Research Article Burkhan T. Kalimbetov* and Olim D ...

Research Article

Burkhan T. Kalimbetov* and Olim D. Tuychiev

Asymptotic solution of the Cauchy problemfor the singularly perturbed partial integro-differential equation with rapidly oscillatingcoefficients and with rapidly oscillatingheterogeneity

https://doi.org/10.1515/math-2021-0021received September 20, 2020; accepted February 8, 2021

Abstract: In this paper, the regularization method of S. A. Lomov is generalized to integro-differentialequations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The maingoal of the work is to reveal the influence of the oscillating components on the structure of the asymptoticsof the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficientand a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observedin the problem. Other cases of the relationship between frequencies can lead to so-called non-identicalresonances, the study of which is nontrivial and requires the development of a new approach. It is supposedto study these cases in our further work.

Keywords: singularly perturbed, integro-partial differential equation, regularization of an integral, spaceof non-resonant solutions, iterative problems, solvability of iterative problems

MSC 2020: 35F10, 35R09

1 Introduction

Singularly perturbed integro-differential equations have been the subject of research for many decades,starting with the work of A. Vasilyeva, V. Butuzov [1–3], and M. Imanaliev [4,5]. These work argue theimportance of such research for theory and applications. However, before the appearance of work related tothe regularization method, S. Lomov [6–8], integro-differential equations were considered under the con-ditions of the spectrum of the matrix of the first variation (on a degenerate solution) lying in the open lefthalf-plane, which significantly narrowed the scope of the above work in problems with purely imaginarypoints of the spectrum. And only after the development of the method of Lomov, it became possible toconsider problems with a spectrum lying on an imaginary axis [9–24]. Note that the Lomov regularizationmethod was mainly used for ordinary singularly perturbed differential and integro-differential equations[25–30]. The development of this method for integro-differential equations with partial derivatives wascarried out by A. Bobodzhanov, V. Safonov, and B. Kalimbetov [31–38]. In the study of problems with a

* Corresponding author: Burkhan T. Kalimbetov, Akhmed Yassawi University, B. Sattarkhanov 29, Turkestan, 161200,Kazakhstan, e-mail: [email protected] D. Tuychiev: Khudjant State University named after B. Gafurov, Movlonbekov Ave., 735700, Khudjant, Tajikistan,e-mail: [email protected]

Open Mathematics 2021; 19: 244–258

Open Access. © 2021 Burkhan T. Kalimbetov and Olim D. Tuychiev, published by De Gruyter. This work is licensed under theCreative Commons Attribution 4.0 International License.

https://doi.org/10.1515/math-2021-0021

mailto:[email protected]

mailto:[email protected]

slowly changing kernel, it turned out that the regularization procedure and the construction of a regular-ized asymptotic solution essentially depend on the type of integral operator. The most difficult case waswhen the upper limit of the integral is not a differentiation variable. For the integral operator with an upperlimit coinciding with the differentiation variable, the scalar case is investigated. The case when the upperlimit of the integral operator coincides with the differentiation variable is studied for equations of partialdifferential integro-differential equations with an integral operator, the kernel of which contains a rapidlychanging exponential factor. Summarizing the results of work for integro-differential equations with one-dimensional integrals, the problem of constructing a regularized asymptotic solution of the problem forintegro-differential equations with two independent variables is investigated. In the present paper, weconsider a singularly perturbed partial differential integro-differential equation with high-frequency coeffi-cients and with rapidly oscillating coefficients, and rapidly oscillating heterogeneity that generates essen-tially special singularities in the solution of the problem.

In this paper,we consider the Cauchyproblem for the integro-differential equationwithpartial derivatives:

∫( ) ≡∂∂

− ( ) − ( ) ( ) − ( )( )

= ( )( )

+ ( )

( ) = ( ) (( ) ∈ [ ] × [ ])

L y x t ε ε yx

A x y K x t s y s t ε s εg x β xε

By

εh x t β xε

h x t

y x t ε y t x t x X T

, , , , , , d cos

, sin , ,

, , , , 0, ,

ε

x

x

1 2

00

0

0 (1)

where ( )A x , ( )g x , ( )h x t,1 , ( )h x t,2 , ( )y t0 , ′( ) > ( ∈ [ ])β x x x X0 ,0 are known scalar functions, =B const,= ( )y y x t ε, , is an unknown function, and >ε 0 is a small parameter.Such an equation in the case ( ) = ( ) ≡β x γ x B2 , 0 for ordinary equations in the absence of an integral

term was considered in [39–44]. The limiting operator ( )A x has a spectrum ( ) = ( )λ x A x1 , ′( )β x is a frequency

of rapidly oscillating ( )cos β tε . In the following, functions ( ) = − ′( ) ( ) = + ′( )λ x iβ x λ x iβ x,2 3 will be called the

spectrum of a rapidly oscillating coefficient.We assume that the following conditions are fulfilled:

(1) ( )g x , ( )β x , ( ) ∈ ([ ] )∞A x C x X R, ,0 , ( )h x t,1 , ( ) ∈ ([ ] × [ ] )∞h x t C x X T R, , 0, ,2 0 , ( ) ∈ ∞y t C0 ([ ]T R0, , ),( ) ∈ ({ ≤ ≤ ≤ ≤ ≤ } )∞K x t s C x s x X t T R, , , 0 ,0 ;

(2) ( ) < (∀ ∈ [ ])A x x x X0 ,0 .We will develop an algorithm for constructing a regularized asymptotic solution [6] of problem (1).

2 Regularization of problem (1)Denote by = ( )σ σ εj j independent of magnitude = =− ( ) + ( )σ e σ e,β x β x

1 2iε

iε0 0 , and rewrite equation (1) as

∫∫ ∫

∫ ∫

( ) ≡∂∂

− ( ) − ( ) ( ) − ( ) +

= ( ) + ( ) − ( ) = ( )

− ′( ) + ′( )

− ′( ) + ′( )

L y x t ε ε yx

A x y K x t s y s t ε s ε g x e σ e σ By

h x t ε h x ti

e σ e σ y x t ε y t

, , , , , , d2

, ,2

, , , .

