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

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., [1–19]).

In this work, the algorithm for construction of an asymptotical solution for linear singular perturbed systems with the irreversible limit operator is given—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 form𝜀𝑦̇=𝐴(𝑡)𝑦+ℎ(𝑡),𝑦(0,𝜀)=𝑦⁰,𝑡∈[0,𝑇],(1.1) where 𝑦={𝑦₁,…,𝑦_𝑛},𝐴(𝑡) is a matrix of order (𝑛×𝑛),ℎ(𝑡)={ℎ₁,…,ℎ_𝑛} is a known function, 𝑦⁰∈𝐶^𝑛 is a constant vector, and 𝜀>0 is a small parameter, in the case of violation of stability of a spectrum {𝜆_𝑗(𝑡)} of the limiting operator 𝐴(𝑡).

Difference of such type problems from similar problems with a stable spectrum (i.e., in the case of 𝜆_𝑖(𝑡)≠0,𝜆_𝑖(𝑡)≠𝜆_𝑗(𝑡),𝑖≠𝑗,𝑖,𝑗=1,𝑛forall𝑡∈[0,𝑇]) is that the limiting system 0=𝐴(𝑡)𝑦+ℎ(𝑡)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,𝑇] solutions 𝑦(𝑡)of the limiting system is not excluded. Under conditions, one can prove (see, e.g., [1, 6]) that the exact solution 𝑦(𝑡,𝜀) of problem (1.1) tends (at 𝜀→+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 {𝜆_𝑗(𝑡)} of the limiting operator 𝐴(𝑡). As it was shown in [3–7], they were induced by instability points 𝑡_𝑗 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)𝐴(𝑡)∈𝐶^∞([0,𝑇],𝐶^𝑛),ℎ(𝑡)∈𝐶^∞[0,𝑇]; for any 𝑡∈[0,𝑇], the spectrum {𝜆_𝑗(𝑡)} of the operator 𝐴(𝑡) satisfies the conditions:(ii)𝜆_𝑖(𝑡)=−(𝑡−𝑡_𝑖)^𝑠𝑖𝑘_𝑖(𝑡),𝑘_𝑖(𝑡)≠0,𝑡_𝑖∈[0,𝑇],𝑖=1,𝑚,𝑚<𝑛 (here 𝑠_𝑖- are even natural numbers),(iii)𝜆_𝑖(𝑡)≠0,𝑗=𝑚+1,𝑛,(iv)𝜆_𝑖(𝑡)≠𝜆_𝑗(𝑡),𝑖≠𝑗,𝑖,𝑗=1,𝑛,(v)Re𝜆_𝑗(𝑡)⩽0,𝑗=1,𝑛.

2. Regularization of the Problem

We introduce basic regularized variables by the spectrum of the limiting operator𝜏_𝑗=𝜀⁻¹^𝑡0𝜆_𝑗(𝑠)𝑑𝑠≡𝜑^𝑗(𝑡)𝜀,𝑗=1,𝑛.(2.1) Instable points 𝑡_𝑖∈[0,𝑇] of the spectrum {𝜆_𝑗(𝑡)} induce additional regularized variables described by the formulas𝜎_𝑖𝑞𝑖=𝑒^{𝜑𝑖(𝑡)/𝜀}^𝑡0𝑒^{−𝜑𝑖(𝑠)/𝜀}𝑠−𝑡_𝑖^𝑞𝑖𝑞_𝑖𝑑𝑠≡𝜓_𝑖𝑞𝑖(𝑡,𝜀),𝑖=1,𝑚,𝑞_𝑖=0,𝑠_𝑖−1.(2.2) We consider a vector function 𝑦(̃𝑡,𝜏,𝜎,𝜀) instead of the solution 𝑦(𝑡,𝜀) to be found for problem (1.1). This vector function is such that𝑦̃(𝑡,𝜏,𝜎,𝜀)∣_{𝜏=𝜑,𝜎=𝜓}≡𝑦(𝑡,𝜀).(2.3)

For 𝑦(̃𝑡,𝜏,𝜎,𝜀), it is natural to set the following problem:𝐿_𝜀𝑦̃(𝑡,𝜏,𝜎,𝜀)≡𝜀𝜕𝑦̃𝜕𝑡+^𝑛_𝑗=1𝜆_𝑗(𝑡)𝜕𝑦̃𝜕𝜏_𝑗+^𝑚_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝜆^𝑖(𝑡)𝜎_𝑖𝑞𝑖+𝜀𝑡−𝑡^𝑖^𝑞𝑖𝑞_𝑖!𝜕𝜕𝜎𝑦̃_𝑖𝑞𝑖−𝐴(𝑡)𝑦̃=ℎ(𝑡),𝑦̃(0,0,0,𝜀)=𝑦⁰.(2.4) We determine the solution of problem (2.4) in the form of a series𝑦̃(𝑡,𝜏,𝜎,𝜀)=^∞_𝑘₌₋₁𝜀^𝑘𝑦_𝑘(𝑡,𝜏,𝜎),(2.5) with coefficients 𝑦_𝑘(𝑡,𝜏,𝜎)∈𝐶^∞[0,𝑇].

If we substitute (2.5) in (2.4) and equate coefficients at identical degrees of 𝜀, we obtain the systems for coefficients 𝑦_𝑘(𝑡,𝜏,𝜎):𝐿𝑦₋₁(𝑡,𝜏,𝜎)≡^𝑛_𝑗=1𝜆_𝑗(𝑡)𝜕𝑦⁻¹𝜕𝜏_𝑗+^𝑚_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝜆_𝑖(𝑡)𝜎_𝑖𝑞𝑖𝜕𝑦₋₁𝜕𝜎_𝑖𝑞𝑖𝐿𝑦₀−𝐴(𝑡)𝑦₋₁=0,𝑦₋₁(0,0,0)=0,(𝜀⁻¹)(𝑡,𝜏,𝜎)=−𝜕𝑦⁻¹𝜕𝑡−^𝑚∑∑_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝑡−𝑡_𝑖^𝑞𝑖𝑞_𝑖!𝜕𝜕𝑦𝜎⁻¹_𝑖𝑞𝑖⋮+ℎ(𝑡),𝑦₀(0,0,0)=𝑦⁰,(𝜀₀𝐿𝑦_𝑘+1)(𝑡,𝜏,𝜎)=−𝜕𝑦^𝑘𝜕𝑡−^𝑚∑∑_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝑡−𝑡_𝑖^𝑞𝑖𝑞_𝑖!𝜕𝜕𝜎𝑦^𝑘_𝑖𝑞𝑖⋮,𝑘⩾1,𝑦_𝑘+1(0,0,0)=0,(𝜀_𝑘+1)

