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 š‘ˆ(1) representable in the form š‘¦(1)(š‘”,šœ,šœŽ)=š‘›ī“š‘›š‘˜=1ī“š‘—=1,š‘—ā‰ š‘˜ā„Žš‘˜š‘—(š‘”)ī€ŗšœ†š‘˜(š‘”)āˆ’šœ†š‘—ī€»š‘(š‘”)š‘—(š‘”)š‘’šœš‘—+š‘›ī“š‘˜=1š›¼š‘˜(š‘”)š‘š‘˜(š‘”)š‘’šœš‘˜+š‘šī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0š›¾š‘–š‘žš‘–(š‘”)š‘š‘–(š‘”)šœŽš‘–š‘žš‘–+š‘›ī“š‘šš‘˜=1ī“š‘ š‘–=1,š‘–ā‰ š‘˜š‘–āˆ’1ī“š‘žš‘–=0ā„Žš‘˜š‘–š‘žš‘–(š‘”)ī€ŗšœ†š‘–(š‘”)āˆ’šœ†š‘˜ī€»š‘(š‘”)š‘˜(š‘”)šœŽš‘–š‘žš‘–,(3.11) where š›¼š‘˜(š‘”),š›¾š‘–š‘žš‘–(š‘”)āˆˆš¶āˆž([0,š‘‡],š¶1) are arbitrary functions.
Consider now system (3.5). As detš“(š‘”)ā‰”0 in points š‘”=š‘”š‘–, š‘–=1,š‘š, this system does not always have a solution in š‘ˆ(0). Introduce the space š‘‰(0)āŠ‚š‘ˆ(0) consisting of vector functions š‘§(0)(š‘”)=š‘›ī“š‘—=1š‘§š‘—(š‘”)š‘š‘—(š‘”),š‘§š‘—(š‘”)āˆˆš¶āˆžī€·[]0,š‘‡,š¶1ī€ø,š‘—=1,š‘›,(3.12) having the properties ī€ŗš·š‘™š‘–ī€·š‘§(0)(š‘”),š‘‘š‘–(š‘”)ī€øī€»š‘”=š‘”š‘–=ī€·š·š‘™š‘–š‘§š‘–š‘”ī€øī€·š‘–ī€ø=0,āˆ€š‘™š‘–=0,š‘ š‘–āˆ’1,š‘–=1,š‘š,(3.13) where š‘‘š‘–(š‘”) are eigenvectors of the operator š“āˆ—(š‘”) with regard to eigenvalues šœ†š‘–(š‘”),š‘–=1,š‘š. Let ā„Ž(0)āˆ‘(š‘”)=š‘›š‘—=1ā„Žš‘—(š‘”)š‘š‘—(š‘”)āˆˆš‘‰(0), that is, ī€·š·š‘™š‘–ā„Žš‘–š‘”ī€øī€·š‘–ī€ø=0āˆ€š‘™š‘–=0,š‘ š‘–āˆ’1,š‘–=1,š‘š.(3.14) Determine a solution of system (3.5) in the š‘¦(0)(š‘”)=š‘›ī“š‘—=1š‘¦š‘—(š‘”)š‘š‘—(š‘”).(3.15) Substituting this function in (3.5), we obtain āˆ’š‘›ī“š‘—=1š‘¦š‘—(š‘”)šœ†š‘—(š‘”)š‘š‘—(š‘”)=š‘›ī“š‘—=1ā„Žš‘—(š‘”)š‘š‘—(š‘”).(3.16) Since {š‘š‘—(š‘”)} is a basis in š¶š‘›, we get āˆ’šœ†š‘–(š‘”)š‘¦š‘–(š‘”)=ā„Žš‘–(š‘”),š‘–=1,š‘š,(3.17)āˆ’šœ†š‘—(š‘”)š‘¦š‘—(š‘”)=ā„Žš‘—(š‘”),š‘—=š‘š+1,š‘›.(3.18) It is easy to see that (3.18) has the unique solution š‘¦š‘—(š‘”)=āˆ’ā„Žš‘—(š‘”)šœ†š‘—(š‘”),š‘—=š‘š+1,š‘›.(3.19) By virtue of conditions (3.14), the function ā„Žš‘–(š‘”) can be represented in the form ā„Žš‘–ī€·(š‘”)=š‘”āˆ’š‘”š‘–ī€øš‘ š‘–īā„Žš‘–(š‘”),š‘–=1,š‘š,(3.20) where īā„Žš‘–(š‘”)āˆˆš¶āˆž([0,š‘‡],š¶1 is the certain scalar function, āˆ’(š‘”āˆ’š‘”š‘–)š‘ š‘–š‘˜š‘–(š‘”)š‘¦š‘–(š‘”)=(š‘”āˆ’š‘”š‘–)š‘ š‘–īā„Žš‘–(š‘”), and we see that š‘¦š‘–āŽ§āŽŖāŽØāŽŖāŽ©āˆ’īā„Ž(š‘”)=š‘–(š‘”)š‘˜š‘–(š‘”),š‘”ā‰ š‘”š‘–,š›¾š‘–,š‘”=š‘”š‘–,(3.21) where š›¾š‘– are arbitrary constants, š‘–=1,š‘š. However, the solution of system (3.5) should belong to the space š‘ˆ(0), and it means that š‘¦š‘–(š‘”)āˆˆš¶āˆž([0,š‘‡],š¶1). Therefore, constants in (3.21) š›¾š‘–īā„Ž=(š‘–(š‘”)/š‘˜š‘–(š‘”))āˆ£š‘”=š‘”š‘– and functions are determined uniquely in the form š‘¦š‘–āˆ’īā„Ž(š‘”)=š‘–(š‘”)š‘˜š‘–[](š‘”),āˆ€š‘”āˆˆ0,š‘‡,š‘–=1,š‘š.(3.22) Thus, under conditions (3.14), system (3.5) has the solution š‘¦(0)(š‘”) in š‘ˆ(0) of š‘¦(0)(š‘”)=āˆ’š‘šī“š‘–=1īā„Žš‘–(š‘”)š‘˜š‘–š‘(š‘”)š‘–(š‘”)āˆ’š‘›ī“š‘—=š‘š+1ā„Žš‘–(š‘”)šœ†š‘–š‘(š‘”)š‘–(š‘”),(3.23) where ā„Žš‘–īā„Ž(š‘”)=š‘–(š‘”)/(š‘”āˆ’š‘”š‘–)š‘ š‘– (in points š‘”=š‘”š‘–,š‘–=1,š‘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 š“(š‘”) satisfy condition (i), and let its spectrum satisfy conditions (ii)ā€“(iv). Then, for any vector function ā„Ž(0)(š‘”)āˆˆš‘‰(0), system (3.5) has the unique solution š‘¦(0)(š‘”) in space š‘ˆ(0).