ε

x

x β θ θ β θ θ

β θ θ β θ θ

d

1

d

2

21

d

1

d

2 00

iε

x

xiε

x

x

iε

x

xiε

x

x

0

0 0

0 0

(2)

Introduce the regularized variables:

∫= ( ) ≡( )

=τε

λ θ θψ x

εj1 d , 1, 3,j

x

x

jj

0

Asymptotic solution of the Cauchy problem for the singularly 245

and instead of problem (2), consider the problem

∫ ( )

( )

∑∂∂

+ ( )∂∂

− ( ) − ( )( )

+ ( ) +

= ( ) − + ( ) ( )∣ =

=

= =

ε yx

λ x yτ

A x y K x t s y s t ψ sε

ε s ε g x e σ e σ By

ε h x ti

e σ e σ h x t y x t τ ε y

˜ ˜ ˜ , , ˜ , , , d2

˜

,2

, , ˜ , , , ,

jj

jx

x

τ τ

τ τx x τ

1

3

1 2

11 2 2 , 0

0

0

2 3

2 30

(3)

for the function = ( )y y x t τ ε˜ ˜ , , , , where = ( )τ τ τ τ, ,1 2 3 , = ( )ψ ψ ψ ψ, ,1 2 3 . It is clear that if = ( )y y x t τ ε˜ ˜ , , , isa solution of problem (3), then the function ( )= ( )y y x t ε˜ ˜ , , ,ψ x

ε is an exact solution to problem (2); therefore,problem (3) is extended with respect to problem (2). However, it cannot be considered fully regularized,since it does not regularize the integral

∫= ( )

( )Jy K x t s y s t ψ sε

ε s˜ , , ˜ , , , d .x

x

0

Definition 1. A class Mε is said to be asymptotically invariant (with → +ε 0) with respect to an operator P0if the following conditions are fulfilled:(1) ⊂ ( )M D Pε 0 for each fixed >ε 0;(2) the image ( )P μ x t ε, ,0 of any element ( ) ∈μ x t ε M, , ε decomposes in a power series

∑( ) = ( ) ( → + ( ) ∈ = …)=

∞

P μ x t ε ε μ x t ε ε μ x t ε M n, , , , 0, , , , 0, 1,n

nn n ε0

0

convergent asymptotically for → +ε 0 (uniformly in ( ) ∈ [ ] × [ ]x t x X T, , 0,0 ).

From this definition, it can be seen that the class Mε depends on the spaceU , in which the operator P0is defined. In our case =P J0 .

Before describing the spaceU , we introduce the sets of resonant multi-indices. We introduce the nota-tions:

∑

∑

( ) = ( ( ) ( ) ( )) ( ( )) = ( )

∣ ∣ = = { ( ( )) ≡ ∀∣ ∣ ≥ }

= { ( ( )) ≡ ( ) ∀∣ ∣ ≥ } =

=

=

λ x λ x λ x λ x m λ x m λ x

m m m m λ x m

m m λ x λ x m j

, , , , ,

, Γ : , 0, 2 ,

Γ : , , 2 , 1, 2, 3,

jj j

jj

j j

1 2 31

3

1

3

0

(the set Γ0 corresponds to a point of the spectrum ( ) ≡λ x 00 , generated by the integral operator (see [45]).For the space U , we take the space of functions ( )y x t τ σ, , , , represented by sums

∑ ∑( ) = ( ) + ( ) + ( )= ≤∣ ∣≤

∗( )y x t τ σ y x t σ y x t σ e y x t σ e, , , , , , , , , ,

jj

τ

m N

m m τ0

1

3

2

,j

y

(4)

( ) ( ) ( ) ∈ ([ ] × [ ] ) = ≤ ∣ ∣ ≤∞y x t σ y x t σ y x t σ C x X T j m NC, , , , , , , , , 0, , , 1, 3, 2 ,jm

y0 0

where asterisk∗ above the sum sign indicates that the summation for ∣ ∣ ≡ + + ≥m m m m 21 2 3 , it occurs onlyon the non-resonant multi-indexes, i.e., ∉ ⋃ =m Γj j0

3 , = ( )σ σ σ,1 2 .Note that here the degree Ny of the polynomial ( )y x t τ σ, , , relative to the exponentials eτj depends on

the element y. In addition, the elements of spaceU depend on bounded in >ε 0 terms of constants = ( )σ σ ε1 1and = ( )σ σ ε2 2 and which do not affect the development of the algorithm described below; therefore, inthe record of element (4) of this spaceU , we omit the dependence on = ( )σ σ σ,1 2 for brevity. We show thatthe class = ∣ = ( ) ∕M Uε τ ψ x ε is asymptotically invariant with respect to the operator J . The image of the integraloperator J on an arbitrary element ( )y x t τ, , of the space U has the form

246 Burkhan T. Kalimbetov and Olim D. Tuychiev

∫ ∫

∫

∑

∑

∫

∫

( ) = ( ) ( ) + ( ) ( )

+ ( ) ( )

=

( )

≤∣ ∣≤

∗ ( ( ))

Jy x t τ K x t s y s t s K x t s y s t e s

K x t s y s t e s

, , , , , d , , , d

, , , d .

x

x

j x

x

j

λ θ θ

m N x

x

mm λ θ θ

01

3 d

2

, d

ε

x

s

j

y

ε

x

s0 0

1

0

0

1

0

Apply the operation of integration by parts to the second term

∫

∫

∫

∫

∫ ∫

∫ ∫

( ) =( ) ( )

( )

=( ) ( )

( )− ∂

∂

( ) ( )

( )

=( ) ( )

( )−

( ) ( )

( )− ∂

∂

( ) ( )

( )

( )

( )

=

=

( )

( ) ( )

J x t ε εK x t s y s t

λ se

εK x t s y s t

λ se

sK x t s y s t

λ se s

εK x t x y x t

λ xe

K x t x y x tλ x

εs

K x t s y s tλ s

e s

, ,, , ,

d

, , , , , ,d

, , , , , , , , ,d .

j

x

xj

j

λ θ θ

j

j

λ θ θ

s x

s x

x

xj

j

λ θ θ

j

j

λ θ θj

jx

xj

j

λ θ θ

d

d d

d0 0

0

d

ε

x

s

j

ε

x

s

j ε

x

s

j

ε

x

x

j ε

x

s

j

0

1

0

1

0

00

1

0

1

0

0

1

0

Continuing this process, we obtain the series

∑∫

( ) = (− ) ( ( ( ) ( ))) − ( ( ( ) ( )))=

∞+

=

( )

=J x t ε ε I K x t s y s t e I K x t s y s t, , 1 , , , , , , ,jν

ν νjν

j s x

λ θ θ

jν

j s x0

1dε

x

x

j1

0 0

where the operators

=( )

=( )

