Asymptotic solutions of scalar integro-differential equations with partial derivatives and with rapidly oscillating coefficients

UDC 517.955.8

B.T. Kalimbetov

¹

, Kh.F. Etmishev

²

1H.A.Yasawi International Kazakh-Turkish University, Turkestan, Kazakhstan;

2Jizzakh Polytechnic Institute, Uzbekistan (E-mail: bkalimbetov@mail.ru)

Asymptotic solutions of scalar integro-differential equations with partial derivatives and with rapidly oscillating coefficients

The work is devoted to the development of an asymptotic integration algorithm for the Cauchy problem for a singularly perturbed partial differential integro-differential equation with rapidly oscillating coefficients, which describe various physical processes in micro-inhomogeneous media.This direction in the theory of partial differential equations is developing intensively and finds numerous applications in radiophysics, electrical engineering, filtering theory, phase transition theory, elasticity theory, and other branches of physics, mechanics, and technology. For studies of such processes, asymptotic methods are usually used. It is known that currently rapidly developing numerical methods do not exclude asymptotic. This happens for a number of reasons. Firstly, a reasonably constructed asymptotics, especially its main term, carries information that is important for applications about the qualitative behavior of the solution and, in this sense, to some extent replaces the exact solution, which most often cannot be found. Secondly, as follows from the above, knowledge of the solution structure helps in the development of numerical methods for solving complex problems; therefore, the development of asymptotic methods contributes to the development of numerical methods. Regularization of the problem is carried out, the normal and unique solvability of general iterative problems is proved.

Keywords:singularly perturbed, partial integro differential equation, regularization of an integral, solvability of iterative problems.

Introduction

A mathematical description of physical processes in micro-inhomogeneousmedia suggests that the local characteristics of the latter depend on a small parameter which is a characteristic scale of the microstructure of the medium. To construct mathematical models of such processes, an asymptotic analysis of the problem is performed for It turns out that the limits of the solutions to the problem are described by some new differential equations that have relatively smoothly varying coefficients and are considered in simple domains.These equations are mathematical models of physical processes in micro-inhomogeneousmedia, and their coefficients are effective characteristics of such media. For mathematical studies of such processes, asymptotic methods are usually used. It is known that currently rapidly developing numerical methods do not exclude asymptotic ones. This happens for a number of reasons. Firstly, a reasonably constructed asymptotics, especially its main term, carries information that is important for applications about the qualitative behavior of the solution and, in this case to some extent replaces the exact solution, which most often cannot be found. Secondly, as follows from the above, knowledge of the solution structure helps in the development of numerical methods for solving complex problems; therefore, the development of asymptotic methods contributes to the development of numerical methods. Thirdly, for some problems, especially those related to fast oscillations, there are simply no effective numerical methods that give a sufficient degree of accuracy. The first of the problems with an irregular dependence in perturbation theory that arose in connection with the problems of celestial mechanics and electrical engineering were nonlinear equations, which are often called oscillating equations at present. Tasks of this kind arise everywhere where certain transient processes take place. Studies of oscillating and singularly perturbed oscillating systems described by ordinary differential equations to the splitting methods were carried out in [1–4] and regularization methods in [5–8]. An analysis of the main results of the study for systems of homogeneous and inhomogeneous differential equations prompted the idea to study singularly perturbed integro-differential equations with rapidly oscillating coefficients.A system of integro-differential equations in the absence of resonance is considered, i.e. when the integer linear combination of frequencies of the rapidly oscillating cosine does not coincide with the frequency of the spectrum of the limit operator [9, 10]. It should be

Ре по зи то ри й Ка рГ У

noted that when developing an algorithm for constructing an asymptotic solution to the problem, the ideas of the regularization method used to study ordinary integro-differential equations [11–21] and integro-differential equations with partial derivatives [23–24] were used.

We consider the Cauchy problem for the integro-differential equation with partial derivatives:

ε^∂y(x,t,ε)_∂x =a(x)y(x, t, ε) +

x

R

x0

K(x, t, s)y(s, t, ε)ds+h(x, t)+

+εg(x)cos^β(x)_ε y(x, t, ε), y(x0, t, ε) =y⁰(t) ( (x, t)∈[x0, X]×[0, T] ),

(1)

where β⁰(x)>0, g(x), a(x) is a scalar functions, y⁰(t) constant, ε >0 is a small parameter. Denote by λ1(x) = −a(x), β⁰(x) is a frequency of rapidly oscillating cosine. In the following, functions λ2(x) =−iβ⁰(x), λ3(x) = +iβ⁰(x)will be calledthe spectrum of a rapidly oscillating coefficient.