3. Resolvability of Iterative Problems

We solve each of the iterative problems (𝜀^𝑘) in the following space of functions:𝑈=⎧⎪⎨⎪⎩𝑦(𝑡,𝜏,𝜎)∶𝑦=^𝑛_𝑛𝑘=1_𝑗=1𝑦_𝑘𝑗(𝑡)𝑐_𝑘(𝑡)𝑒^𝜏𝑗+^𝑛_𝑚𝑘=1_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝑦_{𝑘𝑖𝑞𝑖}(𝑡)𝑐_𝑘(𝑡)𝜎_𝑖𝑞𝑖+^𝑛_𝑘=1𝑦_𝑘(𝑡)𝑐_𝑘(𝑡),𝑦_𝑘𝑗(𝑡),𝑦_{𝑘𝑖𝑞𝑖}(𝑡),𝑦_𝑘(𝑡)∈𝐶^∞[0,𝑇],𝐶¹,(3.1) where 𝑐_𝑘(𝑡) are eigenvectors of the operator 𝐴(𝑡) corresponding eigenvalues 𝜆_𝑘(𝑡),𝑘=1,𝑛. We represent 𝑈 in the form of 𝑈⁽¹⁾⊕𝑈⁽⁰⁾ where𝑈⁽⁰⁾=𝑦⁽⁰⁾(𝑡)∶𝑦⁽⁰⁾=^𝑛_𝑗=1𝑦𝑈^𝑗(0)(₍₁₎𝑡)𝑐_𝑗(𝑡),𝑦^𝑗(0)(𝑡)∈𝐶^∞[0,𝑇],𝐶¹,=𝑈𝑈⁽⁰⁾.(3.2) It is easy to note that each of the systems (𝜀^𝑘+1) can be written in the form𝐿𝑦(𝑡,𝜏,𝜎)=ℎ(𝑡,𝜏,𝜎),(3.3) where ℎ(𝑡,𝜏,𝜎) are the corresponding right hand side. Using representations of space 𝑈, we can write system (3.3) in the equivalent form𝐿−𝑦𝐴⁽¹⁾((𝑡𝑡),𝑦𝜏,𝜎₍₀₎)=ℎ⁽¹⁾(𝑡,𝜏,𝜎),(3.4)(𝑡)=ℎ⁽⁰⁾(𝑡),(3.5) where 𝑦⁽¹⁾(𝑡,𝜏,𝜎),ℎ⁽¹⁾(𝑡,𝜏,𝜎)∈𝑈⁽¹⁾,𝑦⁽⁰⁾(𝑡),ℎ⁽⁰⁾(𝑡)∈𝑈⁽⁰⁾.

We have the following result.

Theorem 3.1. Let ℎ⁽¹⁾(𝑡,𝜏,𝜎)∈𝑈⁽¹⁾ and satisfy conditions (i)–(iv). Then, system (3.4) is solvable in the 𝑈⁽¹⁾ if and only if ℎℎ⁽¹⁾(𝑡,𝜏,𝜎),𝜈_𝑗(𝑡,𝜏,𝜎)≡0∀𝑡∈[0,𝑇],𝑗=1,𝑛,₍₁₎(𝑡,𝜏,𝜎),𝜈_𝑖𝑞𝑖(𝑡,𝜏,𝜎)≡0,𝑖=1,𝑚,𝑞_𝑖=0,𝑠_𝑖−1,(3.6) where 𝜈_𝑗(𝑡,𝜏,𝜎),𝜈_𝑖𝑞𝑖(𝑡,𝜏,𝜎) are basic elements of the kernel of the operator 𝐿^∗≡^𝑛_𝑗=1𝜆_𝑗(𝑡)𝜕𝜕𝜏_𝑗+^𝑚_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝜆_𝑖(𝑡)𝜎_𝑖𝑞𝑖𝜕𝜕𝜎_𝑖𝑞𝑖−𝐴^∗(𝑡).(3.7)