∂∂

( ≥ = )−Iλ s

Iλ s s

I ν j1 , 1 1, 1, 3jj

jν

jjν0 1

are introduced.Applying the integration operation in parts to integrals

∫∫

( ) = ( ) ( )( ( ))

J x t ε K x t s y s t e s, , , , , d ,m

x

x

mm λ θ θ, dε

x

s

0

1

0

we note that for all multi-indices = ( )m m m m, ,1 2 3 , ∉ ⋃ =m Γj j03 inequalities

( ( )) ≠ ∀ ∈ [ ]m λ x x x X, 0 ,0

are satisfied. Therefore, integration by parts in integrals ( )J x t ε, ,m is possible. Performing it, we will have:

∫

∫

∫

∫ ∫

( ) =( ) ( )

( ( ))

=( ) ( )

( ( ))− ∂

∂( ) ( )

( ( ))

( ( ))

( ( ))

=

=

( ( ))

J x t ε ε K x t s y s tm λ s

e

ε K x t s y s tm λ s

es

K x t s y s tm λ s

e s

, , , , ,,

d

, , ,,

, , ,,

d

m

x

xm m λ θ θ

m m λ θ θ

s x

s x

x

xm m λ θ θ

, d

, d , d

ε

x

s

ε

x

s

ε

x

s

0

1

0

1

0

00

1

0


∫

∑

∫ ∫

∫

=( ) ( )

( ( ))−

( ) ( )( ( ))

− ∂∂

( ) ( )( ( ))

= (− ) ( ( ( ) ( ))) − ( ( ( ) ( )))

( ( )) ( ( ))

=

∞+

=

( ( ))

=

ε K x t x y x tm λ x

e K x t x y x tm λ x

εs

K x t s y s tm λ s

e s

ε I K x t s y s t e I K x t s y s t

, , ,,

, , ,,

, , ,,

d

1 , , , , , , ,

m m λ θ θ m

x

xm m λ θ θ

ν

ν νmν m

s x

m λ θ θ

mν m

s x

, d0 0

0

, d

0

1, d

ε

x

x

ε

x

s

ε

x

x

1

0

0

1

0

1

0 0

where the operators

=( ( ))

=( ( ))

∂∂

( ≥ ∣ ∣ ≥ )−Im λ s

Im λ s s

I ν m1,

, 1,

1, 2m mν

mν0 1

are introduced. Therefore, the image of the operator J on the element (4) of the space U is represented asa series

∫

∑ ∑

∑ ∑

∫

∫

( ) = ( ) ( )

+ (− ) ( ( ( ) ( ))) − ( ( ( ) ( )))

+ (− ) ( ( ( ) ( ))) − ( ( ( ) ( )))

= =

∞+

=

( )

=

≤∣ ∣≤

∗

=

∞+

=

( ( ))

=

Jy x t τ K x t s y s t s



, , , , , d

1 , , , , , ,

1 , , , , , , .

x

x

j ν

ν νjν

j s x

λ θ θ

jν

j s x

m N ν

ν νmν m

s x

m λ θ θ

mν m

s x

0

1

3

0

1d

2 0

1, d

ε

x

x

j

y

ε

x

x

0

1

0 0

1

0 0

It is easy to show (see, for example, [46, pp. 291–294]) that this series converges asymptotically for → +ε 0(uniformly in ( ) ∈ [ ] × [ ]x t x X T, , 0,0 ). This means that the class Mε is asymptotically invariant (for → +ε 0)with respect to the operator J .

We introduce operators →R U U:ν , acting on each element ( ) ∈y x t τ U, , of the form (4) according tothe law:

∫( ) = ( ) ( )R y x t τ K x t s y s t s, , , , , d ,x

x

0 0

0

(50)

∑

∑

( ) = [( ( ( ) ( ))) − ( ( ( ) ( ))) ]

+ [( ( ( ) ( ))) − ( ( ( ) ( ))) ]

== =

≤∣ ∣≤

∗

=( )

=( )

R y x t τ I K x t s y s t e I K x t s y s t

I K x t s y s t e I K x t s y s t e

, , , , , , , ,

, , , , , , ,

jj j s x

τj j s x

m Nm

ms x

m τm

ms x

m τ

11

30 0

1

0 , 0 ,

j

y

0

(51)

∑

∑

( ) = (− ) [( ( ( ) ( ))) − ( ( ( ) ( ))) ]

+ (− ) [( ( ( ) ( ))) − ( ( ( ) ( ))) ]

+=

= =

≤∣ ∣≤

∗

=( )

=

R y x t τ I K x t s y s t e I K x t s y s t

I K x t s y s t e I K x t s y s t

, , 1 , , , , , ,

1 , , , , , , .

νj

νjν

j s xτ

jν

j s x

m N

νmν m

s xm τ

mν m

s x

11

3

2

,

j

y

0

0

(5v)

Now let ( )y x t τ ε˜ , , , be an arbitrary continuous function on ( ) ∈ = [ ] × [ ] × { <x t τ G x X T τ τ, , , 0, : Re 0,0 1≤ = }τ jRe 0, 2, 3j , with asymptotic expansion

∑( ) = ( ) ( ) ∈=

∞

y x t τ ε ε y x t τ y x t τ U˜ , , , , , , , ,k

kk k

0(6)


converging as → +ε 0 (uniformly in ( ) ∈x t τ G, , ). Then, the image ( )Jy x t τ ε˜ , , , of this function is decom-posed into an asymptotic series

∑ ∑ ∑( ) = ( ) = ( )∣=

∞

=

∞

=− = ( ) ∕Jy x t τ ε ε Jy x t τ ε R y x t τ˜ , , , , , , , .

k

kk

r

r

s

r

r s s τ ψ x ε0 0 0

This equality is the basis for introducing an extension of an operator J on series of the form (6):

∑ ∑ ∑≡ ( ) = ( )

=

∞

=

∞

=−J y J ε y x t τ ε R y x t τ˜ ˜ ˜ , , , , .

k

kk

r

r

k

r

r k k0 0 0

Although the operator J̃ is formally defined, its utility is obvious; since in practice, it is usual to constructthe N th approximation of the asymptotic solution of problem (2), which impose only N th partial sums of theseries (6), which have not a formal, but a true meaning. Now you can write a problem that is completelyregularized with respect to the original problem (2):

( )

( )

∑( ) ≡∂∂

+ ( )∂∂

− ( ) − − ( ) +

= ( ) − + ( ) ( ) = ( ) (( ) ∈ [ ] × [ ])

=

L y x t τ ε ε yx

λ x yτ

A x y J y ε g x e σ e σ By

ε h x ti

e σ e σ h x t y x t ε y t x t x X T

˜ , , , ˜ ˜ ˜ ˜ ˜2

˜

,2

, , ˜ , , 0, , , , 0, .