For uniquely determination of functions š›¼š‘—(š‘”),š›¾š‘–š‘žš‘–(š‘”), consider system (3.4) with additional conditions:š‘¦(1)(0,0,0)=š‘¦āˆ—,īƒ”āˆ’(3.24)šœ•š‘¦(1)šœ•š‘”,šœˆš‘—īƒ¢[](š‘”,šœ,šœŽ)ā‰”0āˆ€š‘”āˆˆ0,š‘‡,š‘—=īƒ”āˆ’1,š‘›,(3.25)šœ•š‘¦(1)šœ•š‘”,šœˆš‘–š‘žš‘–īƒ¢(š‘”,šœ,šœŽ)ā‰”0,š‘–=1,š‘š,š‘žš‘–=0,š‘ š‘–āˆ’1,(3.26) where š‘¦āˆ—āˆˆš¶š‘› 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 š›¾š‘–š‘žš‘–(š‘”)š‘š‘–(š‘”)šœŽš‘–š‘žš‘–(š‘–=1,š‘š,š‘žš‘–=0,š‘ š‘–āˆ’1). Functions š›¾š‘–š‘žš‘–(š‘”) in the last summand are determined by the formula š›¾š‘–š‘žš‘–(š‘”)=š›¾0š‘–š‘žš‘–ā‹…š‘’š‘ƒš‘–š‘–š‘ž(š‘”)+š‘“š‘–š‘žš‘–(š‘”),(3.27) where š‘ƒš‘–š‘žš‘–(š‘”),š‘“š‘–š‘žš‘–(š‘”) are known functions, and š›¾0š‘–š‘žš‘–arbitrary constants.