Proof. Let ℎ⁽¹⁾(𝑡,𝜏,𝜎)=∑^𝑛𝑘=1∑^𝑛𝑗=1ℎ_𝑘𝑗(𝑡)𝑐_𝑗(𝑡)𝑒^𝜏𝑘+∑^𝑛𝑘=1∑^𝑚𝑖=1∑^{𝑠𝑖𝑞−1}_𝑖=0ℎ_{𝑘𝑖𝑞𝑖}(𝑡)𝑐_𝑘(𝑡)𝜎_𝑖𝑞𝑖.
Determine solutions of system (3.4) in the form 𝑦⁽¹⁾(𝑡,𝜏,𝜎)=^𝑛_𝑛𝑘=1_𝑗=1𝑦_𝑘𝑗(𝑡)𝑐_𝑘(𝑡)𝑒^𝜏𝑗+^𝑛_𝑚𝑘=1_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝑦_{𝑘𝑖𝑞𝑖}(𝑡)𝑐_𝑘(𝑡)𝜎_𝑖𝑞𝑖.(3.8) Substituting (3.8) in (3.4) and equating separately coefficients at 𝑒^𝜏𝑗 and 𝜎_𝑖𝑞𝑖, we obtain the equations 𝜆𝜆_𝑘(𝑡)−𝜆_𝑗(𝑡)𝑦_𝑘𝑗(𝑡)=ℎ_𝑘𝑗(𝑡),𝑘,𝑗=1,𝑛,_𝑖(𝑡)−𝜆_𝑘(𝑡)𝑦_{𝑖𝑞𝑖𝑘}(𝑡)=ℎ_{𝑘𝑖𝑞𝑖}(𝑡),𝑖=1,𝑚,𝑞_𝑖=0,𝑠_𝑖−1,𝑘=1,𝑛.(3.9)
One can see from this that obtained equations are solvable if and only if ℎ_𝑘𝑘(𝑡)≡0,𝑘=1,𝑛,ℎ_{𝑖𝑖𝑞𝑖}(𝑡)≡0,𝑖=1,𝑚,𝑞_𝑖=0,𝑠_𝑖−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 𝑈⁽¹⁾ representable in the form 𝑦⁽¹⁾(𝑡,𝜏,𝜎)=^𝑛_𝑛𝑘=1_{𝑗=1,𝑗≠𝑘}ℎ_𝑘𝑗(𝑡)𝜆_𝑘(𝑡)−𝜆_𝑗(𝑡)𝑐^𝑗(𝑡)𝑒^𝜏𝑗+^𝑛_𝑘=1𝛼_𝑘(𝑡)𝑐_𝑘(𝑡)𝑒^𝜏𝑘+^𝑚_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝛾_𝑖𝑞𝑖(𝑡)𝑐_𝑖(𝑡)𝜎_𝑖𝑞𝑖+^𝑛_𝑚𝑘=1_{𝑠𝑖=1,𝑖≠𝑘}^𝑖−1_𝑞𝑖=0ℎ_{𝑘𝑖𝑞𝑖}(𝑡)𝜆_𝑖(𝑡)−𝜆_𝑘(𝑡)𝑐^𝑘(𝑡)𝜎_𝑖𝑞𝑖,(3.11) where 𝛼_𝑘(𝑡),𝛾_𝑖𝑞𝑖(𝑡)∈𝐶^∞([0,𝑇],𝐶¹) are arbitrary functions.
Consider now system (3.5). As det𝐴(𝑡)≡0 in points 𝑡=𝑡_𝑖, 𝑖=1,𝑚, this system does not always have a solution in 𝑈⁽⁰⁾. Introduce the space 𝑉⁽⁰⁾⊂𝑈⁽⁰⁾ consisting of vector functions 𝑧⁽⁰⁾(𝑡)=^𝑛_𝑗=1𝑧_𝑗(𝑡)𝑐_𝑗(𝑡),𝑧_𝑗(𝑡)∈𝐶^∞[0,𝑇],𝐶¹,𝑗=1,𝑛,(3.12) having the properties 𝐷^𝑙𝑖𝑧⁽⁰⁾(𝑡),𝑑_𝑖(𝑡)_{𝑡=𝑡𝑖}=𝐷^𝑙𝑖𝑧_𝑖𝑡_𝑖=0,∀𝑙_𝑖=0,𝑠_𝑖−1,𝑖=1,𝑚,(3.13) where 𝑑_𝑖(𝑡) are eigenvectors of the operator 𝐴^∗(𝑡) with regard to eigenvalues 𝜆_𝑖(𝑡),𝑖=1,𝑚. Let ℎ⁽⁰⁾(𝑡)=∑^𝑛𝑗=1ℎ_𝑗(𝑡)𝑐_𝑗(𝑡)∈𝑉⁽⁰⁾, that is, 𝐷^𝑙𝑖ℎ_𝑖𝑡_𝑖=0∀𝑙_𝑖=0,𝑠_𝑖−1,𝑖=1,𝑚.(3.14) Determine a solution of system (3.5) in the 𝑦⁽⁰⁾(𝑡)=^𝑛_𝑗=1𝑦_𝑗(𝑡)𝑐_𝑗(𝑡).(3.15) Substituting this function in (3.5), we obtain −^𝑛_𝑗=1𝑦_𝑗(𝑡)𝜆_𝑗(𝑡)𝑐_𝑗(𝑡)=^𝑛_𝑗=1ℎ_𝑗(𝑡)𝑐_𝑗(𝑡).(3.16) Since {𝑐_𝑗(𝑡)} is a basis in 𝐶^𝑛, we get −𝜆_𝑖(𝑡)𝑦_𝑖(𝑡)=ℎ_𝑖(𝑡),𝑖=1,𝑚,(3.17)−𝜆_𝑗(𝑡)𝑦_𝑗(𝑡)=ℎ_𝑗(𝑡),𝑗=𝑚+1,𝑛.(3.18) It is easy to see that (3.18) has the unique solution 𝑦_𝑗(𝑡)=−ℎ^𝑗(𝑡)𝜆_𝑗(𝑡),𝑗=𝑚+1,𝑛.(3.19) By virtue of conditions (3.14), the function ℎ_𝑖(𝑡) can be represented in the form ℎ_𝑖(𝑡)=𝑡−𝑡_𝑖^𝑠𝑖ℎ_𝑖(𝑡),𝑖=1,𝑚,(3.20) where ℎ_𝑖(𝑡)∈𝐶^∞([0,𝑇],𝐶¹ is the certain scalar function, −(𝑡−𝑡_𝑖)^𝑠𝑖𝑘_𝑖(𝑡)𝑦_𝑖(𝑡)=(𝑡−𝑡_𝑖)^𝑠𝑖ℎ_𝑖(𝑡), and we see that 𝑦_𝑖(𝑡)=⎧⎪⎨⎪⎩−ℎ^𝑖(𝑡)𝑘_𝑖(𝑡),𝑡≠𝑡^𝑖𝛾_𝑖,,𝑡=𝑡_𝑖,(3.21) where 𝛾_𝑖 are arbitrary constants, 𝑖=1,𝑚. However, the solution of system (3.5) should belong to the space 𝑈⁽⁰⁾, and it means that 𝑦_𝑖(𝑡)∈𝐶^∞([0,𝑇],𝐶¹). Therefore, constants in (3.21) 𝛾_𝑖=(ℎ_𝑖(𝑡)/𝑘_𝑖(𝑡))∣_{𝑡=𝑡𝑖} and functions are determined uniquely in the form 𝑦_𝑖(𝑡)=−ℎ^𝑖(𝑡)𝑘_𝑖(𝑡),∀𝑡∈[0,𝑇],𝑖=1,𝑚.(3.22) Thus, under conditions (3.14), system (3.5) has the solution 𝑦⁽⁰⁾(𝑡) in 𝑈⁽⁰⁾ of 𝑦⁽⁰⁾(𝑡)=−^𝑚_𝑖=1ℎ_𝑖(𝑡)𝑘_𝑖(𝑡)𝑐^𝑖(𝑡)−^𝑛_𝑗=_𝑚+1ℎ_𝑖(𝑡)𝜆_𝑖(𝑡)𝑐^𝑖(𝑡),(3.23) where ℎ_𝑖(𝑡)=ℎ_𝑖(𝑡)/(𝑡−𝑡_𝑖)^𝑠𝑖 (in points 𝑡=𝑡_𝑖,𝑖=1,𝑚, 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 𝐴(𝑡) satisfy condition (i), and let its spectrum satisfy conditions (ii)–(iv). Then, for any vector function ℎ⁽⁰⁾(𝑡)∈𝑉⁽⁰⁾, system (3.5) has the unique solution 𝑦⁽⁰⁾(𝑡) in space 𝑈⁽⁰⁾.