εj

jj

τ τ

τ τ

1

3

1 2

11 2 2 0

00

2 3

2 3

(7)

3 Iterative problems and their solvability in the space USubstituting the series (6) into (7) and equating the coefficients of the same powers of ε, we obtain the follow-ing iterative problems:

∑( ) ≡ ( )∂∂

− ( ) − = ( ) ( ) = ( )=

Ly x t τ λ xyτ

A x y R y h x t y x t ε y t, , , , , , 0, ;j

jj

01

30

0 0 0 2 0 00 (80)

( ) ( )( ) = −∂∂

+ ( ) + + + ( ) − ( ) =Ly x t τyx

g x e σ e σ By R y h x ti

e σ e σ y x t, ,2

,2

, , , 0 0;τ τ τ τ1

01 2 0 1 0

11 2 1 02 3 2 3 (81)

( )( ) = −∂∂

+ ( ) + + + ( ) =Ly x t τyx

g x e σ e σ By R y R y y x t, ,2

, , , 0 0;τ τ2

11 2 1 1 1 2 0 2 02 3 (82)

⋯

( )( ) = −∂

∂+ ( ) + + + ⋯+ ( ) = ≥−

− −Ly x t τy

xg x e σ e σ By R y R y y x t k, ,

2, , , 0 0, 1.k

k τ τk k k k

11 2 1 0 1 1 02 3 (8k)

Each iterative problem (8k) has the form

∑( ) ≡ ( )∂∂

− ( ) − = ( ) ( ) = ( )=

∗Ly x t τ λ x yτ

A x y R y H x t τ y x t y t, , , , , , , 0 ,j

jj1

3

0 0 (9)

where ( ) = ( ) + ∑ ( ) + ∑ ( ) ∈= ≤∣ ∣≤∗ ( )H x t τ H x t H x t e H x t e U, , , , ,j j

τm N

m m τ0 1

32

,jH

, is the known function of spaceU ,( )∗y t is the known function of the complex space C, and the operator R0 has the form (see ( )50 )

∫∑ ∑( ) ≡ ( ) + ( ) + ( ) = ( ) ( )= ≤∣ ∣≤

∗( )R y x t τ R y x t y x t e y x t e K x t s y s t s, , , , , , , , d .

jj

τ

m N

m m τ

x

x

0 0 01

3

2

,0j

H0

We introduce scalar (for each ∈ [ ] ∈ [ ]x x X t T, , 0,0 ) product in space U :

∑ ∑ ∑ ∑

∑ ∑

⟨ ⟩ ≡ ( ) + ( ) + ( ) ( ) + ( ) + ( )

≡ ( ( ) ( )) + ( ( ) ( )) + ( ( ) ( ))

= ≤∣ ∣≤

∗( )

= ≤∣ ∣≤

∗( )

= ≤∣ ∣≤ ( )

∗

u w u x t u x t e u x t e w x t w x t e w x t e

u x t w x t u x t w x t u x t w x t

, , , , , , , ,

, , , , , , , , , ,

jj

τ

m N

m m τ

jj

τ

m N

m m τ

jj j

m min N N

m m

01

3

2

,0

1

3

2

,

0 01

3

2 ,

j

u

j

w

u w


where we denote by (∗ ∗), the usual scalar product in the complex space ( ) = ⋅u v u vC : , . Let us provethe following statement.

Theorem 1. Let conditions (1) and (2) be fulfilled and the right-hand side ( ) = ( ) + ∑ ( ) +=H x t τ H x t H x t e, , , ,j jτ

0 13 j

∑ ( )≤∣ ∣≤∗ ( )H x t e,m N

m m τ2

,H

of equation (9) belongs to the spaceU . Then, equation (9) is solvable inU , if and only if

⟨ ( ) ⟩ ≡ ∀( ) ∈ [ ] × [ ]H x t τ e x t x X T, , , 0, , , 0, .τ01 (10)

Proof. We will determine the solution of equation (9) as an element (4) of the space U :

∑ ∑( ) = ( ) + ( ) + ( )= ≤∣ ∣≤

∗( )y x t τ y x t y x t e y x t e, , , , , .

jj

τ

m N

m m τ0

1

3

2

,j

H

(11)

Substituting (11) into equation (9), we will have

∫∑ ∑

∑ ∑

[ ( ) − ( )] ( ) + [( ( )) − ( )] ( ) − ( ) ( ) − ( ) ( )

= ( ) + ( ) + ( )

= ≤∣ ∣≤

∗( )

= ≤∣ ∣≤

∗( )

λ x A x y x t e m λ x A x y x t e A x y x t K x t s y s t s

H x t H x t e H x t e

, , , , , , , d

, , , .

jj j

τ

m N

m m τ

x

x

jj

τ

m N

m m τ

1

3

2

,0 0

01

3

2

,

j

H

j

H

0

Equating here the free terms and coefficients separately for identical exponents, we obtain the followingequations:

∫− ( ) ( ) − ( ) ( ) = ( )A x y x t K x t s y s t s H x t, , , , d , ,x

x

0 0 0

0

(12)

[ ( ) − ( )] ( ) = ( ) =λ x A x y x t H x t j, , , 1, 3,j j j (12 j)

[( ( )) − ( )] ( ) = ( ) ≤ ∣ ∣ ≤m λ x A x y x t H x t m N, , , , 2 .m mH (12m)

Since ( ) ≠A x 0, equation (12) can be written as

∫( ) = (− ( ) ( )) ( ) − ( ) ( )− −y x t A x K x t s y s t s A x H x t, , , , d , .x

x

01

01

0

0

(120)

Due to the smoothness of the kernel − ( ) ( )−A x K x t s, ,1 and heterogeneity − ( ) ( )−A x H x t,10 , this Volterra inte-

gral equation has a unique solution ( ) ∈ ([ ] × [ ] )∞y x t C x X T C, , 0, ,0 0 . The equations ( )122 and ( )123 also haveunique solutions

( ) = [ ( ) − ( )] ( ) ∈ ([ ] × [ ] ) =− ∞y x t λ x A x H x t C x X T jC, , , 0, , , 2, 3,j j j1

