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

  • Burkhan T. Kalimbetov EMAIL logo and Olim D. Tuychiev
Published/Copyright: May 10, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

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.

Keywords: singularly perturbed; integro-partial differential equation; regularization of an integral; space of non-resonant solutions; iterative problems; solvability of iterative problems
MSC 2010: 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 [1,2,3], 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 one-dimensional 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:

(1) L ε y ( x , t , ε ) ≡ ε ∂ y ∂ x − A ( x ) y − ∫ x 0 x K ( x , t , s ) y ( s , t , ε ) d s − ε g ( x ) cos β ( x ) ε B y = ε h 1 ( x , t ) sin β ( x ) ε + h 2 ( x , t ) , y ( x 0 , t , ε ) = y 0 ( t ) ( ( x , t ) ∈ [ x 0 , X ] × [ 0 , T ] ) ,

where A ( x ) , g ( x ) , h 1 ( x , t ) , h 2 ( x , t ) , y 0 ( t ) , β ′ ( x ) > 0 ( x ∈ [ x 0 , X ] ) 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 ) = 2 γ ( x ) , B ≡ 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 λ 1 ( x ) = A ( x ) , β ′ ( x ) is a frequency of rapidly oscillating cos β ( t ) ε . In the following, functions λ 2 ( x ) = − i β ′ ( x ) , λ 3 ( x ) = + i β ′ ( x ) 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 0 , X ] , R ) , h 1 ( x , t ) , h 2 ( x , t ) ∈ C ∞ ( [ x 0 , X ] × [ 0 , T ] , R ) , y 0 ( t ) ∈ C ∞ ( [ 0 , T ] , R ),

    K ( x , t , s ) ∈ C ∞ ( { x 0 ≤ s ≤ x ≤ X , 0 ≤ t ≤ T } , R ) ;

  2. A ( x ) < 0 ( ∀ x ∈ [ x 0 , X ] ) .

We will develop an algorithm for constructing a regularized asymptotic solution [6] of problem (1).

2 Regularization of problem (1)

Denote by σ j = σ j ( ε ) independent of magnitude σ 1 = e − i ε β ( x 0 ) , σ 2 = e + i ε β ( x 0 ) , and rewrite equation (1) as

(2) L ε y ( x , t , ε ) ≡ ε ∂ y ∂ x − A ( x ) y − ∫ x 0 x K ( x , t , s ) y ( s , t , ε ) d s − ε g ( x ) 2 e − i ε ∫ x 0 x β ′ ( θ ) d θ σ 1 + e + i ε ∫ x 0 x β ′ ( θ ) d θ σ 2 B y = h 2 ( x , t ) + ε h 1 ( x , t ) 2 i e − i ε ∫ x 0 x β ′ ( θ ) d θ σ 1 − e + i ε ∫ x 0 x β ′ ( θ ) d θ σ 2 , y ( x 0 , t , ε ) = y 0 ( t ) .

Introduce the regularized variables:

τ j = 1 ε ∫ x 0 x λ j ( θ ) d θ ≡ ψ j ( x ) ε , j = 1 , 3 ¯ ,

and instead of problem (2), consider the problem

(3) ε ∂ y ˜ ∂ x + ∑ j = 1 3 λ j ( x ) ∂ y ˜ ∂ τ j − A ( x ) y ˜ − ∫ x 0 x K ( x , t , s ) y ˜ s , t , ψ ( s ) ε , ε d s + ε g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y ˜ = ε h 1 ( x , t ) 2 i ( e τ 2 σ 1 − e τ 3 σ 2 ) + h 2 ( x , t ) , y ˜ ( x , t , τ , ε ) ∣ x = x 0 , τ = 0 = y 0 ,

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

J y ˜ = ∫ x 0 x K ( x , t , s ) y ˜ s , t , ψ ( s ) ε , ε d s .

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:

  1. M ε ⊂ D ( P 0 ) for each fixed ε > 0 ;

  2. the image P 0 μ ( x , t , ε ) of any element μ ( x , t , ε ) ∈ M ε decomposes in a power series

    P 0 μ ( x , t , ε ) = ∑ n = 0 ∞ ε n μ n ( x , t , ε ) ( ε → + 0 , μ n ( x , t , ε ) ∈ M ε , n = 0 , 1 , … )

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

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 0 = J .