Proof. Denote in (3.11) that š‘”š‘˜š‘—ā„Ž(š‘”)=š‘˜š‘—(š‘”)šœ†š‘—(š‘”)āˆ’šœ†š‘˜(š‘”),š‘”š‘˜š‘–š‘žš‘–ā„Ž(š‘”)=š‘˜š‘–š‘žš‘–(š‘”)šœ†š‘–(š‘”)āˆ’šœ†š‘˜.(š‘”)(3.28) Using (3.11) and condition (3.24), we obtain the equality š‘›ī“š‘›š‘˜=1ī“š‘—=1š‘”š‘˜š‘—(0)š‘š‘—(0)+š‘›ī“š‘˜=1š›¼š‘˜(0)š‘š‘˜(0)=š‘¦āˆ—.(3.29) Multiplying this equality scalarly by š‘‘š‘ (0), we get š›¼š‘ ī€·š‘¦(0)=āˆ—,š‘‘š‘ ī€øāˆ’(0)š‘›ī“š‘˜=1,š‘˜ā‰ š‘ š‘”š‘˜š‘ (0)ā‰”š›¼0š‘ ,š‘ =1,š‘›.(3.30) By (3.11) and conditions (3.25), we have āˆ’Ģ‡š›¼š‘ ī€·(š‘”)āˆ’Ģ‡š‘š‘ (š‘”),š‘‘š‘ ī€øš›¼(š‘”)š‘ (š‘”)āˆ’š‘›ī“š‘—=1,š‘—ā‰ š‘ š‘”š‘ š‘—ī€·(š‘”)Ģ‡š‘š‘—(š‘”),š‘‘š‘ ī€ø(š‘”)=0,š‘ =1,š‘›.(3.31) Considering these equations with initial conditions (3.30), we can uniquely obtain functions š›¼š‘ (š‘”),š‘ =1,š‘›.
Now, using (3.11) and conditions (3.26), we get āˆ’Ģ‡š›¾š‘–š‘žš‘–ī€·(š‘”)āˆ’Ģ‡š‘š‘–(š‘”),š‘‘š‘–ī€øš›¾(š‘”)š‘–š‘žš‘–(š‘”)āˆ’š‘›ī“š‘˜=1,š‘˜ā‰ š‘–š‘”š‘˜š‘–š‘žš‘–ī€·(š‘”)Ģ‡š‘š‘˜(š‘”),š‘‘š‘–ī€ø(š‘”)=0,š‘–=1,š‘š,š‘žš‘–=0,š‘ š‘–āˆ’1.(3.32) This implies that š›¾š‘–š‘žš‘–(š‘”) have the form (3.27) where š‘ƒš‘–š‘žš‘–(ī€œš‘”)=āˆ’š‘”š‘”š‘–ī€·Ģ‡š‘š‘–(š‘ ),š‘‘š‘–(ī€øš‘“š‘ )š‘‘š‘ ,š‘–š‘žš‘–(š‘”)=š‘’š‘ƒš‘–š‘–š‘ž(š‘”)ī€œš‘”š‘”š‘–š‘’āˆ’š‘ƒš‘–š‘–š‘žš‘›(š‘ )ī“š‘˜=1,š‘˜ā‰ š‘–š‘”š‘˜š‘–š‘žš‘–ī€·(š‘ )Ģ‡š‘š‘˜(š‘ ),š‘‘š‘–ī€ø(š‘ )š‘‘š‘ .(3.33) Theorem 3.4 is proved.

Remark 3.5. If conditions (3.6) hold for ā„Ž(1)(š‘”,šœ,šœŽ)āˆˆš‘ˆ(1)and ā„Ž(0)(š‘”)āˆˆš‘ˆ(0), then system (3.3) has a solution in the space š‘ˆ, representable in the form of š‘¦(š‘”,šœ,šœŽ)=š‘¦(1)(š‘”,šœ,šœŽ)+š‘¦(0)(š‘”),(3.34) where š‘¦(1)(š‘”,šœ,šœŽ)is a function in the form of (3.11), and š‘¦(0)(š‘”) is a function in the form of (3.23); moreover, functions š›¼š‘˜(š‘”)āˆˆš¶āˆž([0,š‘‡],š¶1) are found uniquely in (3.11), and functions š›¾š‘–š‘žš‘–(š‘”) are determined up to arbitrary constants š›¾0š‘–š‘žš‘– in the form of (3.27).

Let us give the following result.

Theorem 3.6. Let ā„Ž(0)(š‘”)āˆˆš‘ˆ(0),ā„Ž(1)(š‘”,šœ,šœŽ)āˆˆš‘ˆ(1),and conditions (i)ā€“(iv), (3.6), (3.24)ā€“(3.26) hold. Then, there exist unique numbers š›¾0š‘–š‘žš‘– involved in (3.27), such that the function (3.34) satisfies the condition š‘ƒš‘¦ā‰”āˆ’šœ•š‘¦(0)āˆ’šœ•š‘”š‘šī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!šœ•š‘¦(1)šœ•šœŽš‘–š‘žš‘–+š»(0)(š‘”)āˆˆš‘‰(0),(3.35) where š»(0)(š‘”)āˆˆš‘‰(0)is a fixed vector function.