0 (13)

since ( ) ( )λ x λ x,2 3 are not equal to ( )A x . The equation ( )121 is solvable in space ([ ] × [ ] )∞C x X T C, 0, ,0 if andonly ( ( ) ) ≡ ∀( ) ∈ [ ] × [ ]H x t e x t x X T, , 0 , , 0,τ

1 01 hold. It is not difficult to see that these identities coincidewith identities (10). Furthermore, since ( ( )) ≠ ( ) ∀ ∉ ⋃ =m λ x λ x m, , Γj j j0

3 , = ∣ ∣ ≥j m0, 3, 2 (see (4)), the equa-tions ( )12m has a unique solution

( ) = [( ( )) − ( )] ( ) ∈ ([ ] × [ ] ) ∀∣ ∣ ≥ ∉ ⋃− ∞

=y x t m λ x A x H x t C x X T m mC, , , , 0, , , 2, Γ .m m

jj

10

0

3

Thus, condition (10) is necessary and sufficient for the solvability of equations (9) in the space U .The Theorem 1 is proved. □

Remark 1. If identity (10) holds, then under conditions (1), (2), equation (9) has the following solution inthe space U :


∑ ∑

∑

( ) = ( ) + ( ) + ( )

≡ ( ) + ( ) + ( ) + ( ) + ( )

= ≤∣ ∣≤

∗( )

≤∣ ∣≤

∗( )

y x t τ y x t y x t e y x t e

y x t α x t e h x t e h x t e P x t e

, , , , ,

, , , , , ,

jj

τ

m N

m m τ

τ τ τ

m N

m m τ

01

3

2

,

0 1 21 312

,

j

H

H

1 2 3

(14)

where ( ) ∈ ([ ] × [ ] )∞α x t C x X T C, , 0, ,1 0 are arbitrary function, ( )y x t,0 is the solution of an integral equation( )120 , and introduced notations

( ) ≡ ( )( ) − ( )

( ) ≡ ( )( ) − ( )

( ) ≡ [( ( )) − ( )] ( )−h x t H x tλ x λ x

h x t H x tλ x λ x

P x t m λ x λ x H x t, , , , , , , , , .m m21

2

2 131

3

3 11

1

4 The unique solvability of the general iterative problem inthe space U: residual term theorem

Let us proceed to the description of the conditions for the unique solvability of equation (9) in the spaceU .Along with problem (9), we consider the following equation:

( ) = −∂∂

+ ( ) ( + ) + + ( )Ly x t τ yx

g x e σ e σ By R y Q x t τ, ,2

, , ,τ τ1 2 12 3 (15)

where = ( )y y x t τ, , is the solution (14) of equation (9), ( ) ∈Q x t τ U, , is the well-known function of the spaceU . The right part of this equation:

∑ ∑

∑ ∑

∑ ∑

( ) ≡ −∂∂

+ ( ) ( + ) + + ( )

= − ∂∂

( ) + ( ) + ( )

+ ( ) ( + ) ( ) + ( ) + ( )

+ ( ) + ( ) + ( ) + ( )

= ≤∣ ∣≤

∗( )

= ≤∣ ∣≤

∗( )

= ≤∣ ∣≤

∗( )

G x t τ yx

g x e σ e σ By R y Q x t τ

xy x t y x t e y x t e

g x e σ e σ B y x t y x t e y x t e

R y x t y x t e y x t e Q x t τ

, ,2

, ,

, , ,

2, , ,

, , , , , ,

τ τ

jj

τ

m N

m m τ

τ τ

jj

τ

m N

m m τ

jj

τ

m N

m m τ

1 2 1

01

3

2

,

1 2 01

3

2

,

1 01

3

2

,

j

H

j

H

j

H

2 3

2 3

may not belong to the spaceU , if = ( ) ∈y y x t τ U, , . Indeed, taking into account the form (14) of the function= ( ) ∈y y x t τ U, , , we consider in ( )G x t τ, , , for example, the terms

∑ ∑

∑

∑

( ) ≡ ( ) ( + ) ( ) + ( ) + ( )

= ( ) ( )( + ) + ( ) ( )( + )

+ ( ) ( + ) ( )

= ≤∣ ∣≤

∗( )

=

+ +

≤∣ ∣≤

∗( )

Z x t τ g x e σ e σ B y x t y x t e y x t e

g x By x t e σ e σ g x By x t e σ e σ

g x e σ e σ B y x t e

, ,2

, , ,

2,

2,

2, .

τ τ

jj

τ

m N

m m τ

τ τ

jj

τ τ τ τ

τ τ

m N

m m τ

1 2 01

3

2

,

0 1 21

3

1 2

1 22

,

j

H

j j

H

2 3

2 3 2 3

2 3

Here, for example, terms with exponents

= ∣ ( = + = )+ ( )=( )

+( )e e e m m m, if 0, 1 ,τ τ m τm

τ m τ,0,1,1

,1 2 32 3 2

( = + = ) ( = = )+( ) +( )e m m m e m m mif 0, 1 , if 0, ,τ m τ τ m τ,1 3 2

,1 2 33 2 (∗)

( = = ) ( = = )+( ) +( )e m m m e m m mif 0, , if 1, ,τ m τ τ m τ,1 2 3

,1 2 33 2

( = = )+( )e m m mif 1,τ m τ,1 2 33


do not belong to the space U , since multi-indexes( ) ∈ ( + ) ∈ ( + ) ∈ ∀ ∈n n n n n n n N0, , Γ , 0, 1, Γ , 0, , 1 Γ0 1 2

are resonant. Then, according to the well-known theory (see [6, p. 234]), we embed these terms in the spaceU according to the following rule (see (∗)):

= = = = ( = + = )+ +( )e e e e m m m1, 1 if 0, 1 ,τ τ τ m τ0 , 01 2 32 3 2

= = ( = + = )+( )e e m m m1 if 0, 1 ,τ m τ, 01 3 23

= ( = = ) = ( = = )+( ) +( )e e m m m e e m m mif 0, , if 0, ,τ m τ τ τ m τ τ,1 2 3

,1 2 32 2 3 3

( = = ) = = ( = = )+( ) +( )e m m m e e e m m mif 1, , if 1, .τ m τ τ τ m τ τ,1 2 3

,1 2 32 1 3 1

In other words, terms with resonant exponentials ( )e m τ, replaced by members with exponents e e e e, , ,τ τ τ0 1 2 3

according to the following rule:

∣ = = ∣ = ∣ = ∣ =( )∈

( )∈

( )∈

( )∈e e e e e e e e1, , , .m τ

mm τ

mτ m τ

mτ m τ

mτ,

Γ0 ,

Γ,

Γ,

Γ0 11

22

33

After embedding, the right-hand side of equation (15) will look like

∑ ∑ ∑ ∑( ) = − ∂∂

( ) + ( ) + ( ) + ( ) + ( )= ≤∣ ∣≤

∗( )

= ∈

G x t τx

y x t y x t e y x t e y x e Q x t τ, , , , , , , .j

jτ

m N

m m τ

j m

m τ0

1

3

2

,

0

3

Γ

j

H j j

jj

As indicated in [6], the embedding ( ) → ( )G x t τ G x t τ, , , , will not affect the accuracy of the construction ofasymptotic solutions of problem (2), since ( )G x t τ, , at = ( )τ ψ x

ε coincides with ( )G x t τ, , .