We assume that the conditions are fulfilled:

(i) a(x), g(x), β(x) ∈ C^∞[x₀, X]; h(x, t) ∈ C^∞[x₀, X]×[0, T], the kernel K(x, t, s) belongs to the space K(x, t, s)∈C^∞{x0< x < s < X,0< t < T};

(ii)λ₁(x)≡a(x)6=λ_j(x), j= 2,3, λ_i(x)6= 0, (∀x∈[x₀, X]), i= 1,2,3;

(iii)λ₁(x)≤0, (∀x∈[x₀, X]);

(iv) for∀x∈[x0, X]andn26=n3 inequalities

n2λ2(x) +n3λ3(x)6=λ1(x),

λ1(x) +n2λ2(x) +n3λ3(x)6=λ1(x), (∀x∈[x0, X])

for all multi-indicesn= (n₂, n₃)with|n| ≡n₂+n₃≥1 (n₂andn₃ are non-negative integers) are holds.

We will develop an algorithm for constructing a regularized [5] asymptotic solution of problem (1).

1 Regularization of problem (1)

Denote by σj =σj(ε),independent of t magnitudesσ1 =e^−εiβ(t0), σ1 =e^+εiβ(t0),and rewrite system (1) as

ε^∂y(x,t,ε)_∂x −λ1(x)y(x, t, ε)−ε^g(t)₂ e^−εi

Rt t0β⁰(θ)dθ

σ1+e^+iε

Rt t0β⁰(θ)dθ

σ2

y(x, t, ε)−

−Rx

x₀K(x, t, s)y(s, t, ε)ds=h(x, t), y(x0, t, ε) =y⁰, ((x, t)∈[x0, X]×[0, T]).

(2)

We introduce regularizing variables (see [5, 6]):

τ_j =1 ε

Z x x₀

λ_j(θ)dθ≡ ψj(x)

ε , j= 1,3, and instead of problem (2), consider the problem

ε^∂_∂x^y˜+

3

P

j=1

λj(t)_∂τ^∂y˜

j −λ1(x)˜y−ε^g(t)₂ (e^τ2σ1+e^τ3σ2) ˜y−

−Rx

x₀K(x, t, s) ˜y(s, t,^ψ(s)_ε , ε)ds=h(x, t), ,y(x, t, τ, ε)|˜ x=x₀,τ=0=y⁰, ((x, t)∈[x0, X]×[0, T]),

(3)

for the function y˜ = ˜y(x, t, τ, ε), where is indicated: τ = (τ₁, τ₂, τ₃), ψ = (ψ₁, ψ₂, ψ₃). It is clear that if

˜

y= ˜y(x, t, τ, ε)is a solution to problem (3), then the vector functiony = ˜y

x,^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 term Jy˜=Rx

x0K(x, t, s) ˜y(s,^ψ(s)_ε , ε)ds. To regularize the integral operator, we introduce a classM_εthat is asymptotically invariant with respect to the operatorJy˜ (see [5], p. 62). Recall the corresponding concept.

Definition 1. A classM_ε is said to be asymptotically invariant (withε→+0) with respect to an operator P₀if the following conditions are fulfilled:

1) M_ε⊂D(P₀)with each fixedε >0;

2) the image P0g(x, ε)of any element g(x, ε)∈Mεdecomposes in a power series P0g(x, ε) =

∞

X

n=0

εⁿgn(x, ε)(ε→+0, gn(x, ε)∈Mε, n= 0,1, ...), convergent asymptotically forε→+0(uniformly withx∈[x0, X]).

Ре по зи то ри й Ка рГ У

From this definition it can be seen that the class Mεdepends on the space U, in which the operatorP0 is defined. In our case P0 =J. For the space U we take the space of vector functions y(x, t, τ), represented by sums

y(x, t, τ, σ) =y0(x, t, σ) +

3

X

i=1

yi(x, t, σ)e^τi+

∗

X

2≤|m|≤Ny

y^m(x, t, σ)e^(m,τ)+

+

∗

X

1≤|m|≤Ny

y^e1+m(x, t, σ)e^(e1+m,τ), yi(x, t, σ), y^m(x, t, σ), y^e1+m(x, t, σ)∈C^∞[x0, X]×[0, T], (4) m= (0, m₂, m₃),1≤ |m| ≡m₂+m₃≤N_y, i= 1,3,

where is denoted:λ(x)≡(λ1, λ2, λ3), (m, λ(x))≡m2λ2(x) +m3λ3(x),(e1+m, λ(x))≡λ1(x) +m2λ3(x) + +m3λ3(x) ; an asterisk ∗ above the sum sign indicates that the summation for |m| ≥ 1 it occurs only over multi-indices m= (0, m2, m3)withm26=m3, e1= (1,0,0), σ= (σ1, σ2).