Proof. To determine functions uniquely, calculate š‘ƒš‘¦ā‰”āˆ’š‘šī“š‘–=1ī‚øā„Žš‘–(š‘”)š‘˜š‘–š‘(š‘”)š‘–ī‚¹(š‘”)ī…žāˆ’š‘›ī“š‘—=š‘š+1ī‚øā„Žš‘—(š‘”)šœ†š‘—š‘(š‘”)š‘—ī‚¹(š‘”)ī…žāˆ’š‘šī“š‘ š‘–=1š‘–āˆ’1ī“š‘žš‘–=0ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!š›¾š‘–š‘žš‘–(š‘”)š‘š‘–+(š‘”)š‘›ī“š‘šš‘˜=1ī“š‘ š‘–=1š‘–ā‰ š‘˜š‘–āˆ’1ī“š‘žš‘–=0ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!ā‹…ā„Žš‘–š‘žš‘–(š‘”)šœ†š‘–(š‘”)āˆ’šœ†š‘˜š‘(š‘”)š‘˜(š‘”)+š»(0)ī€·(š‘”),š‘ƒš‘¦,š‘‘š‘–ī€øī‚øā„Ž(š‘”)ā‰”āˆ’š‘–(š‘”)š‘˜š‘–ī‚¹(š‘”)ī…žāˆ’š‘šī“š‘–=1ā„Žš‘–(š‘”)š‘˜š‘–ī€ŗ(š‘”)Ģ‡š‘š‘–(š‘”),š‘‘š‘–ī€»āˆ’(š‘”)š‘›ī“š‘—=š‘š+1ī‚øā„Žš‘—(š‘”)š‘˜š‘—ī‚¹ī€·(š‘”)Ģ‡š‘š‘—(š‘”),š‘‘š‘–ī€øāˆ’(š‘”)š‘ š‘–āˆ’1ī“š‘žš‘–š‘š=0ī“š‘–=1ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!š›¾š‘–š‘žš‘–ī€ŗš»(š‘”)+(0)(š‘”),š‘‘š‘–ī€»(š‘”),š‘–=1,š‘š.(3.36) Denote by š‘Ÿš‘–(š‘”)the known function š‘Ÿš‘–ī‚øā„Ž(š‘”)ā‰”āˆ’š‘–(š‘”)š‘˜š‘–ī‚¹(š‘”)ī…žāˆ’š‘šī“š‘–=1ā„Žš‘–(š‘”)š‘˜š‘–ī€ŗ(š‘”)Ģ‡š‘š‘–(š‘”),š‘‘š‘–ī€»āˆ’(š‘”)š‘›ī“š‘—=š‘š+1ī‚øā„Žš‘—(š‘”)š‘˜š‘—ī‚¹ī€·(š‘”)Ģ‡š‘š‘—(š‘”),š‘‘š‘–ī€ø+ī€·š»(š‘”)(0)(š‘”),š‘‘š‘–ī€ø,(š‘”)(3.37) and write the conditions (3.13) for (š‘ƒš‘¦,š‘‘š‘–(š‘”)).Taking into account expression (3.27) for š›¾š‘–š‘žš‘–(š‘”), we get š‘ š‘–āˆ’1ī“š‘žš‘–=0š›¾0š‘–š‘žš‘–īƒ¬š·š‘™š‘–īƒ©ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!š‘’š‘ƒš‘–š‘–š‘ž(š‘„)īƒŖīƒ­š‘”=š‘”š‘–+š‘ š‘–āˆ’1ī“š‘žš‘–=0ī€ŗš·š‘™š‘–š‘“š‘–š‘žš‘–ī€»(š‘”)š‘”=š‘”š‘–=ī€ŗš·š‘™š‘–š‘Ÿš‘–ī€»(š‘”)š‘”=š‘”š‘–,š‘–=1,š‘š,š‘™š‘–=0,š‘ š‘–āˆ’1.(3.38) Using the Leibnitz formula, we obtain that īƒ¬š·š‘™š‘–īƒ©ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!š‘’š‘ƒš‘–š‘–š‘ž(š‘”)īƒŖīƒ­š‘”=š‘”š‘–=āŽ”āŽ¢āŽ¢āŽ£š‘™š‘–ī“šœˆ=0š¶šœˆš‘™š‘–īƒ©ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!īƒŖ(šœˆ)ī€·š‘’š‘ƒš‘–š‘–š‘ž(š‘”)ī€ø(š‘™š‘–āˆ’šœˆ)āŽ¤āŽ„āŽ„āŽ¦š‘”=š‘”š‘–=āŽ”āŽ¢āŽ¢āŽ£š‘žš‘–ī“šœˆ=0š¶šœˆš‘™š‘–īƒ©ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!īƒŖ(šœˆ)ī€·š‘’š‘ƒš‘–š‘–š‘ž(š‘”)ī€ø(š‘™š‘–āˆ’šœˆ)āŽ¤āŽ„āŽ„āŽ¦š‘”=š‘”š‘–=š¶š‘žš‘–š‘™š‘–ī€·š‘’š‘ƒš‘–š‘–š‘ž(š‘”)ī€ø(š‘™š‘–āˆ’š‘žš‘–)š‘”=š‘”š‘–,(3.39) for š‘™š‘–ā‰„š‘žš‘–, īƒ¬š·š‘™š‘–īƒ©ī€·š‘”āˆ’š‘”š‘–ī€øš‘žš‘–š‘žš‘–!š‘’š‘ƒš‘–š‘–š‘ž(š‘”)īƒŖīƒ­š‘”=š‘”š‘–=0,(3.40) for 0ā‰¤š‘™š‘–ā‰¤š‘žš‘–.
Therefore, previous equalities are written in the form of š‘ š‘–āˆ’1ī“š‘žš‘–=0š›¾0š‘–š‘žš‘–š¶š‘žš‘–š‘™š‘–ī€·š‘’š‘ƒš‘–š‘–š‘ž(š‘”)ī€ø(š‘™š‘–āˆ’š‘žš‘–)š‘”=š‘”š‘–=š‘Ÿ0š‘–š‘™š‘–ī‚€š‘–=1,š‘š,š‘™š‘–=0,š‘ š‘–ī‚āˆ’1,(3.41) where š‘Ÿ0š‘–š‘™š‘–=āˆ’š‘ š‘–āˆ’1ī“š‘žš‘–=0ī€ŗš·š‘™š‘–š‘“š‘–š‘žš‘–ī€»(š‘”)š‘”=š‘”š‘–āˆ’ī€ŗš·š‘™š‘–š‘Ÿš‘–ī€»(š‘”)š‘”=š‘”š‘–,forš‘™š‘–=0,wegetš›¾0š‘–0š‘’š‘ƒš‘–š‘–š‘ž(š‘”š‘–)=š‘Ÿ0š‘–0;forš‘™š‘–=1,wegetš›¾0š‘–0š‘01ī€ŗš‘’š‘ƒš‘–š‘–š‘ž(š‘”)ī€»ī…žš‘”=š‘”š‘–+š›¾0š‘–1š‘’š‘ƒš‘–š‘–š‘ž(š‘”š‘–)=š‘Ÿ0š‘–1;ā‹®forš‘™š‘–=š‘ š‘–āˆ’1,wegetš›¾0š‘–0š‘0š‘ š‘–āˆ’1ī€ŗš‘’š‘ƒš‘–š‘–š‘ž(š‘”)ī€»š‘ š‘–āˆ’1š‘”=š‘”š‘–+ā‹Æ+š›¾0š‘–š‘ š‘–āˆ’1š‘’š‘ƒš‘–š‘–š‘ž(š‘”š‘–)=š‘Ÿ0š‘–š‘ š‘–āˆ’1.(3.42)
We obtain from here sequentially the numbers š›¾0š‘–0,ā€¦,š›¾0š‘–š‘ š‘–āˆ’1.Theorem 3.6 is proved.