Before describing the space U , we introduce the sets of resonant multi-indices. We introduce the notations:

λ ( x ) = ( λ 1 ( x ) , λ 2 ( x ) , λ 3 ( x ) ) , ( m , λ ( x ) ) = ∑ j = 1 3 m j λ j ( x ) , ∣ m ∣ = ∑ j = 1 3 m j , Γ 0 = { m : ( m , λ ( x ) ) ≡ 0 , ∀ ∣ m ∣ ≥ 2 } , Γ j = { m : ( m , λ ( x ) ) ≡ λ j ( x ) , ∀ ∣ m ∣ ≥ 2 } , j = 1 , 2 , 3 ,

(the set Γ 0 corresponds to a point of the spectrum λ 0 ( x ) ≡ 0 , 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

(4) y ( x , t , τ , σ ) = y 0 ( x , t , σ ) + ∑ j = 1 3 y j ( x , t , σ ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N y ∗ y m ( x , t , σ ) e ( m , τ ) ,

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

where asterisk ∗ above the sum sign indicates that the summation for ∣ m ∣ ≡ m 1 + m 2 + m 3 ≥ 2 , it occurs only on the non-resonant multi-indexes, i.e., m ∉ ⋃ j = 0 3 Γ j , σ = ( σ 1 , σ 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 1 = σ 1 ( ε ) and σ 2 = σ 2 ( 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 1 , σ 2 ) for brevity. We show that the class M ε = U ∣ τ = ψ ( x ) ∕ ε is asymptotically invariant with respect to the operator J . The image of the integral operator J on an arbitrary element y ( x , t , τ ) of the space U has the form

J y ( x , t , τ ) = ∫ x 0 x K ( x , t , s ) y 0 ( s , t ) d s + ∑ j = 1 3 ∫ x 0 x K ( x , t , s ) y j ( s , t ) e 1 ε ∫ x 0 s λ j ( θ ) d θ d s + ∑ 2 ≤ ∣ m ∣ ≤ N y ∗ ∫ x 0 x K ( x , t , s ) y m ( s , t ) e 1 ε ∫ x 0 s ( m , λ ( θ ) ) d θ d s .

Apply the operation of integration by parts to the second term

J j ( x , t , ε ) = ε ∫ x 0 x K ( x , t , s ) y j ( s , t ) λ j ( s ) d e 1 ε ∫ x 0 s λ j ( θ ) d θ = ε K ( x , t , s ) y j ( s , t ) λ j ( s ) e 1 ε ∫ x 0 s λ j ( θ ) d θ s = x 0 s = x − ∫ x 0 x ∂ ∂ s K ( x , t , s ) y j ( s , t ) λ j ( s ) e 1 ε ∫ x 0 s λ j ( θ ) d θ d s = ε K ( x , t , x ) y j ( x , t ) λ j ( x ) e 1 ε ∫ x 0 x λ j ( θ ) d θ − K ( x , t , x 0 ) y j ( x 0 , t ) λ j ( x 0 ) − ε ∫ x 0 x ∂ ∂ s K ( x , t , s ) y j ( s , t ) λ j ( s ) e 1 ε ∫ x 0 s λ j ( θ ) d θ d s .

Continuing this process, we obtain the series

J j ( x , t , ε ) = ∑ ν = 0 ∞ ( − 1 ) ν ε ν + 1 ( I j ν ( K ( x , t , s ) y j ( s , t ) ) ) s = x e 1 ε ∫ x 0 x λ j ( θ ) d θ − ( I j ν ( K ( x , t , s ) y j ( s , t ) ) ) s = x 0 ,

where the operators

I j 0 = 1 λ j ( s ) , I j ν = 1 λ j ( s ) ∂ ∂ s I j ν − 1 ( ν ≥ 1 , j = 1 , 3 ¯ )

are introduced.

Applying the integration operation in parts to integrals

J m ( x , t , ε ) = ∫ x 0 x K ( x , t , s ) y m ( s , t ) e 1 ε ∫ x 0 s ( m , λ ( θ ) ) d θ d s ,

we note that for all multi-indices m = ( m 1 , m 2 , m 3 ) , m ∉ ⋃ j = 0 3 Γ j inequalities

( m , λ ( x ) ) ≠ 0 ∀ x ∈ [ x 0 , X ]

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

J m ( x , t , ε ) = ε ∫ x 0 x K ( x , t , s ) y m ( s , t ) ( m , λ ( s ) ) d e 1 ε ∫ x 0 s ( m , λ ( θ ) ) d θ = ε K ( x , t , s ) y m ( s , t ) ( m , λ ( s ) ) e 1 ε ∫ x 0 s ( m , λ ( θ ) ) d θ s = x 0 s = x − ∫ x 0 x ∂ ∂ s K ( x , t , s ) y m ( s , t ) ( m , λ ( s ) ) e 1 ε ∫ x 0 s ( m , λ ( θ ) ) d θ d s