Note that here the degree Ny of the polynomialy(x, t, τ, σ)relative to the exponentialse^τ1 depends on the element y. In addition, the elements of space U depend on bounded inε > 0 terms of constantsσ1 = σ1(ε) and σ2 =σ2(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 onσ = (σ1, σ2) for brevity. We show that the classMε=U|τ=ψ(t)/εis asymptotically invariant with respect to the operatorJ.The image of the operator on the element (4) of the spaceU has the form

J y(x, t, τ) =

x

Z

x₀

K(x, t, s)y0(s, t)ds+

3

X

i=1 x

Z

x₀

K(x, t, s)yi(s, t)e

1 ε

s

R

x0

λ_i(θ)dθ

ds+

+

∗

X

2≤|m|≤Nz

x

Z

x0

K(x, t, s)y^m(s, t)e

1 ε

s

R

x0

(m,λ(θ))dθ

ds+

∗

X

1≤|m|≤Nz

x

Z

x0

K(x, t, s)y^e1+m(s, t)e

1 ε

s

R

x0

(e1+m,λ(θ))dθ

ds.

Integrating in parts, we will have

Ji(x, t, ε) =

x

Z

x0

K(x, t, s)yi(s, t)e

1 ε

s

R

x0

λi(θ)dθ

ds=ε

x

Z

x0

K(x, t, s)y_i(s, t) λi(s) de

1 ε

s

R

x0

λi(θ)dθ

=

=ε





K(x, t, s)y_i(s, t) λi(s) e

1 ε

s

R

x0

λi(θ)dθ

s=x

s=x0

−

x

Z

x₀

∂

∂s

K(x, t, s)y_i(s, t) λi(θ)

e

1 ε

s

R

x0

λi(θ)dθ

ds



=

=ε





K(x, t, x)y_i(x, t) λi(x) e

1 ε

x

R

x0

λi(θ)dθ

−K(x, t, x₀)y_i(x₀, t) λi(x0)



−ε

x

Z

x₀

∂

∂s

K(x, t, s)y_i(s, t) λi(s)

e

1 ε

s

R

x0

λi(θ)dθ

ds . Continuing this process further, we obtain the decomposition

Ji(x, t, ε) =

∞

P

ν= 0

(−1)^νε^ν+1



(I_i^ν(K(x, t, s)yi(s, t)))_s=xe

1 ε

Rx x0

λ_i(θ))dθ

−(I_i^ν(K(x, t, s)yi(s, t)))_s=x

0



, I_i⁰ = _λ¹

i(s)., I_i^ν = _λ¹

i(s)I_i^ν−1(ν ≥ 1, i= 1,3).

Applying the integration operation in parts to integrals

Jm(x, t, ε) =

x

R

x₀

K(x, t, s)y^m(s, t)e

1 ε

s

R

x0

(m,λ(θ))dθ

ds,

J_e1+m(x, t, ε) =

x

R

x₀

K(x, t, s)y^e1+m(s, t)e

1 ε

s

R

x0

(e1+m,λ(θ))dθ

ds,

we note that for all multi-indicesm= (0, m2, m3), m26=m3, inequalities

(m, λ(x))≡m₂λ₂(x) +m₃λ₃(x)6= 0 ∀x∈[x₀, X], m₂+m₃≥2.

Ре по зи то ри й Ка рГ У

are satisfied. In addition, for the same multi-indicesm= (0, m2, m3)we have

(e₁+m, λ(x))6= 0∀x∈[x₀, X], m₂6=m₃, |m|=m₂+m₃≥1.

Indeed, if (e₁+m, λ(x)) = 0for some x∈[x₀, X]andm₂6=m₃, m₂+m₃≥1,thenm₂λ₂(x) +m₃λ₃(x) =

=−λ₁(x), m₂+m₃≥1,which contradicts condition (iv). Therefore, integration by parts in integrals is possible.

Performing it, we will have:

Jm(x, t, ε) =

x

Z

t0

K(x, t, s)y^m(s, t)e

1 ε

s

R

x0

(m,λ(θ))dθ

ds=ε

x

Z

x0

K(x, t, s)y^m(s, t) (m, λ(s)) de

1 ε

s

R

x0

(m,λ(θ))dθ

=

=ε





K(x, t, s)y^m(x, t) (m, λ(s)) e

1 ε

s

R

x0

(m,λ(θ))dθ

s=x

s=x₀

−ε

x

Z

x₀

∂

∂s

K(x, t, s)y^m(s, t) (m, λ(s))

e

1 ε

s

R

x0

(m,λ(θ))dθ



−

=ε





K(x, t, x)y^m(x, t) (m, λ(x)) e

1 ε

x

R

x0

(m,λ(θ))dθ

−K(x, t, x0)y^m(x0, t) (m, λ(x0))



−

−ε

x

Z

x₀

∂

∂s

K(x, t, s)y^m(s, t) (m, λ(s))

e

1 ε

s

R

x0

(m,λ(θ))dθ

ds.