For uniquely determination of functions 𝛼_𝑗(𝑡),𝛾_𝑖𝑞𝑖(𝑡), consider system (3.4) with additional conditions:−𝜕𝑦₍₁₎𝑦⁽¹⁾(0,0,0)=𝑦_∗,(3.24)𝜕𝑡,𝜈^𝑗−𝜕𝑦₍₁₎(𝑡,𝜏,𝜎)≡0∀𝑡∈[0,𝑇],𝑗=1,𝑛,(3.25)𝜕𝑡,𝜈^𝑖𝑞𝑖(𝑡,𝜏,𝜎)≡0,𝑖=1,𝑚,𝑞_𝑖=0,𝑠_𝑖−1,(3.26) where 𝑦_∗∈𝐶^𝑛 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 𝛾_𝑖𝑞𝑖(𝑡)𝑐_𝑖(𝑡)𝜎_𝑖𝑞𝑖(𝑖=1,𝑚,𝑞_𝑖=0,𝑠_𝑖−1). Functions 𝛾_𝑖𝑞𝑖(𝑡) in the last summand are determined by the formula 𝛾_𝑖𝑞𝑖(𝑡)=𝛾^0𝑖𝑞_𝑖⋅𝑒^{𝑃𝑖𝑖𝑞(𝑡)}+𝑓_𝑖𝑞𝑖(𝑡),(3.27) where 𝑃_𝑖𝑞𝑖(𝑡),𝑓_𝑖𝑞𝑖(𝑡) are known functions, and 𝛾^0𝑖𝑞_𝑖arbitrary constants.

Proof. Denote in (3.11) that 𝑔_𝑘𝑗(𝑡)=ℎ^𝑘𝑗(𝑡)𝜆_𝑗(𝑡)−𝜆_𝑘(𝑡),𝑔^{𝑘𝑖𝑞𝑖}(𝑡)=ℎ^{𝑘𝑖𝑞𝑖}(𝑡)𝜆_𝑖(𝑡)−𝜆_𝑘(𝑡).(3.28) Using (3.11) and condition (3.24), we obtain the equality ^𝑛_𝑛𝑘=1_𝑗=1𝑔_𝑘𝑗(0)𝑐_𝑗(0)+^𝑛_𝑘=1𝛼_𝑘(0)𝑐_𝑘(0)=𝑦_∗.(3.29) Multiplying this equality scalarly by 𝑑_𝑠(0), we get 𝛼_𝑠(0)=𝑦_∗,𝑑_𝑠(0)−^𝑛_𝑘=_{1,𝑘≠𝑠}𝑔_𝑘𝑠(0)≡𝛼^0𝑠,𝑠=1,𝑛.(3.30) By (3.11) and conditions (3.25), we have −𝛼̇_𝑠(𝑡)−𝑐_𝑠̇(𝑡),𝑑_𝑠(𝑡)𝛼_𝑠(𝑡)−^𝑛_𝑗=_{1,𝑗≠𝑠}𝑔_𝑠𝑗(𝑡)𝑐_𝑗̇(𝑡),𝑑_𝑠(𝑡)=0,𝑠=1,𝑛.(3.31) Considering these equations with initial conditions (3.30), we can uniquely obtain functions 𝛼_𝑠(𝑡),𝑠=1,𝑛.
Now, using (3.11) and conditions (3.26), we get −𝛾̇_𝑖𝑞𝑖(𝑡)−𝑐_𝑖̇(𝑡),𝑑_𝑖(𝑡)𝛾_𝑖𝑞𝑖(𝑡)−^𝑛_𝑘=_{1,𝑘≠𝑖}𝑔_{𝑘𝑖𝑞𝑖}(𝑡)𝑐_𝑘̇(𝑡),𝑑_𝑖(𝑡)=0,𝑖=1,𝑚,𝑞_𝑖=0,𝑠_𝑖−1.(3.32) This implies that 𝛾_𝑖𝑞𝑖(𝑡) have the form (3.27) where 𝑓𝑃_𝑖𝑞𝑖(𝑡)=−^𝑡𝑡_𝑖𝑐_𝑖̇(𝑠),𝑑_𝑖(𝑠)𝑑𝑠,_𝑖𝑞𝑖(𝑡)=𝑒^{𝑃𝑖𝑖𝑞(𝑡)}^𝑡𝑡_𝑖𝑒^{−𝑃𝑖𝑖𝑞𝑛(𝑠)}_𝑘=_{1,𝑘≠𝑖}𝑔_{𝑘𝑖𝑞𝑖}(𝑠)𝑐_𝑘̇(𝑠),𝑑_𝑖(𝑠)𝑑𝑠.(3.33) Theorem 3.4 is proved.

Remark 3.5. If conditions (3.6) hold for ℎ⁽¹⁾(𝑡,𝜏,𝜎)∈𝑈⁽¹⁾and ℎ⁽⁰⁾(𝑡)∈𝑈⁽⁰⁾, then system (3.3) has a solution in the space 𝑈, representable in the form of 𝑦(𝑡,𝜏,𝜎)=𝑦⁽¹⁾(𝑡,𝜏,𝜎)+𝑦⁽⁰⁾(𝑡),(3.34) where 𝑦⁽¹⁾(𝑡,𝜏,𝜎)is a function in the form of (3.11), and 𝑦⁽⁰⁾(𝑡) is a function in the form of (3.23); moreover, functions 𝛼_𝑘(𝑡)∈𝐶^∞([0,𝑇],𝐶¹) are found uniquely in (3.11), and functions 𝛾_𝑖𝑞𝑖(𝑡) are determined up to arbitrary constants 𝛾^0𝑖𝑞_𝑖 in the form of (3.27).

Let us give the following result.