= ε K ( x , t , x ) y m ( x , t ) ( m , λ ( x ) ) e 1 ε ∫ x 0 x ( m , λ ( θ ) ) d θ − K ( x , t , x 0 ) y m ( x 0 , t ) ( m , λ ( x 0 ) ) − ε ∫ x 0 x ∂ ∂ s K ( x , t , s ) y m ( s , t ) ( m , λ ( s ) ) e 1 ε ∫ x 0 s ( m , λ ( θ ) ) d θ d s = ∑ ν = 0 ∞ ( − 1 ) ν ε ν + 1 ( I m ν ( K ( x , t , s ) y m ( s , t ) ) ) s = x e 1 ε ∫ x 0 x ( m , λ ( θ ) ) d θ − ( I m ν ( K ( x , t , s ) y m ( s , t ) ) ) s = x 0 ,

where the operators

I m 0 = 1 ( m , λ ( s ) ) , I m ν = 1 ( m , λ ( s ) ) ∂ ∂ s I m ν − 1 ( ν ≥ 1 , ∣ m ∣ ≥ 2 )

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

J y ( x , t , τ ) = ∫ x 0 x K ( x , t , s ) y 0 ( s , t ) d s + ∑ j = 1 3 ∑ ν = 0 ∞ ( − 1 ) ν ε ν + 1 ( I j ν ( K ( x , t , s ) y j ( s , t ) ) ) s = x e 1 ε ∫ x 0 x λ j ( θ ) d θ − ( I j ν ( K ( x , t , s ) y j ( s , t ) ) ) s = x 0 + ∑ 2 ≤ ∣ m ∣ ≤ N y ∗ ∑ ν = 0 ∞ ( − 1 ) ν ε ν + 1 ( I m ν ( K ( x , t , s ) y m ( s , t ) ) ) s = x e 1 ε ∫ x 0 x ( m , λ ( θ ) ) d θ − ( I m ν ( K ( x , t , s ) y m ( s , t ) ) ) s = x 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 0 , X ] × [ 0 , T ] ). This means that the class M ε is asymptotically invariant (for ε → + 0 ) with respect to the operator J .

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

(50) R 0 y ( x , t , τ ) = ∫ x 0 x K ( x , t , s ) y 0 ( s , t ) d s ,

(51) R 1 y ( x , t , τ ) = ∑ j = 1 3 [ ( I j 0 ( K ( x , t , s ) y j ( s , t ) ) ) s = x e τ j − ( I j 0 ( K ( x , t , s ) y j ( s , t ) ) ) s = x 0 ] + ∑ 1 ≤ ∣ m ∣ ≤ N y ∗ [ ( I m 0 ( K ( x , t , s ) y m ( s , t ) ) ) s = x e ( m , τ ) − ( I m 0 ( K ( x , t , s ) y m ( s , t ) ) ) s = x e ( m , τ ) ] ,

(5 v ) R ν + 1 y ( x , t , τ ) = ∑ j = 1 3 ( − 1 ) ν [ ( I j ν ( K ( x , t , s ) y j ( s , t ) ) ) s = x e τ j − ( I j ν ( K ( x , t , s ) y j ( s , t ) ) ) s = x 0 ] + ∑ 2 ≤ ∣ m ∣ ≤ N y ∗ ( − 1 ) ν [ ( I m ν ( K ( x , t , s ) y m ( s , t ) ) ) s = x e ( m , τ ) − ( I m ν ( K ( x , t , s ) y m ( s , t ) ) ) s = x 0 ] .

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

(6) y ˜ ( x , t , τ , ε ) = ∑ k = 0 ∞ ε k y k ( x , t , τ ) , y k ( x , t , τ ) ∈ U

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

J y ˜ ( x , t , τ , ε ) = ∑ k = 0 ∞ ε k J y k ( x , t , τ ) = ∑ r = 0 ∞ ε r ∑ s = 0 r R r − s y s ( x , t , τ ) ∣ τ = ψ ( x ) ∕ ε .

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

J ˜ y ˜ ≡ J ˜ ∑ k = 0 ∞ ε k y k ( x , t , τ ) = ∑ r = 0 ∞ ε r ∑ k = 0 r R r − k y k ( x , t , τ ) .

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):

(7) L ε y ˜ ( x , t , τ , ε ) ≡ ε ∂ y ˜ ∂ x + ∑ j = 1 3 λ j ( x ) ∂ y ˜ ∂ τ j − A ( x ) y ˜ − J ˜ y ˜ − ε g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y ˜ = ε h 1 ( x , t ) 2 i ( e τ 2 σ 1 − e τ 3 σ 2 ) + h 2 ( x , t ) , y ˜ ( x 0 , t , 0 , ε ) = y 0 ( t ) , ( ( x , t ) ∈ [ x 0 , X ] × [ 0 , T ] ) .