Therefore, the image of the operatorJ on the element (4) of the spaceU is represented as a series J z(t, τ) =Rt

t₀K(t, s)z0(s)ds+

4

P

i=1

∞

P

ν=0

(−1)^νε^ν+1[(I_i^ν(K(t, s)zi(s)))_s=te^1ε

Rt t0λ_i(θ))dθ

−

−(I_i^ν(K(t, s)zi(s)))_s=t

0] +

∞

P

ν=0

(−1)^νε^ν+1[(I_m^ν (K(t, s)z^m(s)))_s=te^1ε

Rt

t0(m,λ(θ))dθ

−

−(I_m^ν (K(t, s)z^m(s)))_s=t

0] +

2

P

j=1

P

1≤|m|≤N_z

∞

P

ν=0

(−1)^νε^ν+1[ I_j,m^ν (K(t, s)z^ej+m(s))

s=t×

×e^{1εRtt0(ej+m,λ(θ))}− I_j,m^ν (K(t, s)z^ej+m(s))

s=t0]_τ=ψ(t)/ε. Continuing this process, we obtain the series

Jm(x, t, ε) =

∞

P

ν= 0

(−1)^νε^ν+1



(I_m^ν (K(x, t, s)y^m(s, t)))_s=te

1 ε

x

R

x0

(m,λ(θ))dθ

−

−(I_m^ν (K(x, t, s)y^m(s, t)))_s=t

0

,

I_m⁰ = 1

(m, λ(s)) ., I_m^ν = 1 (m, λ(s))

∂

∂sI_m^ν−1(ν ≥1,|m| ≥2), Je₁+m(x, t, ε) =

x

R

x₀

K(x, t, s)y^e1+m(s, t)e

1 ε

s

R

x0

(e1+m,λ(θ))dθ

ds=

=ε

s

R

x₀

K(x,t,s)y^{e1 +m}(s,t) (e1+m,λ(s)) de

1 ε

s

R

x0

(e1+m,λ(θ))dθ

=

=ε





K(x,t,s)y^{e1 +m}(s,t) (e1+m,λ(s)) e

1 ε

s

R

x0

(e1+m,λ(θ))dθ

s=x

s=x0

−

−ε

x

R

x₀

∂

∂s

K(x,t,s)y^{e1 +m}(s,t) (e1+m,λ(s))

e

1 ε

s

R

x0

(e1+m,λ(θ))dθ

ds



−

=ε





K(x, t, x)y^e1+m(x, t) (e1+m, λ(x)) e

1 ε

x

R

x0

(e1+m,λ(θ))dθ

−K(x, t, x0)y^e1+m(x0, t) (e1+m, λ(x0))



−

Ре по зи то ри й Ка рГ У

−ε

x

Z

x₀

∂

∂s

K(x, t, s)y^e1+m(s, t) (e1+m, λ(s))

e

1 ε

s

R

x0

(e₁+m,λ(θ))dθ

ds.

Continuing this process, we obtain the series

Je₁+m(x, t, ε) =

∞

P

ν= 0

(−1)^νε^ν+1



 I_e^ν_1+m(K(x, t, s)y^e1+m(s, t))

s=xe

1 ε

Rx x0

(e₁+m,λ(θ))dθ

,

− I_e^ν

1+m(K(x, t, s)y^e1+m(s, t))

s=x₀

i,

I_e⁰

1+m = 1

(e1+m, λ(s)) ., I_j,m^ν = 1 (e1+m, λ(s))

∂

∂sI_m^ν−1(ν ≥ 1,|m| ≥1),

Therefore, the image of the operatorj on the element (4) of the space U is represented as a series

J y(x, t, τ) =

x

Z

x₀

K(x, t, s)y0(s, t)ds+

3

X

i=1

∞

X

ν= 0

(−1)^νε^ν+1



(I_i^ν(K(x, t, s)yi(s, t)))_s=te

1 ε

Rx x0

λ_i(θ))dθ

−

−(I_i^ν(K(x, t, s)yi(s, t)))_s=t

0

+

+

∗

X

2≤|m|≤NY

∞

X

ν= 0

(−1)^νε^ν+1



(I_m^ν (K(x, t, s)y^m(s, t)))_s=te

1 ε

x

R

x0

(m, λ(θ))dθ

−(I_m^ν (K(x, t, s)y^m(s, t)))_s=t

0



+

+ P

1≤|m|≤NY

∞

P

ν= 0

(−1)^νε^ν+1



 I_e^ν

1+m(K(x, t, s)y^e1+m(s, t))

s=xe

1 ε

x

R

x0

(e1+m,λ(θ))dθ

−

− I_e^ν_1+m(K(x, t, s)y^e1+m(s, t))

s=x0

i .

It is easy to show (see, for example, [25], pp. 291-294) that this series converges asymptotically for ε→+0 (uniformly in(x, t)∈[x0, X]×[0, T]). This means that the class Mεis asymptotically invariant (for ε→+0) with respect to the operatorJ.