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

So, if ā„Ž(0)(š‘”)āˆˆš‘ˆ(0),ā„Ž(1)(š‘”,šœ,šœŽ)āˆˆš‘ˆ(1),and conditions (3.6), (3.24)ā€“(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 š‘ˆ=š‘ˆ(1)āŠ•š‘ˆ(0).Two sequential problems (šœ€š‘˜) and (šœ€š‘˜+1) are connected uniquely by conditions (3.23)ā€“(3.25), (3.30); therefore, by virtue of Theorems 3.1ā€“3.6, they are solvable uniquely in the space š‘ˆ.

4. Asymptotical Character of Formal Solutions

Let š‘¦āˆ’1(š‘”,šœ,šœŽ),ā€¦,š‘¦š‘˜(š‘”,šœ,šœŽ) be solutions of formal problems (šœ€āˆ’1), ā€¦,(šœ€š‘˜) in the space š‘ˆ,respectively. Compose the partial sum for series (2.4):š‘†š‘›(š‘”,šœ,šœŽ)=š‘›ī“š‘˜=āˆ’1šœ€š‘˜š‘¦š‘˜(š‘”,šœ,šœŽ),(4.1) and take its restriction š‘¦šœ€š‘›(š‘”)=š‘†š‘›(š‘”,šœ‘(š‘”)/šœ€,šœ“(š‘”,šœ€)).

We have the following result.

Theorem 4.1. Let conditions (i)ā€“(v) hold. Then, for sufficiently small šœ€(0ā‰¤šœ€ā‰¤šœ€0), the estimates ā€–ā€–š‘¦(š‘”,šœ)āˆ’š‘¦šœ€š‘›ā€–ā€–(š‘”)š¶[0,š‘‡]ā‰¤š¶š‘›šœ€š‘›+1,š‘›=āˆ’1,0,1,ā€¦,(4.2) hold. Here, š‘¦(š‘”,šœ€) is the exact solution of problem (1.1), and š‘¦šœ€š‘›(š‘”) is the states above restriction of the š‘›th partial sum of series (2.4).

