Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro- differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity

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


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][2][3], and M. Imanaliev [4,5]. These work argue the importance of such research for theory and applications. However, before the appearance of work related to the regularization method, S. Lomov [6][7][8], integro-differential equations were considered under the conditions of the spectrum of the matrix of the first variation (on a degenerate solution) lying in the open left half-plane, which significantly narrowed the scope of the above work in problems with purely imaginary points of the spectrum. And only after the development of the method of Lomov, it became possible to consider problems with a spectrum lying on an imaginary axis [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Note that the Lomov regularization method was mainly used for ordinary singularly perturbed differential and integro-differential equations [25][26][27][28][29][30]. The development of this method for integro-differential equations with partial derivatives was carried out by A. Bobodzhanov, V. Safonov, and B. Kalimbetov [31][32][33][34][35][36][37][38]. In the study of problems with a slowly changing kernel, it turned out that the regularization procedure and the construction of a regularized asymptotic solution essentially depend on the type of integral operator. The most difficult case was when the upper limit of the integral is not a differentiation variable. For the integral operator with an upper limit coinciding with the differentiation variable, the scalar case is investigated. The case when the upper limit of the integral operator coincides with the differentiation variable is studied for equations of partial differential integro-differential equations with an integral operator, the kernel of which contains a rapidly changing exponential factor. Summarizing the results of work for integro-differential equations with onedimensional integrals, the problem of constructing a regularized asymptotic solution of the problem for integro-differential equations with two independent variables is investigated. In the present paper, we consider a singularly perturbed partial differential integro-differential equation with high-frequency coefficients and with rapidly oscillating coefficients, and rapidly oscillating heterogeneity that generates essentially special singularities in the solution of the problem.
In this paper, we consider the Cauchy problem for the integro-differential equation with partial derivatives: L y x t ε ε y x A x y K x t s y s t ε s εg x β x ε By 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 B 2 , 0 for ordinary equations in the absence of an integral term was considered in [39][40][41][42][43][44]. The limiting operator ( ) A x has a spectrum ( ) = ( ) 2 3 will be called the spectrum of a rapidly oscillating coefficient. We assume that the following conditions are fulfilled: We will develop an algorithm for constructing a regularized asymptotic solution [6] of problem (1).

Regularization of problem (1)
Introduce the regularized variables: and instead of problem (2), consider the problem s y s t  ψ s  ε  ε s ε  g x e σ e σ By   ε  h x t  i  e σ e σ  h x t  y x t τ ε  ỹ˜˜,   ,˜, ,  , d  2,   2 , ,˜, , , , 3 . It is clear that if = ( ) y y x t τ ε˜, , , is a solution of problem (3), then the function 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 Definition 1. A class M ε is said to be asymptotically invariant (with → + ε 0) with respect to an operator P 0 if the following conditions are fulfilled: .
From this definition, it can be seen that the class M ε depends on the space U , in which the operator P 0 is defined. In our case = P J 0 . Before describing the space U , we introduce the sets of resonant multi-indices. We introduce the nota- , 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 , it occurs only on the non-resonant multi-indexes, i.e., ∉ ⋃ = m Γ j j 0 3 , = ( ) σ σ σ , 1 2 . Note that here the degree N y of the polynomial ( ) y x t τ σ , , , relative to the exponentials e τ j depends on the element y. In addition, the elements of space U depend on bounded in > ε 0 terms of constants = ( ) σ σ ε and which do not affect the development of the algorithm described below; therefore, in the record of element (4) of this space U , we omit the dependence on = ( ) σ σ σ ,

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 . Apply the operation of integration by parts to the second term  Continuing this process, we obtain the series Applying the integration operation in parts to integrals is possible. Performing it, we will have:   where the operators are introduced. Therefore, the image of the operator J on the element (4) of the space U is represented as a series

Jy x t τ K x t s y s t s ε I K x t s y s t e I K x t s y s t ε I K x t s y s t e I K x t s y s t
, , , It is easy to show (see, for example, [46, pp. 291-294]) that this series converges asymptotically for → + ε 0 ). This means that the class M ε is asymptotically invariant (for → + ε 0) with respect to the operator J .
We introduce operators of the form (4) according to the law:

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
, , , Now let ( ) y x t τ ε , , , be an arbitrary continuous function on ( ). Then, the image ( ) Jy x t τ ε , , , of this function is decomposed into an asymptotic series This equality is the basis for introducing an extension of an operator J on series of the form (6): Although the operator J is formally defined, its utility is obvious; since in practice, it is usual to construct the N th approximation of the asymptotic solution of problem (2), which impose only N th partial sums of the series (6), which have not a formal, but a true meaning. Now you can write a problem that is completely regularized with respect to the original problem (2): 3 Iterative problems and their solvability in the space U Substituting the series (6) into (7) and equating the coefficients of the same powers of ε, we obtain the following iterative problems:  , is the known function of space U , ( ) * y t is the known function of the complex space C, and the operator R 0 has the form (see ( ) Proof. We will determine the solution of equation (9) as an element (4)

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
, , , Equating here the free terms and coefficients separately for identical exponents, we obtain the following equations: , , , d , ,

y x t A x K x t s y s t s A x H x t ,
,, , d , .
hold. It is not difficult to see that these identities coincide with identities (10). Furthermore, since ( 2 (see (4)), the equations ( ) 12 m has a unique solution

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
, , , where are arbitrary function, ( ) y x t , 0 is the solution of an integral equation ( ) 12 0 , and introduced notations Let us proceed to the description of the conditions for the unique solvability of equation (9) in the space U . Along with problem (9), we consider the following equation: is the well-known function of the space U . The right part of this equation: In other words, terms with resonant exponentials ( ) e m τ , replaced by members with exponents e e e e , , , according to the following rule: After embedding, the right-hand side of equation (15) will look like As indicated in [6], the embedding , , will not affect the accuracy of the construction of asymptotic solutions of problem (2), since ( ) Theorem 2. Let conditions (1) and (2) be fulfilled and the right-hand side ( )= ( )+∑ , , Proof. Since the right-hand side of equation ( (14) to the orthogonality condition (16). We write ( ) G t τ , in more detail the right side of equation (9):

G x t τ x y x t α x t e h x t e h x t e P x t e 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 , , , Embedding this function into the space U , we will have where are some functions linearly dependent on ( ) α x t , 1 . Performing scalar multiplication here, we obtain a linear ordinary differential equation (relative x) for a function ( ) α x t , 1 . Given the initial condition ( ) = ( ) * α x t y t , 1 0 , found above, we find uniquely the function , and therefore, we will uniquely construct a solution to equation (9) in the space U . The theorem is proved.
As mentioned earlier, the right-hand sides of iterative problems ( ) 8 k (if solved sequentially) may not belong to the space U . Then, according to [6, p. 234], the right-hand sides of these problems must be embedded into U , according to the above rule. As a result, we obtain the following problems: , , , , 0, ; 8 and R ν do not need to be embedding in the space U , since these operators operate from U to U ). Such a change will not affect the construction of the asymptotic solution of the original problem (1) (or the equivalent problem (2)), so on the restriction = ( ) τ ψ x ε series of problems ( ) 8 k will coincide with a series of problems ( ) 8 k (see [6, pp. 234-235]. □ Applying Theorems 1 and 2 to iterative problems ( ) 8 k , we find uniquely their solutions in the space U and 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).
, in this case, the estimate

Construction of the solution of the first iteration problem
Using Theorem 1, we will try to find a solution to the first iteration problem ( ) 8 0 . Since the right side ( ) h x t , 2 of the equation ( ) 8 0 satisfies condition (10), this equation has (according to (14)) a solution in the space U in the form:

Conclusion
The function ( ) y x t , ε0 shows that when passing from a differential equation of type (1) ( ( )≡ K x t s , , 0) to an integro-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 ,
Funding information: This work was supported by grant no. AP05133858 "Contrast structures in singularly perturbed equations and their application in the theory of phase transitions" of the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan.

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