Theorem 3.6. Let ℎ⁽⁰⁾(𝑡)∈𝑈⁽⁰⁾,ℎ⁽¹⁾(𝑡,𝜏,𝜎)∈𝑈⁽¹⁾,and conditions (i)–(iv), (3.6), (3.24)–(3.26) hold. Then, there exist unique numbers 𝛾^0𝑖𝑞_𝑖 involved in (3.27), such that the function (3.34) satisfies the condition 𝑃𝑦≡−𝜕𝑦⁽⁰⁾𝜕𝑡−^𝑚_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝑡−𝑡_𝑖^𝑞𝑖𝑞_𝑖!𝜕𝜕𝑦𝜎⁽¹⁾_𝑖𝑞𝑖+𝐻⁽⁰⁾(𝑡)∈𝑉⁽⁰⁾,(3.35) where 𝐻⁽⁰⁾(𝑡)∈𝑉⁽⁰⁾is a fixed vector function.

Proof. To determine functions uniquely, calculate 𝑃𝑦≡−^𝑚_𝑖=1ℎ^𝑖(𝑡)𝑘_𝑖(𝑡)𝑐^𝑖(𝑡)^−^𝑛_𝑗=_𝑚+1ℎ^𝑗(𝑡)𝜆_𝑗(𝑡)𝑐^𝑗(𝑡)^−^𝑚_𝑠𝑖=1^𝑖−1_𝑞𝑖=0𝑡−𝑡_𝑖^𝑞𝑖𝑞_𝑖!𝛾^𝑖𝑞𝑖+(𝑡)𝑐_𝑖(𝑡)_𝑛_𝑚𝑘=1_{𝑠𝑖=1𝑖≠𝑘}^𝑖−1_𝑞𝑖=0𝑡−𝑡_𝑖^𝑞𝑖𝑞_𝑖!⋅ℎ^𝑖𝑞𝑖(𝑡)𝜆_𝑖(𝑡)−𝜆_𝑘(𝑡)𝑐^𝑘𝑃𝑦,𝑑_𝑖(𝑡)+𝐻⁽⁰⁾(𝑡),(𝑡)≡−ℎ^𝑖(𝑡)𝑘_𝑖(𝑡)^−^𝑚_𝑖=1ℎ_𝑖(𝑡)𝑘_𝑖(𝑡)𝑐^𝑖̇(𝑡),𝑑_𝑖(𝑡)−^𝑛_𝑗=_𝑚+1ℎ^𝑗(𝑡)𝑘_𝑗(𝑡)𝑐^𝑗̇−(𝑡),𝑑_𝑖(𝑡)_𝑠𝑖−1_𝑞_𝑖𝑚=0_𝑖=1𝑡−𝑡_𝑖^𝑞𝑖𝑞_𝑖!𝛾^𝑖𝑞𝑖(𝑡)+𝐻⁽⁰⁾(𝑡),𝑑_𝑖(𝑡),𝑖=1,𝑚.(3.36) Denote by 𝑟_𝑖(𝑡)the known function 𝑟_𝑖(𝑡)≡−ℎ^𝑖(𝑡)𝑘_𝑖(𝑡)^−^𝑚_𝑖=1ℎ_𝑖(𝑡)𝑘_𝑖(𝑡)𝑐^𝑖̇(𝑡),𝑑_𝑖(𝑡)−^𝑛_𝑗=_𝑚+1ℎ^𝑗(𝑡)𝑘_𝑗(𝑡)𝑐^𝑗̇(𝑡),𝑑_𝑖(𝑡)+𝐻⁽⁰⁾(𝑡),𝑑_𝑖(𝑡),(3.37) and write the conditions (3.13) for (𝑃𝑦,𝑑_𝑖(𝑡)).Taking into account expression (3.27) for 𝛾_𝑖𝑞𝑖(𝑡), we get ^𝑠𝑖⁻¹_𝑞𝑖=0𝛾^0𝑖𝑞_𝑖𝐷^𝑙𝑖𝑡−𝑡^𝑖^𝑞𝑖𝑞_𝑖!𝑒^{𝑃𝑖𝑖𝑞(𝑥)}_{𝑡=𝑡𝑖}+^𝑠𝑖−1_𝑞_𝑖=0𝐷^𝑙𝑖𝑓_𝑖𝑞𝑖(𝑡)_{𝑡=𝑡𝑖}=𝐷^𝑙𝑖𝑟_𝑖(𝑡)_{𝑡=𝑡𝑖},𝑖=1,𝑚,𝑙_𝑖=0,𝑠_𝑖−1.(3.38) Using the Leibnitz formula, we obtain that 𝐷^𝑙𝑖𝑡−𝑡^𝑖^𝑞𝑖𝑞_𝑖!𝑒^{𝑃𝑖𝑖𝑞(𝑡)}_{𝑡=𝑡𝑖}=⎡⎢⎢⎣^𝑙𝑖_𝜈=0𝐶^𝜈𝑙_𝑖𝑡−𝑡^𝑖^𝑞𝑖𝑞_𝑖!^(𝜈)𝑒^{𝑃𝑖𝑖𝑞(𝑡)}^{(𝑙𝑖−𝜈)}⎤⎥⎥⎦_{𝑡=𝑡𝑖}=⎡⎢⎢⎣^𝑞𝑖_𝜈=0𝐶^𝜈𝑙_𝑖𝑡−𝑡^𝑖^𝑞𝑖𝑞_𝑖!^(𝜈)𝑒^{𝑃𝑖𝑖𝑞(𝑡)}^{(𝑙𝑖−𝜈)}⎤⎥⎥⎦_{𝑡=𝑡𝑖}=𝐶^𝑞𝑖_𝑙𝑖𝑒^{𝑃𝑖𝑖𝑞(𝑡)}^{(𝑙𝑖−𝑞𝑖)𝑡=𝑡}_𝑖,(3.39) for 𝑙_𝑖≥𝑞_𝑖, 𝐷^𝑙𝑖𝑡−𝑡^𝑖^𝑞𝑖𝑞_𝑖!𝑒^{𝑃𝑖𝑖𝑞(𝑡)}_{𝑡=𝑡𝑖}=0,(3.40) for 0≤𝑙_𝑖≤𝑞_𝑖.
Therefore, previous equalities are written in the form of ^𝑠𝑖⁻¹_𝑞𝑖=0𝛾^0𝑖𝑞_𝑖𝐶^𝑞𝑖_𝑙𝑖𝑒^{𝑃𝑖𝑖𝑞(𝑡)}^{(𝑙𝑖−𝑞𝑖)𝑡=𝑡}_𝑖=𝑟^0𝑖𝑙_𝑖𝑖=1,𝑚,𝑙_𝑖=0,𝑠_𝑖−1,(3.41) where 𝑟^0𝑖𝑙_𝑖=−^𝑠𝑖−1_𝑞_𝑖=0𝐷^𝑙𝑖for𝑙_𝑖𝑓_𝑖𝑞𝑖(𝑡)_{𝑡=𝑡𝑖}−𝐷^𝑙𝑖𝑟_𝑖(𝑡)_{𝑡=𝑡𝑖},for𝑙_𝑖=0,weget𝛾^0𝑖0𝑒^{𝑃𝑖𝑖𝑞(𝑡𝑖)}=𝑟^0𝑖0;fo⋮r𝑙_𝑖=1,weget𝛾^0𝑖0𝑐⁰¹𝑒^{𝑃𝑖𝑖𝑞(𝑡)}^𝑡=𝑡_𝑖+𝛾^0𝑖1𝑒^{𝑃𝑖𝑖𝑞(𝑡𝑖)}=𝑟^0𝑖1;=𝑠_𝑖−1,weget𝛾^0𝑖0𝑐^0𝑠_𝑖−1𝑒^{𝑃𝑖𝑖𝑞(𝑡)}^{𝑠𝑖−1𝑡=𝑡}_𝑖+⋯+𝛾^0𝑖𝑠_𝑖−1𝑒^{𝑃𝑖𝑖𝑞(𝑡𝑖)}=𝑟^0𝑖𝑠_𝑖−1.(3.42)
We obtain from here sequentially the numbers 𝛾^0𝑖0,…,𝛾^0𝑖𝑠_𝑖−1.Theorem 3.6 is proved.