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:

(80) L y 0 ( x , t , τ ) ≡ ∑ j = 1 3 λ j ( x ) ∂ y 0 ∂ τ j − A ( x ) y 0 − R 0 y 0 = h 2 ( x , t ) , y 0 ( x 0 , t , 0 , ε ) = y 0 ( t ) ;

(81) L y 1 ( x , t , τ ) = − ∂ y 0 ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y 0 + R 1 y 0 + h 1 ( x , t ) 2 i ( e τ 2 σ 1 − e τ 3 σ 2 ) , y 1 ( x 0 , t , 0 ) = 0 ;

(82) L y 2 ( x , t , τ ) = − ∂ y 1 ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y 1 + R 1 y 1 + R 2 y 0 , y 2 ( x 0 , t , 0 ) = 0 ;

⋯

(8 k ) L y k ( x , t , τ ) = − ∂ y k − 1 ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y k − 1 + R k y 0 + ⋯ + R 1 y k − 1 , y k ( x 0 , t , 0 ) = 0 , k ≥ 1 .

Each iterative problem (8 k ) has the form

(9) L y ( x , t , τ ) ≡ ∑ j = 1 3 λ j ( x ) ∂ y ∂ τ j − A ( x ) y − R 0 y = H ( x , t , τ ) , y ( x 0 , t , 0 ) = y ∗ ( t ) ,

where H ( x , t , τ ) = H 0 ( x , t ) + ∑ j = 1 3 H j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ H m ( x , t ) e ( m , τ ) ∈ U , 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 ( 5 0 ) )

R 0 y ( x , t , τ ) ≡ R 0 y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) = ∫ x 0 x K ( x , t , s ) y 0 ( s , t ) d s .

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

⟨ u , w ⟩ ≡ u 0 ( x , t ) + ∑ j = 1 3 u j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N u ∗ u m ( x , t ) e ( m , τ ) , w 0 ( x , t ) + ∑ j = 1 3 w j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N w ∗ w m ( x , t ) e ( m , τ ) ≡ ( u 0 ( x , t ) , w 0 ( x , t ) ) + ∑ j = 1 3 ( u j ( x , t ) , w j ( x , t ) ) + ∑ 2 ≤ ∣ m ∣ ≤ m i n ( N u , N w ) ∗ ( u m ( x , t ) , w m ( x , t ) ) ,

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

Theorem 1

Let conditions (1) and (2) be fulfilled and the right-hand side H ( x , t , τ ) = H 0 ( x , t ) + ∑ j = 1 3 H j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ H m ( x , t ) e ( m , τ ) of equation (9) belongs to the space U . Then, equation (9) is solvable in U , if and only if

(10) ⟨ H ( x , t , τ ) , e τ 1 ⟩ ≡ 0 , ∀ ( x , t ) ∈ [ x 0 , X ] × [ 0 , T ] .

Proof

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

(11) y ( x , t , τ ) = y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) .

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

∑ j = 1 3 [ λ j ( x ) − A ( x ) ] y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ [ ( m , λ ( x ) ) − A ( x ) ] y m ( x , t ) e ( m , τ ) − A ( x ) y 0 ( x , t ) − ∫ x 0 x K ( x , t , s ) y 0 ( s , t ) d s = H 0 ( x , t ) + ∑ j = 1 3 H j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ H m ( x , t ) e ( m , τ ) .

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

(12) − A ( x ) y 0 ( x , t ) − ∫ x 0 x K ( x , t , s ) y 0 ( s , t ) d s = H 0 ( x , t ) ,

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

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

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

(120) y 0 ( x , t ) = ∫ x 0 x ( − A − 1 ( x ) K ( x , t , s ) ) y 0 ( s , t ) d s − A − 1 ( x ) H 0 ( x , t ) .

Due to the smoothness of the kernel − A − 1 ( x ) K ( x , t , s ) and heterogeneity − A − 1 ( x ) H 0 ( x , t ) , this Volterra integral equation has a unique solution y 0 ( x , t ) ∈ C ∞ ( [ x 0 , X ] × [ 0 , T ] , C ) . The equations ( 1 2 2 ) and ( 1 2 3 ) also have unique solutions