Proof. The restriction š‘¦šœ€š‘›(š‘”) of series (2.4) satisfies the initial condition š‘¦šœ€š‘›(0)=š‘¦0 and system (1.1) up to terms containing šœ€š‘›+1, that is, šœ€š‘‘š‘¦šœ€š‘›(š‘”)š‘‘š‘”=š“(š‘”)š‘¦šœ€š‘›(š‘”)+šœ€š‘›+1š‘…š‘›(š‘”,šœ€)+ā„Ž(š‘”),(4.3) where š‘…š‘›(š‘”,š‘ ) is a known function satisfying the estimate (ā€–š‘…š‘”,šœ€)ā€–š¶[0,š‘‡]ā‰¤š‘…š‘›,š‘…š‘›---const.(4.4) Under conditions of Theorem 4.1 on the spectrum of the operator š“(š‘”) for the fundamental matrix š‘Œ(š‘”,š‘ ,šœ€)ā‰”š‘Œ(š‘”,šœ€)š‘Œāˆ’1(š‘”,šœ€) of the system šœ€Ģ‡š‘Œ=š“(š‘”)š‘Œ, the estimate ī€ŗā€–š‘Œ(š‘”,š‘ ,šœ€)ā€–ā‰¤constāˆ€(š‘”,šœ€)āˆˆš‘„ā‰”{0ā‰¤š‘ ā‰¤š‘”ā‰¤š‘‡},āˆ€šœ€>0āˆˆ0,šœ€0ī€»,(4.5) is valid. Here, šœ€0>0āˆ’is sufficiently small. Now, write the equation šœ€š‘‘Ī”(š‘”,šœ€)š‘‘š‘”=š“(š‘”)Ī”(š‘”,šœ€)āˆ’šœ€š‘›+1š‘…š‘›(š‘”,šœ€),Ī”(0,šœ€)=0,(4.6) for the remainder term Ī”(š‘”,šœ€)ā‰”š‘¦(š‘”,šœ€)āˆ’š‘¦šœ€š‘›(š‘”).We obtain from this equation that Ī”(š‘”,šœ€)=āˆ’šœ€š‘›ī€œš‘”0š‘Œ(š‘„,š‘ ,šœ€)š‘…š‘›(š‘ ,šœ€)š‘‘š‘ ,(4.7) whence we get the estimate (ā€–Ī”š‘”,šœ€)ā€–š¶[0,š‘‡]ā‰¤āˆ’šœ€š‘›š‘…š‘›,(4.8) where š‘…š‘›=max(š‘”,š‘ )āˆˆš‘„ā€–š‘Œ(š‘”,š‘ ,šœ€)ā€–ā‹…ā€–š‘…š‘›(š‘”,š‘ )ā€–ā‹…š‘‡. So, we obtain the estimate ā€–ā€–š‘¦(š‘”,šœ€)āˆ’š‘¦šœ€š‘›(ā€–ā€–š‘”)š¶[0,š‘‡]ā‰¤šœ€š‘›š‘…š‘›,š‘›=āˆ’1,0,1,ā€¦.(4.9) Taking instead of š‘¦šœ€š‘›(š‘”) the partial sum š‘¦šœ€,š‘›+1(š‘”)ā‰”š‘¦šœ€š‘›(š‘”)+šœ€š‘›+1š‘¦š‘›+1ī‚µš‘”,šœ‘(š‘”)šœ€ī‚¶,šœ“(š‘”,šœ€),(4.10) we get ā€–ā€–ā€–ī€·š‘¦(š‘”,šœ€)āˆ’š‘¦šœ€š‘›ī€ø(š‘”)āˆ’šœ€š‘›+1š‘¦š‘›+1ī‚µš‘”,šœ‘(š‘”)šœ€ī‚¶ā€–ā€–ā€–,šœ“(š‘”,šœ€)ā‰¤šœ€š‘›+1š‘…š‘›+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 problemšœ€āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽĢ‡š‘¦Ģ‡š‘§āˆ’5š‘”2+42š‘”2āˆ’2āˆ’10š‘”2+104š‘”2āŽžāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽš‘¦š‘§āŽžāŽŸāŽŸāŽ +āŽ›āŽœāŽœāŽš‘”āˆ’52ā„Ž10āŽžāŽŸāŽŸāŽ (š‘”),š‘¦(0,šœ€)=š‘¦0,š‘§(0,šœ€)=š‘§0,(5.1) where ā„Ž1(š‘”)āˆˆš¶āˆž[0,2],šœ€>0is a small parameter. Eigenvalues of the matrix ī‚€š“(š‘”)=āˆ’5š‘”2+42š‘”2āˆ’2āˆ’10š‘”2+104š‘”2āˆ’5ī‚are šœ†1(š‘”)=āˆ’š‘”2,šœ†2(š‘”)=āˆ’1.Eigenvectors of matrices š“(š‘”)and š“āˆ—(š‘”), are, respectively,šœ‘1=āŽ›āŽœāŽœāŽ12āŽžāŽŸāŽŸāŽ ,šœ‘2=āŽ›āŽœāŽœāŽ25āŽžāŽŸāŽŸāŽ ,šœ“1=āŽ›āŽœāŽœāŽ5āŽžāŽŸāŽŸāŽ āˆ’2,šœ“2=āŽ›āŽœāŽœāŽ1āŽžāŽŸāŽŸāŽ āˆ’2.(5.2) We get (ā„Ž(š‘”),šœ“1(š‘”))ā‰”5š‘”2ā„Ž1(š‘”). Therefore,ī€·ā„Ž(0),šœ“1ī€øš‘‘(0)=0,ī€·š‘‘š‘”ā„Ž(0),šœ“1ī€ø(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 š¾š‘—š‘–(š‘”).Since šœ“(š‘”)ā‰”šœ†1(š‘”)=āˆ’š‘”2,there will be two such polynomials: š¾00(š‘”)and š¾01(š‘”).