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

So, if ℎ⁽⁰⁾(𝑡)∈𝑈⁽⁰⁾,ℎ⁽¹⁾(𝑡,𝜏,𝜎)∈𝑈⁽¹⁾,and conditions (3.6), (3.24)–(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 𝑈=𝑈⁽¹⁾⊕𝑈⁽⁰⁾.Two sequential problems (𝜀^𝑘) and (𝜀^𝑘+1) are connected uniquely by conditions (3.23)–(3.25), (3.30); therefore, by virtue of Theorems 3.1–3.6, they are solvable uniquely in the space 𝑈.

4. Asymptotical Character of Formal Solutions

Let 𝑦₋₁(𝑡,𝜏,𝜎),…,𝑦_𝑘(𝑡,𝜏,𝜎) be solutions of formal problems (𝜀⁻¹), …,(𝜀^𝑘) in the space 𝑈,respectively. Compose the partial sum for series (2.4):𝑆_𝑛(𝑡,𝜏,𝜎)=^𝑛_𝑘₌₋₁𝜀^𝑘𝑦_𝑘(𝑡,𝜏,𝜎),(4.1) and take its restriction 𝑦_𝜀𝑛(𝑡)=𝑆_𝑛(𝑡,𝜑(𝑡)/𝜀,𝜓(𝑡,𝜀)).

We have the following result.

Theorem 4.1. Let conditions (i)–(v) hold. Then, for sufficiently small 𝜀(0≤𝜀≤𝜀₀), the estimates ‖‖𝑦(𝑡,𝜏)−𝑦_𝜀𝑛(𝑡)‖‖_𝐶[0,𝑇]≤𝐶_𝑛𝜀^𝑛+1,𝑛=−1,0,1,…,(4.2) hold. Here, 𝑦(𝑡,𝜀) is the exact solution of problem (1.1), and 𝑦_𝜀𝑛(𝑡) is the states above restriction of the 𝑛th partial sum of series (2.4).