Theorem 2. Let conditions (1) and (2) be fulfilled and the right-hand side ( ) = ( ) + ∑ ( ) +=H x t τ H x t H x t e, , , ,j jτ

0 13 j

∑ ( ) ∈≤∣ ∣≤∗ ( )H x t e U,m N

m m τ2

,H

of equation (9) satisfy condition (10). Then, problem (9) under additional conditions⟨ ( ) ⟩ ≡ ∀( ) ∈ [ ] × [ ]G x t τ e x t x X T, , , 0 , , 0, ,τ

01 (16)

where ( ) = ( ) + ∑ ( ) + ∑ ( )= ≤∣ ∣≤∗ ( )Q x t τ Q x t Q x t e Q x t e, , , , ,k k

τm N

m m τ0 1

32

,iz

is the known function of the space U ,

is uniquely solvable inU .

Proof. Since the right-hand side of equation (9) satisfies condition (10), this equation has a solution inthe spaceU in the form (14), where ( ) ∈ ([ ] × [ ] )∞α x t C x X T C, , 0, ,1 0 is the arbitrary function. Submit (14) tothe initial condition ( ) = ( )∗y x t y t, , 00 . We get ( ) = ( )∗α x t y t,1 0 , where denoted

∑( ) = ( ) + ( ) ( ) − ( )( ) − ( )

− ( )( ) − ( )

− [( ( )) − ( )] ( )∗∗ −

≤∣ ∣≤

∗−y t y t A x H x t H x t

λ x λ xH x t

λ x λ xm λ x A x H x t, , , , , .

m N

m10 0 0

2 0

2 0 1 0

3 0

3 0 1 0 20 0

10

H

Now we subordinate the solution (14) to the orthogonality condition (16). We write ( )G t τ, in more detailthe right side of equation (9):

∑

∑

∑

( ) ≡ − ∂∂

( ) + ( ) + ( ) + ( ) + ( )

+ ( ) ( + ) ( ) + ( ) + ( ) + ( ) + ( )

+ ( ) + ( ) + ( ) + ( ) + ( ) + ( )

≤∣ ∣≤

∗( )

≤∣ ∣≤

∗( )

≤∣ ∣≤

∗( )

G x t τx


g x e σ e σ B y x t α x t e h x t e h x t e P x t e

R y x t α x t e h x t e h x t e P x t e Q x t τ

, , , , , , ,

2, , , , ,

, , , , , , , .

τ τ τ

m N

m m τ

τ τ τ τ τ

m N

m m τ

τ τ τ

m N

m m τ

0 1 21 312

,

1 2 0 1 21 312

,

1 0 1 21 312

,

H

H

H

1 2 3

2 3 1 2 3

1 2 3

Embedding this function into the space U , we will have


∑

∑

∑

( ) ≡ − ∂∂

( ) + ( ) + ( ) + ( ) + ( )

+ ( ) ( + ) ( ) + ( ) + ( ) + ( ) + ( )

+ ( ) + ( ) + ( ) + ( ) + ( ) + ( )

≤∣ ∣≤

∗( )

≤∣ ∣≤

∗( )

≤∣ ∣≤

∗( )

G x t τx


g x e σ e σ B y x t α x t e h x t e h x t e P x t e


, , , , , , ,

2, , , , ,

, , , , , , , .

τ τ τ

m N

m m τ

τ τ τ τ τ

m N

m m τ

τ τ τ

m N

m m τ

0 1 21 312

,

1 2 0 1 21 312

,

1 0 1 21 312

,

H

H

H

1 2 3

2 3 1 2 3

1 2 3

Embedding this function into the space U , we will have

∑

∑ ∑

∑ ∑

∑

∑

∑ ∑

∑

( ) ≡ − ∂∂

( ) + ( ) + ( ) + ( ) + ( )

+ ( ) ( ) + ( ) ( ) ( ) ( ) + ( ) ( )

+ ( ) ( ) + ( ) ( )

+ ( ) + ( ) + ( ) + ( ) + ( ) + ( )

= − ∂∂

( ) + ( ) + ( ) + ( ) + ( )

+ ( ) ( ) + ( ) + ( ) + ( ) + ( )

+ ( ) + ( ) + ( )

+ ( ) + ( )

+ ( ) + ( ) + ( ) + ( ) + ( ) + ( )

≤∣ ∣≤

∗( )

=

+

=

+

≤∣ ∣≤

∗( )+

≤∣ ∣≤

∗( )+

∧

≤∣ ∣≤

∗( )

≤∣ ∣≤

∗( )

+ +

+ +

≤∣ ∣≤

∗( )+

≤∣ ∣≤

∗( )+

∧

≤∣ ∣≤

∗( )

G x t τx


g x By x t e σ g x By x t e σ g x By x t e σ g x By x t e σ

g x By x t e σ g x By x t e σ


xy x t α x t e h x t e h x t e P x t e

B g x y x t e σ y x t e σ α x t e σ h x t e σ h x t e σ

α x t e σ h x t e σ h x t e σ

P x t e σ P x t e σ


ˆ , , , , , , ,

2,

2,

2,

2,

2,

2,

, , , , , , ,

, , , , ,

2, , , , ,

, , ,

, ,

, , , , , , , .