We introduce operatorsRν : U →U,acting on each elementy(x, t, τ)∈U of the form (4) according to the law:

R0y(x, t, τ) =

x

Z

x0

K(x, t, s)y0(s, t)ds, (50)

R1y(x, t, τ) =

3

X

i= 1

I_i⁰(K(x, t, s)yi(s, t))

s=xe^τi− I_i⁰(K(x, t, s)yi(s, t))

s=x₀

i +

+

∗

X

1≤|m|≤Nz

h

I_m⁰ (K(x, t, s)y^m(s, t))

s=xe^(m,τ)− I_m⁰ (K(x, t, s)y^m(s, t))

s=x0

i

+ (51)

+

∗

X

1≤|m|≤Nz

I_e⁰_1+m

K(x, t, s)y^{e1 +m}(s, t)

s=xe^(e1+m,τ)− I_e⁰_1+m

K(x, t, s)y^{e1 +m}(s, t)

s=x₀

,

Rν+1y(x, t, τ) =

3

X

i= 1

(I_i^ν(K(x, t, s)yi(s, t)))_s=xe^τi−(I_i^ν(K(x, t, s)yi(s, t)))_s=x

0

+

+

∗

X

2≤|m|≤Ny

h

(I_m^ν (K(x, t, s)y^m(s, t)))_s=xe^(m,τ)−(I_m^ν (K(x, t, s)y^m(s, t)))_s=x

0

i

+ (5ν+1)

+

∗

X

1≤|m|≤Nz

h I_e^ν

1+m K(x, t, s)z^e1+m(s, t)

s=xe^(e1+m,τ)− I_e^ν

1+m K(x, t, s)z^e1+m(s, t)

s=x₀

i , ν≥1.

Ре по зи то ри й Ка рГ У

Now let y˜(x, t, τ, ε) be an arbitrary continuous function on (x, t, τ) ∈ [x0, X]×[0, T]× {Reλ1(x)} with asymptotic expansion

˜

y(x, t, τ, ε) =

∞

X

k=0

ε^kyk(x, t, τ), yk(x, t, τ)∈U, (6) converging asε → +0 (uniformly in (x, t, τ)∈ [x0, X]×[0, T]× {Reλ1(x)}). Then the image Jy˜(x, t, τ, ε) of this function is decomposed into an asymptotic series

Jz˜(x, t, τ, ε) =

∞

X

k=0

ε^kJ yk(x, t, τ) =

∞

X

r=0

ε^r

r

X

s=0

R_r−sys(x, t, τ)|_τ=ψ(x)/ε. This equality is the basis for introducing an extension of an operatorJ on series of the form (6):

J˜y˜(x, t, τ, ε)≡J˜

∞

X

k=0

ε^ky_k(x, t, τ)

!

=∆

∞

X

r=0

ε^r

r

X

s=0

R_r−sy_s(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 the problem (2), in which impose onlyN-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):

Lεy(x, t, τ, ε)˜ ≡ε^∂_∂x^y˜+

3

P

j=1

λj(x)_∂τ^∂y˜

j −λ1(x)˜y−J˜y˜−ε^g(x)₂ (e^τ2σ1+e^τ3σ2)˜y=

=h(x, t), y(x˜ 0, t,0, ε) =y⁰, ((x, t)∈[x0, X]×[0, T]).

(7)

2 Solvability of iterative problems

Substituting the series (6) into (7) and equating the coefficients with the same degrees ε, we obtain the following iterative problems:

L y0(x, t, τ)≡

3

X

j=1

λj(x)∂y0

∂τ_j −λ1(x)y0−R0y0=h(x, t), y0(x0, t,0) =y⁰(t); (80) L y₁(x, t, τ) =−∂y₀

∂x +g(x)

2 (e^τ2σ₁+e^τ3σ₂)y₀+R₁y₀, y₁(x₀, t,0) = 0; (8₁) L y2(x, t, τ) =−∂y1

∂x +g(x)

2 (e^τ2σ1+e^τ3σ2)y1+R1y1+R2y0, y0(x0, t,0) = 0; (82)

· · · L y_k(x, t, τ) =−∂y_k−1

∂x +g(x)

2 (e^τ2σ₁+e^τ3σ₂)y_k−1+R_ky₀+...+R₁y_k−1, y_k(x₀, t,0) = 0, k≥1. (8_k) Each of the iterative problems(8_k)can be written as

L z(x, t, τ)≡

3

X

j=1

λj(x) ∂y

∂τ_j −λ1(x)y−R0y=h(x, t, τ), y(x0, t,0) =y^∗, (9) where

h(x, t, τ) =h₀(x, t) +

3

X

i=1

h_i(x, t)e^τi+

∗

X

2≤|m|≤Ny

h^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤Ny

h^e1+m(x, t)e^(e1+m,τ)∈U, is the known vector function of space U, y^∗ is the known constant vector of the complex space C, and the operatorR0has the form (see(50))

R0y(x, t, τ)≡R0

"

y0(x, t) +

3

P

i=1

yi(x, t)e^τi+

∗

P

2≤|m|≤Ny

y^m(x, t)e^(m,τ)+

∗

P

1≤|m|≤Ny

y^e1+m(x, t)e^(e+m,τ)

#

=∆

=∆ x

R

x₀

K(x, t, s)y₀(s, t)ds.