Take the arbitrary numbers š‘Ž00(š‘”)and š‘Ž01(š‘”),and set the interpolation conditions for the polynomial š‘Ÿ(š‘”),š‘Ÿ(š‘”)=š‘Ž00,Ģ‡š‘Ÿ(1)=š‘Ž01.(5.4) Expand š‘Ÿ(š‘”)onto partial fractionsš‘Ÿ(š‘”)=š“šœ“(š‘”)š‘”2+šµš‘”,(5.5) from whereš‘Ÿ(š‘”)ā‰”š“+šµš‘”.(5.6) Use the interpolation polynomial (5.4). We get š“=š‘Ž00,šµ=š‘Ž01. Hence, (5.6) takes the formš‘Ÿ(š‘”)ā‰”š‘Ž00+š‘”š‘Ž01.(5.7) Since numbers š‘Ž00and š‘Ž01 are arbitrary, basic Lagrange-Silvestre polynomials will be coefficients standing before them, that is,š¾00(š‘”)ā‰”1,š¾01(š‘”)ā‰”š‘”.(5.8)

Introduce the regularizing variablesšœŽ00=š‘’āˆ«(1/šœ€)š‘”0šœ†1š‘‘š‘ ī€œš‘”0š‘’āˆ«āˆ’(1/šœ€)š‘ 0šœ†1š‘‘š‘„ā‹…š¾00(š‘ )š‘‘š‘ =š‘’āˆ’š‘”3/3šœ€ī€œš‘”0š‘’š‘ 3/3šœ€š‘‘š‘ ā‰”š‘00(šœŽš‘”),01=š‘’āˆ«(1/šœ€)š‘”0šœ†1š‘‘š‘ ī€œš‘”0š‘’āˆ«āˆ’(1/šœ€)š‘ 0šœ†1š‘‘š‘„ā‹…š¾01(š‘ )š‘‘š‘ =š‘’āˆ’š‘”3/3šœ€ī€œš‘”0š‘’š‘ 3/3šœ€ā‹…š‘ š‘‘š‘ ā‰”š‘01šœ(š‘”),1=1šœ€ī€œš‘”0šœ†1š‘”š‘‘š‘ =āˆ’33šœ€ā‰”š‘ž1(š‘”),šœ2=1šœ€ī€œš‘”0šœ†2š‘”š‘‘š‘ =āˆ’šœ€ā‰”š‘ž2(š‘”).(5.9)

Construct the extended problem corresponding to problem (5.1):šœ€šœ•š‘¤šœ•š‘”+šœ†1(š‘”)šœ•š‘¤šœ•šœ1+šœ†2(š‘”)šœ•š‘¤šœ•šœ2+šœ†1(š‘”)šœŽ00šœ•š‘¤šœ•šœŽ00+šœ†1(š‘”)šœŽ01šœ•š‘¤šœ•šœŽ01+šœ€šœ•š‘¤šœ•šœŽ00+šœ€š‘”šœ•š‘¤šœ•šœŽ01āˆ’š“(š‘”)š‘¤=ā„Ž(š‘”),š‘¤(0,0,0,šœ€)=š‘¤0,(5.10) where šœā‰”(šœ1,šœ2),šœŽ=(šœŽ00,šœŽ01),š‘¤=š‘¤(š‘”,šœ,šœŽ,šœ€).

Determining solutions of problem (5.10) in the form of a seriesš‘¤(š‘”,šœ,šœŽ,šœ€)=āˆžī“š‘˜=0šœ€š‘˜š‘¤š‘˜(š‘”,šœ,šœŽ),(5.11) we obtain the following iteration problems: šæš‘¤0ā‰”šœ†1ī‚ø(š‘”)šœ•š‘¤0+šœ•šœšœ•š‘¤0šœ•šœŽ00+š‘”ā‹…šœŽ01šœ•š‘¤0šœ•šœŽ01ī‚¹+šœ†2(š‘”)šœ•š‘¤0šœ•šœ2āˆ’š“(š‘”)š‘¤0=ā„Ž(š‘”),š‘¤0(0,0,0)=š‘¤0,(5.12)šæš‘¤1=āˆ’šœ•š‘¤0āˆ’šœ•š‘”šœ•š‘¤0šœ•šœŽ00āˆ’š‘”šœ•š‘¤0šœ•šœŽ01,š‘¤1ā‹®(0,0,0)=0,(5.13)

We determine solutions of iteration problems (5.12), (5.13), and so on in the space š‘ˆ of functions in the form ofš‘¤(š‘”,šœ,šœŽ)=š‘¤1(š‘”)š‘’šœ1+š‘¤2(š‘”)š‘’šœ2+š‘¤00(š‘”)šœŽ00+š‘¤01(š‘”)šœŽ01+š‘¤0š‘¤(š‘”),0(š‘”),š‘¤1(š‘”),š‘¤2(š‘”),š‘¤00(š‘”),š‘¤01(š‘”)āˆˆš¶āˆžī€·[]0,2,š¶2ī€ø.(5.14)

Directly calculating, we obtain the solution of system (5.12) in the form ofš‘¤0(š‘”,šœ,šœŽ)=š›¼1(š‘”)šœ‘1š‘’šœ1+š›¼2(š‘”)šœ‘2š‘’šœ2+š›¾00(š‘”)šœ‘1šœŽ00+š›¾01(š‘”)šœ‘1šœŽ01+5ā„Ž1(š‘”)šœ‘1āˆ’2š‘”2ā„Ž1(š‘”)šœ‘2,(5.15) where š›¼š‘—(š‘”),š›¾š‘—š‘–(š‘”)āˆˆš¶āˆž[0,2]are for now arbitrary functions.