Proof. The restriction 𝑦_𝜀𝑛(𝑡) of series (2.4) satisfies the initial condition 𝑦_𝜀𝑛(0)=𝑦⁰ and system (1.1) up to terms containing 𝜀^𝑛+1, that is, 𝜀𝑑𝑦^𝜀𝑛(𝑡)𝑑𝑡=𝐴(𝑡)𝑦^𝜀𝑛(𝑡)+𝜀^𝑛+1𝑅_𝑛(𝑡,𝜀)+ℎ(𝑡),(4.3) where 𝑅_𝑛(𝑡,𝑠) is a known function satisfying the estimate ‖𝑅(𝑡,𝜀)‖_𝐶[0,𝑇]≤𝑅_𝑛,𝑅_𝑛---const.(4.4) Under conditions of Theorem 4.1 on the spectrum of the operator 𝐴(𝑡) for the fundamental matrix 𝑌(𝑡,𝑠,𝜀)≡𝑌(𝑡,𝜀)𝑌⁻¹(𝑡,𝜀) of the system 𝜀𝑌̇=𝐴(𝑡)𝑌, the estimate ‖𝑌(𝑡,𝑠,𝜀)‖≤const∀(𝑡,𝜀)∈𝑄≡{0≤𝑠≤𝑡≤𝑇},∀𝜀>0∈0,𝜀₀,(4.5) is valid. Here, 𝜀₀>0−is sufficiently small. Now, write the equation 𝜀𝑑Δ(𝑡,𝜀)𝑑𝑡=𝐴(𝑡)Δ(𝑡,𝜀)−𝜀^𝑛+1𝑅_𝑛(𝑡,𝜀),Δ(0,𝜀)=0,(4.6) for the remainder term Δ(𝑡,𝜀)≡𝑦(𝑡,𝜀)−𝑦_𝜀𝑛(𝑡).We obtain from this equation that Δ(𝑡,𝜀)=−𝜀^𝑛^𝑡0𝑌(𝑥,𝑠,𝜀)𝑅_𝑛(𝑠,𝜀)𝑑𝑠,(4.7) whence we get the estimate ‖Δ(𝑡,𝜀)‖_𝐶[0,𝑇]≤−𝜀^𝑛𝑅_𝑛,(4.8) where 𝑅_𝑛=max_{(𝑡,𝑠)∈𝑄}‖𝑌(𝑡,𝑠,𝜀)‖⋅‖𝑅_𝑛(𝑡,𝑠)‖⋅𝑇. So, we obtain the estimate ‖‖𝑦(𝑡,𝜀)−𝑦_𝜀𝑛(𝑡)‖‖_𝐶[0,𝑇]≤𝜀^𝑛𝑅_𝑛,𝑛=−1,0,1,….(4.9) Taking instead of 𝑦_𝜀𝑛(𝑡) the partial sum 𝑦_𝜀,𝑛+1(𝑡)≡𝑦_𝜀𝑛(𝑡)+𝜀^𝑛+1𝑦_𝑛+1𝑡,𝜑𝜀(𝑡),𝜓(𝑡,𝜀),(4.10) we get ‖‖‖𝑦(𝑡,𝜀)−𝑦_𝜀𝑛(𝑡)−𝜀^𝑛+1𝑦_𝑛+1𝑡,𝜑𝜀(𝑡),𝜓(𝑡,𝜀)‖‖‖≤𝜀^𝑛+1𝑅_𝑛+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 problem𝜀⎛⎜⎜⎝𝑧𝑦⎞⎟⎟⎠̇̇=⎛⎜⎜⎝−5𝑡²−10𝑡₂+42𝑡²−2+104𝑡²−5⎞⎟⎟⎠⎛⎜⎜⎝𝑧𝑦⎞⎟⎟⎠+⎛⎜⎜⎝𝑡²ℎ₁0(𝑡)⎞⎟⎟⎠,𝑦(0,𝜀)=𝑦₀,𝑧(0,𝜀)=𝑧⁰,(5.1) where ℎ₁(𝑡)∈𝐶^∞[0,2],𝜀>0is a small parameter. Eigenvalues of the matrix 𝐴(𝑡)=^{−5𝑡2+42𝑡2−2−10𝑡}_{2+104𝑡2−5}are 𝜆₁(𝑡)=−𝑡²,𝜆₂(𝑡)=−1.Eigenvectors of matrices 𝐴(𝑡)and 𝐴^∗(𝑡), are, respectively,𝜑₁=⎛⎜⎜⎝12⎞⎟⎟⎠,𝜑₂=⎛⎜⎜⎝25⎞⎟⎟⎠,𝜓₁=⎛⎜⎜⎝−52⎞⎟⎟⎠,𝜓₂=⎛⎜⎜⎝−12⎞⎟⎟⎠.(5.2) We get (ℎ(𝑡),𝜓₁(𝑡))≡5𝑡²ℎ₁(𝑡). Therefore,ℎ(0),𝜓₁(0)=0,𝑑𝑑𝑡ℎ(0),𝜓¹(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 𝐾_𝑗𝑖(𝑡).Since 𝜓(𝑡)≡𝜆₁(𝑡)=−𝑡²,there will be two such polynomials: 𝐾₀₀(𝑡)and 𝐾₀₁(𝑡).

Take the arbitrary numbers 𝑎₀₀(𝑡)and 𝑎₀₁(𝑡),and set the interpolation conditions for the polynomial 𝑟(𝑡),𝑟(𝑡)=𝑎₀₀,𝑟̇(1)=𝑎₀₁.(5.4) Expand 𝑟(𝑡)onto partial fractions𝑟(𝑡)𝜓(𝑡)=𝐴𝑡²+𝐵𝑡,(5.5) from where𝑟(𝑡)≡𝐴+𝐵𝑡.(5.6) Use the interpolation polynomial (5.4). We get 𝐴=𝑎₀₀,𝐵=𝑎₀₁. Hence, (5.6) takes the form𝑟(𝑡)≡𝑎₀₀+𝑡𝑎₀₁.(5.7) Since numbers 𝑎₀₀and 𝑎₀₁ are arbitrary, basic Lagrange-Silvestre polynomials will be coefficients standing before them, that is,𝐾₀₀(𝑡)≡1,𝐾₀₁(𝑡)≡𝑡.(5.8)

Introduce the regularizing variables𝜎₀₀=𝑒^{∫(1/𝜀)𝑡0𝜆1𝑑𝑠}^𝑡0𝑒^{∫−(1/𝜀)𝑠0𝜆1𝑑𝑥}⋅𝐾₀₀(𝑠)𝑑𝑠=𝑒^{−𝑡3/3𝜀}^𝑡0𝜎𝑒^𝑠3/3𝜀𝑑𝑠≡𝑝₀₀(𝑡),₀₁=𝑒^{∫(1/𝜀)𝑡0𝜆1𝑑𝑠}^𝑡0𝑒^{∫−(1/𝜀)𝑠0𝜆1𝑑𝑥}⋅𝐾₀₁(𝑠)𝑑𝑠=𝑒^{−𝑡3/3𝜀}^𝑡0𝜏𝑒^𝑠3/3𝜀⋅𝑠𝑑𝑠≡𝑝₀₁(𝑡),₁=1𝜀^𝑡0𝜆₁𝑑𝑠=−𝑡³3𝜀≡𝑞¹(𝑡),𝜏₂=1𝜀^𝑡0𝜆₂𝑑𝑠=−𝑡𝜀≡𝑞²(𝑡).(5.9)

Construct the extended problem corresponding to problem (5.1):𝜀𝜕𝑤𝜕𝑡+𝜆¹(𝑡)𝜕𝑤𝜕𝜏₁+𝜆₂(𝑡)𝜕𝑤𝜕𝜏₂+𝜆₁(𝑡)𝜎₀₀𝜕𝑤𝜕𝜎₀₀+𝜆₁(𝑡)𝜎₀₁𝜕𝑤𝜕𝜎₀₁+𝜀𝜕𝑤𝜕𝜎₀₀+𝜀𝑡𝜕𝑤𝜕𝜎₀₁=ℎ(𝑡),𝑤(0,0,0,𝜀)=𝑤₀−𝐴(𝑡)𝑤,(5.10) where 𝜏≡(𝜏₁,𝜏₂),𝜎=(𝜎₀₀,𝜎₀₁),𝑤=𝑤(𝑡,𝜏,𝜎,𝜀).