Ре по зи то ри й Ка рГ У

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

y(x, t, τ) =y0(x, t) +

3

X

i=1

yi(x, t)e^τi+

∗

X

2≤|m|≤Nz

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤NH

y^e1+m(x, t)e^(e1+m,τ)≡

≡y₀(x, t) +

3

X

i=1

y_i(x, t)e^τi+

∗

X

2≤|m|≤Ny

y^m(x, t)e^(m,τ)+

∗

X

2≤|m¹|≤NH

y^m1(x, t)e^(m1,τ), (10)

where for convenience introduced multi-indices

m¹=e1+m≡(1, m2, m3), m¹

= 1 +m2+m3≥2,

m2 and m3 are non-negative integer numbers. Substituting (10) into equation (9), and equating here the free terms and coefficients separately for identical exponents, we obtain the following equations:

−λ1(x)y0(x, t)−

x

Z

x0

K(x, t, s)y0(s, t)ds=h0(x, t), (11)

[λi(x)−λ1(x)]yi(x, t) =hi(x, t), i= 1,3, (11i) [(m, λ(x))−λ1(x)]y^m(x, t) =h^m(x, t), m26=m3, 2≤ |m| ≤Nh, (11m) m¹, λ(x)

−λ1(x)

y^m1(x, t) =h^m1(x, t), m26=m3, 1≤ m¹

≤Nh. (12) The equation (11) can be written as

y₀(x, t) =

t

Z

t₀

−λ⁻¹₁ (x)K(x, t, s)

y₀(s, t)ds−λ⁻¹₁ (x)h₀(x, t). (11₀)

Due to the smoothness of the kernel−λ⁻¹₁ (x)K(x, t, s)and heterogenei ty−λ⁻¹₁ (x)h₀(x, t), this Volterra integral system has a unique solutiony0(x, t)∈C^∞[x0, X]×[0, T].The equations(112)-(113)also have unique solutions

yi(x, t) = [λi(x)−λ1(x)]⁻¹hi(x, t)∈C^∞[x0, X]×[0, T], i= 2,3,

sinceλ_i(x)6=λ₁(x), i= 2,3.Equation(11₁)are solvable in spaceC^∞[x₀, X]×[0, T]if and only

h₁(x, t)≡0 (∀(x, t)∈[x₀, X]×[0, T]). (13) Further, since(m, λ(x))≡m2λ2(x) +m3λ3(x)6=λ1(x), |m|=m2+m3≥2 (see condition (iv) the absence of resonance), the equation(11m)has a unique solution

y^m(x, t) = [(m, λ(x))−λ₁(x)]⁻¹h^m(x, t)∈C^∞[x₀, X]×[0, T], 2≤ |m| ≤N_h. We now equation (12). Let us show that when

m¹

≥ 1 the functions m¹, λ(x)

6= λ1(x). Indeed, let m¹, λ(t)

=λ1(t), m¹

≥1.Then

λ₁(x) +m₂λ₂(x) +m₃λ₃(x) =λ₁(x) ⇔ m₂λ₂(x) +m₃λ₃(x) = 0 ⇔ m₂6=m₃, m₂+m₃≥1, which cannot be (see definition of classU). Thus, equation (12) for

m¹

≥1 has a unique solution z^m1(x, t) =

m¹, λ(x)

−λ1(x)−1

h^m1(x, t), 1≤ m¹

≤Nh, inn classC^∞[x0, X]×[0, T].

We have proved the following statement.

Theorem 1. Let conditions (i)-(ii), (iv) be fulfilled and the right-hand side

h(x, t, τ) =h₀(x, t) +

3

X

i=1

h_i(x, t)e^τi+

∗

X

2≤|m|≤Nz

h^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤NH

h^e1+m(x, t)e^(e1+m,τ)∈U

Ре по зи то ри й Ка рГ У

of equation (9) belongs to the space U. Then, for the solvability of equation (9) in spaceU, it is necessary and sufficient that condition (13) is satisfied.

Under constraint (13), equation (9) has the following solution in space U:

y(x, t, τ) =y0(x, t) +ξ(x, t)e^τ1+

3

X

i=2

[λi(x)−λ1(x)]⁻¹Hi(x, t)e^τi+

+

∗

X

2≤|m|≤NH

[(m, λ(x))−λ1(x)]⁻¹H^m(x, t)e^(m,τ)+ (14)

∗

X

1≤|m|≤NH

m¹, λ(x)

−λ1(x)⁻¹

H^e1+m(x, t)e^(e1+m,τ),

where ξ(x, t)∈C^∞[x₀, X]×[0, T] are arbitrary function, y₀(x, t) is the solution of an integral equation(11₀), m≡(0, m₂, m₃), m₂6=m₃, |m|=m₂+m₃≥1.