(13) y j ( x , t ) = [ λ j ( x ) − A ( x ) ] − 1 H j ( x , t ) ∈ C ∞ ( [ x 0 , X ] × [ 0 , T ] , C ) , j = 2 , 3 ,

since λ 2 ( x ) , λ 3 ( x ) are not equal to A ( x ) . The equation ( 1 2 1 ) is solvable in space C ∞ ( [ x 0 , X ] × [ 0 , T ] , C ) if and only ( H 1 ( x , t ) , e τ 1 ) ≡ 0 ∀ ( x , t ) ∈ [ x 0 , X ] × [ 0 , T ] hold. It is not difficult to see that these identities coincide with identities (10). Furthermore, since ( m , λ ( x ) ) ≠ λ j ( x ) , ∀ m ∉ ⋃ j = 0 3 Γ j , j = 0 , 3 ¯ , ∣ m ∣ ≥ 2 (see (4)), the equations ( 1 2 m ) has a unique solution

y m ( x , t ) = [ ( m , λ ( x ) ) − A ( x ) ] − 1 H m ( x , t ) ∈ C ∞ ( [ x 0 , X ] × [ 0 , T ] , C ) , ∀ ∣ m ∣ ≥ 2 , m ∉ ⋃ j = 0 3 Γ j .

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 in the space U :

(14) y ( x , t , τ ) = y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) ≡ y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) ,

where α 1 ( x , t ) ∈ C ∞ ( [ x 0 , X ] × [ 0 , T ] , C ) are arbitrary function, y 0 ( x , t ) is the solution of an integral equation ( 1 2 0 ) , and introduced notations

h 21 ( x , t ) ≡ H 2 ( x , t ) λ 2 ( x ) − λ 1 ( x ) , h 31 ( x , t ) ≡ H 3 ( x , t ) λ 3 ( x ) − λ 1 ( x ) , P m ( x , t ) ≡ [ ( m , λ ( x ) ) − λ 1 ( x ) ] − 1 H m ( x , t ) .

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

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:

(15) L y ( x , t , τ ) = − ∂ y ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y + R 1 y + Q ( x , t , τ ) ,

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 space U . The right part of this equation:

G ( x , t , τ ) ≡ − ∂ y ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y + R 1 y + Q ( x , t , τ ) = − ∂ ∂ x y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) + R 1 y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) + Q ( x , t , τ ) ,

may not belong to the space U , 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 ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) = g ( x ) 2 B y 0 ( x , t ) ( e τ 2 σ 1 + e τ 3 σ 2 ) + ∑ j = 1 3 g ( x ) 2 B y j ( x , t ) ( e τ j + τ 2 σ 1 + e τ j + τ 3 σ 2 ) + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) .

Here, for example, terms with exponents

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

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

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

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

do not belong to the space U , since multi-indexes

( 0 , n , n ) ∈ Γ 0 , ( 0 , n + 1 , n ) ∈ Γ 1 , ( 0 , n , n + 1 ) ∈ Γ 2 ∀ n ∈ N

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

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

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

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

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

In other words, terms with resonant exponentials e ( m , τ ) replaced by members with exponents e 0 , e τ 1 , e τ 2 , e τ 3 according to the following rule:

e ( m , τ ) ^ ∣ m ∈ Γ 0 = e 0 = 1 , e ( m , τ ) ^ ∣ m ∈ Γ 1 = e τ 1 , e ( m , τ ) ^ ∣ m ∈ Γ 2 = e τ 2 , e ( m , τ ) ^ ∣ m ∈ Γ 3 = e τ 3 .

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

G ^ ( x , t , τ ) = − ∂ ∂ x y 0 ( x , t ) + ∑ j = 1 3 y j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ y m ( x , t ) e ( m , τ ) + ∑ j = 0 3 ∑ m j ∈ Γ j y m j ( x ) e τ j + Q ( x , t , τ ) .

As indicated in [6], the embedding G ( x , t , τ ) → G ^ ( x , t , τ ) will not affect the accuracy of the construction of asymptotic 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 0 ( x , t ) + ∑ j = 1 3 H j ( x , t ) e τ j + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ H m ( x , t ) e ( m , τ ) ∈ U of equation (9) satisfy condition (10). Then, problem (9) under additional conditions

(16) ⟨ G ^ ( x , t , τ ) , e τ 1 ⟩ ≡ 0 ∀ ( x , t ) ∈ [ x 0 , X ] × [ 0 , T ] ,

where Q ( x , t , τ ) = Q 0 ( x , t ) + ∑ k = 1 3 Q k ( x , t ) e τ i + ∑ 2 ≤ ∣ m ∣ ≤ N z ∗ Q m ( x , t ) e ( m , τ ) is the known function of the space U , is uniquely solvable in U .

Proof

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

y ∗ ( t ) = y ∗ ( t ) + A − 1 ( x 0 ) H 0 ( x 0 , t ) − H 2 ( x 0 , t ) λ 2 ( x 0 ) − λ 1 ( x 0 ) − H 3 ( x 0 , t ) λ 3 ( x 0 ) − λ 1 ( x 0 ) − ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ [ ( m , λ ( x 0 ) ) − A ( x 0 ) ] − 1 H m ( x 0 , t ) .