Determining solutions of problem (5.10) in the form of a series𝑤(𝑡,𝜏,𝜎,𝜀)=^∞_𝑘=0𝜀^𝑘𝑤_𝑘(𝑡,𝜏,𝜎),(5.11) we obtain the following iteration problems: 𝐿𝑤₀≡𝜆₁(𝑡)𝜕𝑤⁰𝜕𝜏+𝜕𝜕𝜎𝑤⁰₀₀+𝑡⋅𝜎₀₁𝜕𝑤₀𝜕𝜎₀₁+𝜆²(𝑡)𝜕𝑤⁰𝜕𝜏₂𝐿𝑤₁−𝐴(𝑡)𝑤₀=ℎ(𝑡),𝑤₀(0,0,0)=𝑤⁰,(5.12)=−𝜕𝑤⁰𝜕𝑡−𝜕𝜕𝜎𝑤⁰₀₀−𝑡𝜕𝑤⁰𝜕𝜎₀₁⋮,𝑤₁(0,0,0)=0,(5.13)

We determine solutions of iteration problems (5.12), (5.13), and so on in the space 𝑈 of functions in the form of𝑤𝑤(𝑡,𝜏,𝜎)=𝑤₁(𝑡)𝑒^𝜏1+𝑤₂(𝑡)𝑒^𝜏2+𝑤₀₀(𝑡)𝜎₀₀+𝑤₀₁(𝑡)𝜎₀₁+𝑤₀(𝑡),₀(𝑡),𝑤₁(𝑡),𝑤₂(𝑡),𝑤₀₀(𝑡),𝑤₀₁(𝑡)∈𝐶^∞[0,2],𝐶².(5.14)

Directly calculating, we obtain the solution of system (5.12) in the form of𝑤₀(𝑡,𝜏,𝜎)=𝛼₁(𝑡)𝜑₁𝑒^𝜏1+𝛼₂(𝑡)𝜑₂𝑒^𝜏2+𝛾₀₀(𝑡)𝜑₁𝜎₀₀+𝛾₀₁(𝑡)𝜑₁𝜎₀₁+5ℎ₁(𝑡)𝜑₁−2𝑡²ℎ₁(𝑡)𝜑₂,(5.15) where 𝛼_𝑗(𝑡),𝛾_𝑗𝑖(𝑡)∈𝐶^∞[0,2]are for now arbitrary functions.

To calculate the functions 𝛼_𝑗(𝑡)and 𝛾_𝑖𝑗(𝑡),we pass to the following iteration problem (5.13). Taking into account (5.15), it will be written in the form of𝐿𝑤₁=−𝛼̇₁(𝑡)𝜑₁𝑒^𝜏1−𝛼̇₂(𝑡)𝜑₂𝑒^𝜏2−𝛾̇₀₀(𝑡)𝜑₁𝜎₀₀−𝛾̇₀₁(𝑡)𝜑₁𝜎₀₁−5ℎ̇₁(𝑡)𝜑₁−2𝑡²ℎ₁(𝑡)^𝜑₂−𝛾₀₀(𝑡)𝜑₁−𝑡𝛾₀₁(𝑡)𝜑₁.(5.16)

For solvability of problem (5.13) in the space 𝑈, it is necessary and sufficient to fulfill the conditions−−𝛼̇₁5(ℎ𝑡̇₁)=0,−𝛼̇₂(𝑡)=0,−𝛾̇₀₀(𝑡)=0,−𝛾̇₀₁(𝑡)=0,(0)−𝛾₀₀(0)=0,−5ℎ̈₁(0)−𝛾̇₀₀(0)−𝛾₀₁(0)=0.(5.17)

Using solution (5.15) and the initial condition 𝑤₀(0,0,0)=𝑤⁰,we obtain the equation𝛼₁(0)𝜑₁+𝛼₂(0)𝜑₂+5ℎ₁(0)𝜑₁=𝑤⁰.(5.18)

Multiplying it (scalar) on 𝜓₁and 𝜓₂, we obtain the values𝛼₁(0)=𝑤𝛼⁰,𝜓₁−5ℎ₁(0)≡5𝑦⁰−2𝑧⁰−5ℎ₁(0),₂(0)=𝑤⁰,𝜓₂=𝑧⁰−2𝑦⁰.(5.19)

Using equalities (5.17), and also the initial data (5.19), we obtain uniquely the functions 𝛼_𝑗(𝑡)and 𝛾_𝑗𝑖(𝑡):𝛼₁(𝑡)𝛾=₀₀5𝑦⁰−2𝑧⁰−5ℎ₁(0),𝛼₂(𝑡)=𝑧⁰−2𝑦⁰.(𝑡)=−5ℎ̇₁(0),𝛾₀₁(𝑡)=−5ℎ̈₁(0).(5.20)

Substituting these functions into (5.15), we obtain uniquely the solution of problem (5.12) in the space 𝑈,𝑤₀(𝑡,𝜏,𝜎)=5𝑦⁰−2𝑧⁰−5ℎ₁(0)𝜑₁𝑒^𝜏1+𝑧⁰−2𝑦⁰𝜑₂𝑒^𝜏2−5ℎ̇₁(0)𝜑₁𝜎₀₀−5ℎ̈₁(0)𝜑₁𝜎₀₁+5ℎ₁(𝑡)𝜑₁−2𝑡²ℎ₁(𝑡)𝜑₂.(5.21)

Producing here restriction on the functions 𝜏=𝑞(𝑡),𝜎=𝑝(𝑡),we obtain the principal term of the asymptotics for the solution of problem (5.1):𝑤_0𝜀(𝑡)=5𝑦⁰−2𝑧⁰−5ℎ₁(0)𝜑₁𝑒^{−𝑡3/3𝜀}+𝑧⁰−2𝑦⁰𝜑₂𝑒^−𝑡/𝜀−5ℎ̇₁(0)𝜑₁𝑒^{−𝑡3/3𝜀}^𝑡0𝑒^𝑠3/3𝜀𝑑𝑠−5ℎ̈₁(0)𝜑₁𝑒^{−𝑡3/3𝜀}^𝑡0𝑒^𝑠3/3𝜀𝑠𝑑𝑠+5ℎ₁(𝑡)𝜑₁−2𝑡²ℎ₁(𝑡)𝜑₂.(5.22)

The zero-order asymptotical solution is obtained: it satisfies the estimate‖‖𝑤(𝑡,𝜀)−𝑤_0𝜀(𝑡)‖‖_𝐶[0,2]≤𝐶₁⋅𝜀,(5.23) where 𝑤(𝑡,𝜀)is an exact solution of problem (1.1), and 𝐶₁>0is a constant independent of 𝜀at sufficiently small 𝜀(0<𝜀≤𝜀₀).