Subject the solution (14) to the initial conditiony(x0, t,0) =y_∗(t).Then we have

ξ(x₀, t)−λ⁻¹₁ (x₀)h₀(x₀, t) +

3

X

i=2

[λ_i(x₀)−λ₁(x₀)]⁻¹h_i(x₀, t)+

+

∗

X

2≤|m|≤Nh

[(m, λ(x₀))−λ₁(x₀)]⁻¹h^m(x₀, t)+

∗

X

1≤|m|≤Nh

m¹, λ(x₀)

−λ₁(x₀)−1

h^e1+m(x₀, t) =y_∗ ⇔

⇔ ξ(x0, t) =y∗+λ⁻¹₁ (x0)h0(x0, t)−

3

X

i=2

[λi(x0)−λ1(x0)]⁻¹hi(x0, t)− (15)

−

∗

X

2≤|m|≤N_h

[(m, λ(x₀))−λ₁(x₀)]⁻¹h^m(x₀, t)−

∗

X

1≤|m|≤N_h

m¹, λ(x₀)

−λ₁(x₀)−1

h^e1+m(x₀, t).

However, the functions ξ(x, t) were not found completely. An additional requirement is required to solve problem (13). Such a requirement is dictated by iterative problems (8_k), from which it can be seen that the natural additional constraint is the condition

−∂y

∂x +g(x)

2 (e^τ2+e^τ3)y+R₁y+p(x, t, τ)≡0, (∀(x, t)∈[x₀, X]×[0, T]), (16) where p(x, t, τ) =p0(x, t) +

3

P

i=1

pi(x, t)e^τi+

∗

P

2≤|m|≤Nz

p^m(x, t)e^(m,τ)+

∗

P

1≤|m|≤NH

p^e1+m(x, t)e^(e1+m,τ)∈U is the known vector-function. The right part of this equation:

G(x, t, τ)≡ −∂y

∂t +g(x)

2 (e^τ2σ1+e^τ3σ2)y+Q(x, t, τ) =

=− ∂

∂x



y0(x, t) +

3

X

i=1

yi(x, t)e^τi+

∗

X

2≤|m|≤Ny

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤Ny

y^e1+m(x, t)e^(e1+m,τ)



+

+g(x)

2 (e^τ2σ₁+e^τ3σ₂)



y₀(x, t) +

3

X

i=1

y_i(x, t)e^τi+

∗

X

2≤|m|≤Ny

y^m(x, t)e^(m,τ)+

+

∗

X

1≤|m|≤Ny

y^e1+m(x, t)e^(e1+m,τ)



+p(x, t, τ),

Ре по зи то ри й Ка рГ У

may not belong to 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 will have

Z(x, t, τ)≡G(x, t, τ) +∂y

∂x =g(x)

2 (e^τ2σ₁+e^τ3σ₂)

"

y₀(x, t) +

3

X

i=1

y_i(x, t)e^τi+

+

∗

X

2≤|m|≤Nz

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤Nz

y^e1+m(x, t)e^(ej+m,τ)



=

= g(x)

2 y0(x, t) (e^τ2σ1+e^τ3σ2) +

3

X

i=2

g(x)

2 yi(x, t) e^τ1+τ2σ1+e^τ1+τ3σ2

+

+g(x)

2 (e^τ2σ1+e^τ3σ2)





∗

X

2≤|m|≤Nz

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤Nz

y^e1+m(x, t)e^(e1+m,τ)



+p(x, t, τ).

Here are terms with exponents

e^τ2+τ3 =e^(m,τ)|m=(0,1,1), e^τ2+(m,τ)(m2+ 1 =m3), e^τ3+(m,τ)(m3+ 1 =m2), e^{τ2+(e1+m,τ)}(m2+ 1 =m3), e^{τ3+(e1+m,τ)}(m3+ 1 =m2)

(∗)

do not belong to spaceU, since in multi-indexm= (0, m2, m3)of the spaceU must bem26=m3, m2+m3≥1.

Then, according to the well-known theory (see [5], p. 234), we embed these terms in the spaceU according to the following rule (see(∗)):

e\^τ3+τ2 = 1, e^τ\^2+(m,τ)= 1 (m2+ 1 =m3, m26=m3),e^τ\^3+(m,τ)= 1 (m3+ 1 =m2, m26=m3), e^τ2\^+(e1+m,τ)=e^τ1(m2+ 1 =m3, m26=m3), e^τ3+(e\^1+m,τ)=e^τ2(m3+ 1 =m2, m26=m3).

(∗∗)

InZ(x, t, τ)need of embedding only the terms

M(x, t, τ)≡

3

X

i=2

g(x)

2 y_i(x, t) e^τi+τ2σ₁+e^τi+τ3σ₂ +g(x)

2 y₁(x, t) e^τ1+τ2σ₁+e^τ1+τ3σ₂ ,

S(x, t, τ)≡ g(x)

2 (e^τ2σ1+e^τ3σ2)





∗

X

2≤|m|≤Nz

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤Nz

y^e1+m(x, t)e^(e1+m,τ)



. We describe this embedding in more detail, taking into account formulas(∗∗) :

M(x, t, τ)≡

2

X

k=1

g(x)

2 yk(x, t) e^τk+τ2σ1+e^τk+τ3σ2

+g(x)

2 y1(x, t) e^τ1+τ2σ1+e^τ1+τ3σ2

=

= g(x) 2

y1(x, t)e^τ1+τ2σ1+y1(x, t)e^τ1+τ3σ2+y2(x, t)e^2τ2σ1+ +y₂(x, t)e^τ2+τ3σ₂+y₃(x, t)e^τ3+τ2σ₁+y₃(x, t)e^2τ3σ₂

⇒

⇒Mc(x, t, τ) =g(x) 2

y1(x, t)e^τ1+τ2σ1+y1(x, t)e^τ1+τ3σ2+ +y₂(x, t)e^2τ2σ₁+y₂(x, t)σ₂+y₃(x, t)σ₁+y₃(x, t)e^2τ3σ₂

.