Now we subordinate the solution (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 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + R 1 y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + Q ( x , t , τ ) .

Embedding this function into the space U , we will have

G ( x , t , τ ) ≡ − ∂ ∂ x y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + R 1 y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + Q ( x , t , τ ) .

Embedding this function into the space U , we will have

G ˆ ( x , t , τ ) ≡ − ∂ ∂ x y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + g ( x ) 2 B y 0 ( x , t ) e τ 2 σ 1 + g ( x ) 2 B y 0 ( x , t ) e τ 3 σ 2 ∑ j = 1 3 g ( x ) 2 B y j ( x , t ) e τ j + τ 2 σ 1 + ∑ j = 1 3 g ( x ) 2 B y j ( x , t ) e τ j + τ 3 σ 2 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ g ( x ) 2 B y m ( x , t ) e ( m , τ ) + τ 2 σ 1 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ g ( x ) 2 B y m ( x , t ) e ( m , τ ) + τ 3 σ 2 ∧ + R 1 y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + Q ( x , t , τ ) = − ∂ ∂ x y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + B 2 g ( x ) { y 0 ( x , t ) e τ 2 σ 1 + y 0 ( x , t ) e τ 3 σ 2 + α 1 ( x , t ) e τ 1 + τ 2 σ 1 ̲ + h 21 ( x , t ) e 2 τ 2 σ 1 + h 31 ( x , t ) e τ 3 + τ 2 σ 1 + α 1 ( x , t ) e τ 1 + τ 3 σ ̲ 2 + h 21 ( x , t ) e τ 2 + τ 3 σ 2 + h 31 ( x , t ) e 2 τ 3 σ 2 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + τ 2 σ 1 ̲ + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + τ 3 σ 2 ̲ ∧ + R 1 y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) + Q ( x , t , τ ) .

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

R 1 y 0 ( x , t ) + α 1 ( x , t ) e τ 1 + h 21 ( x , t ) e τ 2 + h 31 ( x , t ) e τ 3 + ∑ 2 ≤ ∣ m ∣ ≤ N H ∗ P m ( x , t ) e ( m , τ ) ,

linearly depends on α 1 ( x , t ) (see formula ( 5 1 ) ), we also conclude that after the embedding operation the function G ˆ ( t , τ ) will linearly depend on the scalar function α 1 ( x , t ) . Given that in condition (16) scalar multiplication by functions e τ 1 , containing only the exponent e τ 1 , in the expression for G ^ ( t , τ ) , it is necessary to keep only the term with the exponent e τ 1 . Then, condition (16) takes the form

− ∂ ∂ x ( α 1 ( x , t ) e τ 1 ) + ∑ ∣ m 1 ∣ = 2 : m 1 ∈ Γ 1 N w m 1 ( α 1 ( t ) , t ) e τ 1 + Q 1 ( x , t ) e τ 1 , e τ 1 = 0 ∀ ( x , t ) ∈ [ x 0 , X ] × [ 0 , T ] ,

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

Performing scalar multiplication here, we obtain a linear ordinary differential equation (relative x ) for a function α 1 ( x , t ) . Given the initial condition α 1 ( x 0 , t ) = y ∗ ( t ) , found above, we find uniquely the function α 1 ( x , t ) ∈ C ∞ ( [ x 0 , X ] × [ 0 , T ] ) , 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:

L y 0 ( x , t , τ ) ≡ ∑ j = 1 3 λ j ( x ) ∂ y 0 ∂ τ j − A ( x ) y 0 − R 0 y 0 = h 2 ( x , t ) , y 0 ( x 0 , t , 0 , ε ) = y 0 ( t ) ; ( 8 ¯ 0 )

L y 1 ( x , t , τ ) = − ∂ y 0 ∂ x + g ( x ) 2 ( e 2 τ σ 1 + e 3 τ σ 2 ) B y 0 ∧ + R 1 y 0 + h 1 ( x , t ) 2 i ( e 2 τ σ 1 − e 3 τ σ 2 ) , y 1 ( x 0 , t , 0 ) = 0 ; ( 8 ¯ 1 )

L y 2 ( x , t , τ ) = − ∂ y 1 ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y 1 ∧ + R 1 y 1 + R 2 y 0 , y 2 ( x 0 , t , 0 ) = 0 ; ( 8 ¯ 2 )

⋯

L y k ( x , t , τ ) = − ∂ y k − 1 ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B y k − 1 ∧ + R k y 0 + ⋯ + R 1 y k − 1 , y k ( x 0 , t , 0 ) = 0 , k ≥ 1 ( 8 ¯ k )