τ τ τ

m N

m m τ

τ τ

jj

τ τ

jj

τ τ

m N

m m τ τ

m N

m m τ τ

τ τ τ

m N

m m τ

τ τ τ

m N

m m τ

τ τ τ τ τ τ τ

τ τ τ τ τ

m N

m m τ τ

m N

m m τ τ

τ τ τ

m N

m m τ

0 1 21 312

,

0 1 0 21

3

11

3

2

2

,1

2

,2

1 0 1 21 312

,

0 1 21 312

,

0 1 0 2 1 1 212

1 31 1

1 2 21 2 312

2

2

,1

2

,2

1 0 1 21 312

,

H

j j

H H

H

H

H H

H

1 2 3

2 3 2 3

2 3

1 2 3

1 2 3

2 3 1 2 2 3 2

1 3 2 3 3

2 3

1 2 3

The embedding operation acts only on resonant exponentials, leaving the coefficients unchanged at theseexponents. Given that the expression

∑( ) + ( ) + ( ) + ( ) + ( )≤∣ ∣≤

∗( )R y x t α x t e h x t e h x t e P x t e, , , , , ,τ τ τ

m N

m m τ1 0 1 21 31

2

,

H

1 2 3

linearly depends on ( )α x t,1 (see formula ( )51 ), we also conclude that after the embedding operation thefunction ( )G t τˆ , will linearly depend on the scalar function ( )α x t,1 . Given that in condition (16) scalarmultiplication by functions eτ1, containing only the exponent eτ1, in the expression for ( )G t τ, , it is necessaryto keep only the term with the exponent eτ1. Then, condition (16) takes the form

( ) ∑− ∂

∂( ) + ( ( ) ) + ( ) = ∀( ) ∈ [ ] × [ ]

∣ ∣= ∈xα x t e w α t t e Q x t e e x t x X T, , , , 0 , , 0, ,τ

m m

Nm τ τ τ

12: Γ

1 1 01

1 1 1

11 1 1

where ( ( ) )w α t t,m1

1are some functions linearly dependent on ( )α x t,1 .

Performing scalar multiplication here, we obtain a linear ordinary differential equation (relative x) for afunction ( )α x t,1 . Given the initial condition ( ) = ( )∗α x t y t,1 0 , found above, we find uniquely the function

( ) ∈ ([ ] × [ ])∞α x t C x X T, , 0,1 0 , and therefore, we will uniquely construct a solution to equation (9) in thespace U . The theorem is proved.


As mentioned earlier, the right-hand sides of iterative problems ( )8k (if solved sequentially) may notbelong to the space U . Then, according to [6, p. 234], the right-hand sides of these problems must beembedded into U , according to the above rule. As a result, we obtain the following problems:

∑( ) ≡ ( )∂∂

− ( ) − = ( ) ( ) = ( ) ( )=

Ly x t τ λ xyτ

A x y R y h x t y x t ε y t, , , , , , 0, ; 8j

jj

01

30

0 0 0 2 0 00

0

( ) = −∂∂

+ ( )( + ) + + ( ) ( − ) ( ) = ( )∧

Ly x t τyx

g x e σ e σ By R y h x ti

e σ e σ y x t, ,2

,2

, , , 0 0; 8τ τ τ τ1

02 1 3 2 0 1 0

12 1 3 2 1 0 1

( )( ) = −∂∂

+ ( ) + + + ( ) = ( )∧

Ly x t τyx

g x e σ e σ By R y R y y x t, ,2

, , , 0 0; 8τ τ2

11 2 1 1 1 2 0 2 0 22 3

⋯

( )( ) = −∂

∂+ ( ) + + + ⋯+ ( ) = ≥ ( )−

−

∧

−Ly x t τy

xg x e σ e σ By R y R y y x t k, ,

2, , , 0 0, 1 8k

k τ τk k k k k

11 2 1 0 1 1 02 3

(images of linear operators ∂∂t

and Rν do not need to be embedding in the space U , since these operatorsoperate from U to U ). Such a change will not affect the construction of the asymptotic solution of theoriginal problem (1) (or the equivalent problem (2)), so on the restriction = ( )τ ψ x

ε series of problems ( )8̄k willcoincide with a series of problems ( )8k (see [6, pp. 234–235]. □

Applying Theorems 1 and 2 to iterative problems ( )8̄k , we find uniquely their solutions in the space Uand construct series (6). Just as in [6], we prove the following statement:

Theorem 3. Suppose that conditions (1) and (2) are satisfied for equation (2). Then, when ∈ ( ]ε ε0, 0 ( >ε 00

is sufficiently small), equation (2) has a unique solution ( ) ∈ ([ ] × [ ] )y x t ε C x X T C, , , 0, ,10 , in this case,

the estimate

∥ ( ) − ( )∥ ≤ = …([ ]×[ ])+y x t ε y x t c ε N, , , , 0, 1, 2, ,εN C x X T N

N, 0,

10

holds true, where ( )y x t,εN is the restriction ( )= ( )for τ ψ xε of the N -partial sum of series (6) with coefficients

( ) ∈y x t τ U, ,k , satisfying the iteration problems ( )8̄k , and the constant >c 0N does not depend on ∈ ( ]ε ε0, 0 .

5 Construction of the solution of the first iteration problem

Using Theorem 1, we will try to find a solution to the first iteration problem ( )8̄0 . Since the right side ( )h x t,2

of the equation ( )8̄0 satisfies condition (10), this equation has (according to (14)) a solution in the space Uin the form:

( ) = ( ) + ( )( ) ( )y x t τ y x t α x t e, , , , ,τ0 0

01

0 1 (17)

where ( ) ∈ ([ ] × [ ] )( ) ∞α x t C x X T C, , 0, ,10

0 are arbitrary function and ( )( )y x t,00 is the solution of the integral

equation

∫( ) = (− ( ) ( )) ( ) − ( ) ( )( ) − ( ) −y x t A x K x t s y s t s A x h x t, , , , d , .x

x

00 1

00 1

2

0

(18)

Subordinating (17) to the initial condition ( ) = ( )y x t y t, , 00 00 , we have

( ) + ( ) = ( ) ⇔ ( ) = ( ) − ( ) ⇔ ( ) = ( ) + ( ) ( )( ) ( ) ( ) ( ) ( ) −y x t α x t y t α x t y t y x t α x t y t A x h x t, , , , , , .00

0 10

00

10

00

00

0 10

00 1

0 2 0

To fully compute the function ( )( )α x t,10 , we proceed to the next iteration problem ( )8̄1 . Substituting into it

the solution (17) of the equation ( )8̄0 , we arrive at the following equation:


( ) = − ∂∂

( ) − ∂∂

( ( )) + ( ) ( + ) ( ( ) + ( ) )

+( ) ( )

( )−

( ) ( )( )

+ ( )( − )

( ) ( ) ( ) ( )∧

( ) ( )

Ly x t τx

y x tx

α x t e g x e σ e σ B y x t α x t e

K x t x α x tλ x

e K x t x α x tλ x i

h x t e σ e σ

, , , ,2

, ,

, , , , , , 12

, ,

τ τ τ τ

τ τ τ

1 00

10

1 2 00

10

10

1

0 10

0

1 01 1 2

1 2 3 1

1 2 3

(here we used the expression ( )51 for ( )R y x t τ, ,1 and took into account that when ( ) = ( )y x t τ y x t τ, , , ,0 thesum ( )51 contains only terms with eτ1).