(note that inMc(x, t, τ)there are no members containinge^τ1 measurement exponents|m|= 1);

S(x, t, τ)≡g(x)

2 (e^τ2σ1+e^τ3σ2)





∗

X

2≤|m|≤Nz

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤Nz

y^e1+m(x, t)e^(e1+m,τ)



=

Ре по зи то ри й Ка рГ У

= ^g(x)₂

"

∗

P

2≤|m|≤Nz

y^m(x, t) e^τ2+(m,τ)σ₁+e^τ3+(m,τ)σ₂ +

+

∗

P

1≤|m|≤Nz

y^e1+m(x, t) e^{(e1+m,τ)+τ2}σ1+e^{(e1+m,τ)+τ3}σ2

#

⇒

⇒S(x, t, τ) =b g(x) 2







 X

2≤|m|≤Nz, m₂+1=m₃

y^m(x, t)σ1+ X

2≤|m|≤Nz, m₃+1=m₂

z^m(x, t)σ2+

∗

X

2≤|m|≤Nz, m2+16=m3,m3+16=m2

y^m(x, t)e^(m,τ)+

+







 X

1≤|m|≤Nz, m2+1=m3

y^e1+m(x, t)σ₁+ X

1≤|m|≤Nz, m3+1=m2

y^e1+m(x, t)σ₂







 e^τ1+

∗

X

1≤|m|≤Nz, m₂+16=m3,m₃+16=m2

y^e1+m(x, t)e^(e1+m,τ).

After embedding, the right-hand side of equation (16) will look like

G(x, t, τ) =b − ∂

∂x



y0(x, t) +

3

X

i=1

yi(x, t)e^τi+

∗

X

2≤|m|≤Nz

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤Nz

y^e1+m(x, t)e^(e1+m,τ)



+

+Mc(x, t, τ) +Sb(x, t, τ) +R1y(x, t, τ) +p(x, t, τ),

moreover, in S(x, t, τb ) the coefficient at e^τ1 do not depend on y1(x, t). As indicated in [5], the embedding G(x, t, τ)→G(x, t, τ)b will not affect the accuracy of the construction of asymptotic solutions of problem (2), sinceZ(x, t, τ)|b _τ=ψ(x)/ε≡Z(x, t, τ)|_τ=ψ(x)/ε.

We show that the problem (9) has the unique solution in the spaceU if (16) is satisfied.

Theorem 2. Let the conditions (i)-(iv) take place and the right-hand side

h(x, t, τ) =h₀(x, t) +

3

X

i=1

h_i(x, t)e^τi+

∗

X

2≤|m|≤Nz

h^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤NH

h^e1+m(x, t)e^(e1+m,τ)∈U

satisfy the condition (13). Then the problem (9) is uniquely solvable in the space U under the additional condition (16).

Proof.To use the condition (16), we calculate the expression−^∂y_∂x+^g(x)₂ (e^τ2+e^τ3)y+R₁y+p(x, t, τ).Since

−∂

∂x



y₀(x, t) +

3

X

i=1

y_i(x, t)e^τi+

∗

X

2≤|m|≤N_z

y^m(x, t)e^(m,τ)+

∗

X

1≤|m|≤N_z

y^e1+m(x, t)e^(e1+m,τ)



+

+Mc(x, t, τ) +S(x, t, τ) +b

3

X

i= 1

I_i⁰(K(x, t, s)y_i(s, t))

s=xe^τi− I_i⁰(K(x, t, s)y_i(s, t))

s=x₀

i +

+

∗

X

1≤|m|≤Nz

h

I_m⁰ (K(x, t, s)y^m(s, t))

s=xe^(m,τ)− I_m⁰ (K(x, t, s)y^m(s, t))

s=x0

i +

+

∗

X

1≤|m|≤Nz

I_e⁰_1+m

K(x, t, s)y^{e1 +m}(s, t)

s=xe^(e1+m,τ)− I_e⁰_1+m

K(x, t, s)y^{e1 +m}(s, t)

s=x₀

+

+p(x, t, τ).

therefore(16) takes the form

∂(ξ(x, t))

∂x +K(x, t, x)

λ1(x) ξ(x, t) +p1(x, t)e^τ1+p0(x, t) = 0.

Ре по зи то ри й Ка рГ У