(images of linear operators ∂ ∂ t 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). Then, when ε ∈ ( 0 , ε 0 ] ( ε 0 > 0 is sufficiently small), equation (2) has a unique solution y ( x , t , ε ) ∈ C 1 ( [ x 0 , X ] × [ 0 , T ] , C ) , in this case, the estimate

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

holds true, where y ε N ( x , t ) is the restriction for τ = ψ ( x ) ε of the N -partial sum of series (6) with coefficients y k ( x , t , τ ) ∈ U , satisfying the iteration problems ( 8 ¯ k ) , and the constant c N > 0 does not depend on ε ∈ ( 0 , ε 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 ( 8 ¯ 0 ) . Since the right side h 2 ( x , t ) of the equation ( 8 ¯ 0 ) satisfies condition (10), this equation has (according to (14)) a solution in the space U in the form:

(17) y 0 ( x , t , τ ) = y 0 ( 0 ) ( x , t ) + α 1 ( 0 ) ( x , t ) e τ 1 ,

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

(18) y 0 ( 0 ) ( x , t ) = ∫ x 0 x ( − A − 1 ( x ) K ( x , t , s ) ) y 0 ( 0 ) ( s , t ) d s − A − 1 ( x ) h 2 ( x , t ) .

Subordinating (17) to the initial condition y 0 ( x 0 , t , 0 ) = y 0 ( t ) , we have

y 0 ( 0 ) ( x 0 , t ) + α 1 ( 0 ) ( x 0 , t ) = y 0 ( t ) ⇔ α 1 ( 0 ) ( x 0 , t ) = y 0 ( t ) − y 0 ( 0 ) ( x 0 , t ) ⇔ α 1 ( 0 ) ( x 0 , t ) = y 0 ( t ) + A − 1 ( x 0 ) h 2 ( x 0 , t ) .

To fully compute the function α 1 ( 0 ) ( x , t ) , we proceed to the next iteration problem ( 8 ¯ 1 ) . Substituting into it the solution (17) of the equation ( 8 ¯ 0 ) , we arrive at the following equation:

L y 1 ( x , t , τ ) = − ∂ ∂ x y 0 ( 0 ) ( x , t ) − ∂ ∂ x ( α 1 ( 0 ) ( x , t ) ) e τ 1 + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B ( y 0 ( 0 ) ( x , t ) + α 1 ( 0 ) ( x , t ) e τ 1 ) ∧ + K ( x , t , x ) α 1 ( 0 ) ( x , t ) λ 1 ( x ) e τ 1 − K ( x , t , x 0 ) α 1 ( 0 ) ( x 0 , t ) λ 1 ( x 0 ) + 1 2 i h 1 ( x , t ) ( e τ 2 σ 1 − e τ 3 σ 2 ) ,

(here we used the expression ( 5 1 ) for R 1 y ( x , t , τ ) and took into account that when y ( x , t , τ ) = y 0 ( x , t , τ ) the sum ( 5 1 ) contains only terms with e τ 1 ).

Let us calculate

M = B g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) ( y 0 ( 0 ) ( x , t ) + α 1 ( 0 ) ( x , t ) e τ 1 ) ∧ = B 2 g ( x ) { σ 1 y 0 ( 0 ) ( x , t ) e τ 2 + σ 2 y 0 ( 0 ) ( x , t ) e τ 3 + σ 1 α 1 ( 0 ) ( x , t ) e τ 2 + τ 1 + σ 2 α 1 ( 0 ) ( x , t ) e τ 3 + τ 1 } ∧ .

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

e τ 2 + τ 1 ∣ τ = ψ ( x ) ∕ ε = e 1 ε ∫ t 0 t ( − i β ′ ( θ ) + A ( θ ) ) d θ , − i β ′ + A = 0 , A , − i β ′ , + i β ′ , ⇔ ∅ ; e τ 3 + τ 1 ∣ τ = ψ ( x ) ∕ ε = e 1 ε ∫ t 0 t ( + i β ′ ( θ ) + A ( θ ) ) d θ , + i β ′ + A = 0 , A , − i β ′ , + i β ′ , ⇔ ∅ .