To calculate the functions š›¼š‘—(š‘”)and š›¾š‘–š‘—(š‘”),we pass to the following iteration problem (5.13). Taking into account (5.15), it will be written in the form ofšæš‘¤1=āˆ’Ģ‡š›¼1(š‘”)šœ‘1š‘’šœ1āˆ’Ģ‡š›¼2(š‘”)šœ‘2š‘’šœ2āˆ’Ģ‡š›¾00(š‘”)šœ‘1šœŽ00āˆ’Ģ‡š›¾01(š‘”)šœ‘1šœŽ01Ģ‡ā„Žāˆ’51(š‘”)šœ‘1āˆ’ī€·2š‘”2ā„Ž1ī€ø(š‘”)ī…žšœ‘2āˆ’š›¾00(š‘”)šœ‘1āˆ’š‘”š›¾01(š‘”)šœ‘1.(5.16)

For solvability of problem (5.13) in the space š‘ˆ, it is necessary and sufficient to fulfill the conditionsāˆ’Ģ‡š›¼1(š‘”)=0,āˆ’Ģ‡š›¼2(š‘”)=0,āˆ’Ģ‡š›¾00(š‘”)=0,āˆ’Ģ‡š›¾01Ģ‡ā„Ž(š‘”)=0,āˆ’51(0)āˆ’š›¾00Ģˆā„Ž(0)=0,āˆ’51(0)āˆ’Ģ‡š›¾00(0)āˆ’š›¾01(0)=0.(5.17)

Using solution (5.15) and the initial condition š‘¤0(0,0,0)=š‘¤0,we obtain the equationš›¼1(0)šœ‘1+š›¼2(0)šœ‘2+5ā„Ž1(0)šœ‘1=š‘¤0.(5.18)

Multiplying it (scalar) on šœ“1and šœ“2, we obtain the valuesš›¼1ī€·š‘¤(0)=0,šœ“1ī€øāˆ’5ā„Ž1(0)ā‰”5š‘¦0āˆ’2š‘§0āˆ’5ā„Ž1š›¼(0),2ī€·š‘¤(0)=0,šœ“2ī€ø=š‘§0āˆ’2š‘¦0.(5.19)

Using equalities (5.17), and also the initial data (5.19), we obtain uniquely the functions š›¼š‘—(š‘”)and š›¾š‘—š‘–(š‘”):š›¼1(š‘”)=5š‘¦0āˆ’2š‘§0āˆ’5ā„Ž1(0),š›¼2(š‘”)=š‘§0āˆ’2š‘¦0.š›¾00Ģ‡ā„Ž(š‘”)=āˆ’51(0),š›¾01Ģˆā„Ž(š‘”)=āˆ’51(0).(5.20)

Substituting these functions into (5.15), we obtain uniquely the solution of problem (5.12) in the space š‘ˆ,š‘¤0ī€·(š‘”,šœ,šœŽ)=5š‘¦0āˆ’2š‘§0āˆ’5ā„Ž1ī€øšœ‘(0)1š‘’šœ1+ī€·š‘§0āˆ’2š‘¦0ī€øšœ‘2š‘’šœ2Ģ‡ā„Žāˆ’51(0)šœ‘1šœŽ00Ģˆā„Žāˆ’51(0)šœ‘1šœŽ01+5ā„Ž1(š‘”)šœ‘1āˆ’2š‘”2ā„Ž1(š‘”)šœ‘2.(5.21)

Producing here restriction on the functions šœ=š‘ž(š‘”),šœŽ=š‘(š‘”),we obtain the principal term of the asymptotics for the solution of problem (5.1):š‘¤0šœ€ī€·(š‘”)=5š‘¦0āˆ’2š‘§0āˆ’5ā„Ž1ī€øšœ‘(0)1š‘’āˆ’š‘”3/3šœ€+ī€·š‘§0āˆ’2š‘¦0ī€øšœ‘2š‘’āˆ’š‘”/šœ€Ģ‡ā„Žāˆ’51(0)šœ‘1š‘’āˆ’š‘”3/3šœ€ī€œš‘”0š‘’š‘ 3/3šœ€Ģˆā„Žš‘‘š‘ āˆ’51(0)šœ‘1š‘’āˆ’š‘”3/3šœ€ī€œš‘”0š‘’š‘ 3/3šœ€š‘ š‘‘š‘ +5ā„Ž1(š‘”)šœ‘1āˆ’2š‘”2ā„Ž1(š‘”)šœ‘2.(5.22)

The zero-order asymptotical solution is obtained: it satisfies the estimateā€–ā€–š‘¤(š‘”,šœ€)āˆ’š‘¤0šœ€ā€–ā€–(š‘”)š¶[0,2]ā‰¤š¶1ā‹…šœ€,(5.23) where š‘¤(š‘”,šœ€)is an exact solution of problem (1.1), and š¶1>0is a constant independent of šœ€at sufficiently small šœ€(0<šœ€ā‰¤šœ€0).