Let us calculate

= ( ) ( + )( ( ) + ( ) )

= ( ){ ( ) + ( ) + ( ) + ( ) }

( ) ( )∧

( ) ( ) ( ) + ( ) + ∧

M B g x e σ e σ y x t α x t e

B g x σ y x t e σ y x t e σ α x t e σ α x t e

2, ,

2, , , , .

τ τ τ

τ τ τ τ τ τ

1 2 00

10

1 00

2 00

1 10

2 10

2 3 1

2 3 2 1 3 1

Let us analyze the exponents of the second dimension included here for their resonance:

∫ ∫∣ =

− ′ + = − ′+ ′

⇔ ∅

∣ =

+ ′ + = − ′+ ′

⇔ ∅

+= ( ) ∕

(− ′( )+ ( )) += ( ) ∕

(+ ′( )+ ( ))e e

iβ AA

iβiβ

e e

iβ AA

iβiβ

,0,

,,,

;

,0,

,,,

.

τ ττ ψ x ε

iβ θ A θ θ τ ττ ψ x ε

iβ θ A θ θd dεt

tε

t

t

2 11

0 3 11

0

Thus, exponents +eτ τ2 1 and +eτ τ3 1 are not resonant. Then, for solvability, equation (18), it is necessary andsufficient that the condition

− ∂∂

( ( )) +( ) ( )

( )=( )

( )

xα x t K x t x α x t

λ x, , , , 01

0 10

1

is satisfied. Attaching the initial condition

( ) = ( ) + ( ) ( )( ) −α x t y t A x h x t, , ,10

00 1

0 2 0

to this equation, we find

( )∫( ) = ( )( ) ( )( )

( )α x t α x t e, , ,s

10

10

0d

x

x K s t sλ s

0

, ,1

and therefore, we uniquely calculate the solution (17) of the problem ( )8̄0 in the spaceU . Moreover, the mainterm of the asymptotic of the solution to problem (2) has the form

( )∫ ∫( ) = ( ) + ( )( ) ( ) + ( )( ( ))( )y x t y x t α x t e, , , ,ε

s λ θ θ0 0

01

00

d dx

x K s t sλ s ε

x

x

0

, ,1

1

01 (19)

where ( ) = ( ) + ( ) ( )( ) −α x t y t A x h x t, ,10

00 1

0 2 0 , ( )( )y x t,00 is the solution of the integrated equation (18). From

expression (19) for ( )y x t,ε0 , it is clear that ( )y x t,ε0 is independent of rapidly oscillating terms. However,already in the next approximation, their influence on the asymptotic solution of problem (1) is revealed.Indeed, in view of condition (10), equation ( )8̄1 will be written as

( )

= −∂ ( )

∂+ ( ) ( + ) ( ( ) + ( ) )

−( ) ( )

( )+ ( ) −

= −∂ ( )

∂+ ( ) ( ( ) + ( ) )

+ ( ) ( ( ) + ( ) )

−( ) ( )

( )+ ( ) − ( )

( )( ) ( )

∧

( )

( )( ) ( )

( ) + ( ) +∧

( )

Lyy x t

xg x e σ e σ B y x t α x t e

K x t x α x tλ x

h x ti

e σ e σ

y x tx

g x B y x t σ e y x t σ e

g x B α x t σ e α x t σ e

K x t x α x tλ x

σi

h x t e σi

h x t e

,2

, ,

, , , ,2

,2

, ,

2, ,

, , ,2

,2

, .

τ τ τ

τ τ

τ τ

τ τ τ τ

τ τ

10

0

1 2 00

10

0 10

0

1 0

11 2

00

00

1 00

2

10

1 10

2

0 10

0

1 0

11

21

2 3 1

2 3

2 3

1 2 1 3

2 3

(20)


Since there are no resonance exponents on the right side ( )( )H x t τ, ,1 of this equation, then ( ) ∈( )H x t τ U, ,1 .By Theorem 1, equation (20) has the following solution in the space U (see (14)):

( ) = ( ) + ( ) +( )

( ) − ( )+

( )( ) − ( )

+ ( )( )

( ) + ( )( )

( )

( ) ( )( ) ( )

( ) + ( ) +

y x t τ y x t α x t e H x tλ x λ x

eH x t

λ x λ xe

g xλ x

Bσ α x t e g xλ x

Bσ α x t e

, , , , , ,

2,

2, ,

τ τ τ

τ τ τ τ

1 10

11 2

1

2 1

31

3 1

21 1

0

32 1

0

1 2 3

1 2 1 3

(21)

where ( ) ∈ ([ ] × [ ])( ) ∞α x t C x X T, , 0,11

0 is an arbitrary function determined from the solvability condition (10)of the equation ( )8̄2 in the space U , the function ( )( )y x t,1

0 and the function

( ) =[ ( ) ( ) − ( )]

( ) =[ ( ) ( ) + ( )]

( )( )

( )( )

H x tσ g x By x t ih x t

H x tσ g x By x t ih x t

,, ,

2,

,, ,

2

21 1 0

01

31 2 0

01

are computed uniquely. From (21), it is seen that ( )( )y x t, , ψ xε1 depends on rapidly oscillating exponents

( ( )− ( ))±e β x β xiε 0 , i.e., an already asymptotic solution

( ) =

( )+

( )y x t y x t ψ xε

εy x t ψ xε

, , , , ,ε1 0 1

of the first order depends on rapidly oscillating terms in equation (1).

6 Conclusion

The function ( )y x t,ε0 shows that when passing from a differential equation of type (1) ( ( ) ≡K x t s, , 0) to anintegro-differential one ( ( ) ≠K x t s, , 0), the main term of the asymptotic is influenced by the kernel ( )K x t s, ,of the integral operator. Their effects are detected when constructing the next approximation ( )y x t,ε1 .

Funding information: This work was supported by grant no. AP05133858 “Contrast structures in singularlyperturbed equations and their application in the theory of phase transitions” of the Committee of Science ofthe Ministry of Education and Science of the Republic of Kazakhstan.

Conflict of interest: The authors state no conflict of interest.

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