Thus, exponents e τ 2 + τ 1 and e τ 3 + τ 1 are not resonant. Then, for solvability, equation (18), it is necessary and sufficient that the condition

− ∂ ∂ x ( α 1 ( 0 ) ( x , t ) ) + K ( x , t , x ) α 1 ( 0 ) ( x , t ) λ 1 ( x ) = 0

is satisfied. Attaching the initial condition

α 1 ( 0 ) ( x 0 , t ) = y 0 ( t ) + A − 1 ( x 0 ) h 2 ( x 0 , t ) ,

to this equation, we find

α 1 ( 0 ) ( x , t ) = α 1 ( 0 ) ( x 0 , t ) e ∫ x 0 x K ( s , t , s ) λ 1 ( s ) d s ,

and therefore, we uniquely calculate the solution (17) of the problem ( 8 ¯ 0 ) in the space U . Moreover, the main term of the asymptotic of the solution to problem (2) has the form

(19) y ε 0 ( x , t ) = y 0 ( 0 ) ( x , t ) + α 1 ( 0 ) ( x 0 , t ) e ∫ x 0 x ( K ( s , t , s ) ) λ 1 ( s ) d s + 1 ε ∫ x 0 x λ 1 ( θ ) d θ ,

where α 1 ( 0 ) ( x 0 , t ) = y 0 ( t ) + A − 1 ( x 0 ) h 2 ( x 0 , t ) , y 0 ( 0 ) ( x , t ) is the solution of the integrated equation (18). From expression (19) for y ε 0 ( x , t ) , it is clear that y ε 0 ( x , t ) 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 ( 8 ¯ 1 ) will be written as

(20) L y 1 = − ∂ y 0 ( 0 ) ( x , t ) ∂ x + g ( x ) 2 ( e τ 2 σ 1 + e τ 3 σ 2 ) B ( y 0 ( 0 ) ( x , t ) + α 1 ( 0 ) ( x , t ) e τ 1 ) ∧ − K ( x , t , x 0 ) α 1 ( 0 ) ( x 0 , t ) λ 1 ( x 0 ) + h 1 ( x , t ) 2 i ( e τ 2 σ 1 − e τ 3 σ 2 ) = − ∂ y 0 ( 0 ) ( x , t ) ∂ x + g ( x ) 2 B ( y 0 ( 0 ) ( x , t ) σ 1 e τ 2 + y 0 ( 0 ) ( x , t ) σ 2 e τ 3 ) + g ( x ) 2 B ( α 1 ( 0 ) ( x , t ) σ 1 e τ 1 + τ 2 + α 1 ( 0 ) ( x , t ) σ 2 e τ 1 + τ 3 ) ∧ − K ( x , t , x 0 ) α 1 ( 0 ) ( x 0 , t ) λ 1 ( x 0 ) + σ 1 2 i h 1 ( x , t ) e τ 2 − σ 2 2 i h 1 ( x , t ) e τ 3 .

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

(21) y 1 ( x , t , τ ) = y 1 ( 0 ) ( x , t ) + α 1 ( 1 ) ( x , t ) e τ 1 + H 2 ( 1 ) ( x , t ) λ 2 ( x ) − λ 1 ( x ) e τ 2 + H 3 ( 1 ) ( x , t ) λ 3 ( x ) − λ 1 ( x ) e τ 3 + g ( x ) 2 λ 2 ( x ) B σ 1 α 1 ( 0 ) ( x , t ) e τ 1 + τ 2 + g ( x ) 2 λ 3 ( x ) B σ 2 α 1 ( 0 ) ( x , t ) e τ 1 + τ 3 ,

where α 1 ( 1 ) ( x , t ) ∈ C ∞ ( [ x 0 , X ] × [ 0 , T ] ) is an arbitrary function determined from the solvability condition (10) of the equation ( 8 ¯ 2 ) in the space U , the function y 1 ( 0 ) ( x , t ) and the function

H 2 ( 1 ) ( x , t ) = σ 1 [ g ( x ) B y 0 ( 0 ) ( x , t ) − i h 1 ( x , t ) ] 2 , H 3 ( 1 ) ( x , t ) = σ 2 [ g ( x ) B y 0 ( 0 ) ( x , t ) + i h 1 ( x , t ) ] 2

are computed uniquely. From (21), it is seen that y 1 x , t , ψ ( x ) ε depends on rapidly oscillating exponents e ± i ε ( β ( x ) − β ( x 0 ) ) , i.e., an already asymptotic solution

y ε 1 ( x , t ) = y 0 x , t , ψ ( x ) ε + ε y 1 x , t , ψ ( x ) ε

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

6 Conclusion

The function y ε 0 ( x , t ) 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 ε 1 ( x , t ) .

  1. 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.

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

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Received: 2020-09-20
Revised: 2021-01-15
Accepted: 2021-02-08
Published Online: 2021-05-10

© 2021 Burkhan T. Kalimbetov and Olim D. Tuychiev, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
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  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
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  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0021
Keywords for this article
singularly perturbed; integro-partial differential equation; regularization of an integral; space of non-resonant solutions; iterative problems; solvability of iterative problems
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Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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