1. Introduction.
This work consists of seven parts.
The first part is an introduction.
The second part is the nomenclature.
In the third part, the problem is formulated and an asymptotic solution is constructed for a singularly perturbed boundary value problem for a parabolic equation on the semiaxis when the stability conditions for the spectrum of the limit operator are violated—the presence of a “simple” rational turning point.
In the fourth part, an estimate for the remainder of the asymptotic series is given.
In the fifth part (Application), auxiliary lemmas and theorems are proved.
The sixth part is an Conclusion.
The seventh part is an Bibliography.
Among singularly perturbed problems, problems with an unstable spectrum of the limit operator are of particular interest. Various asymptotic methods have been developed to construct solutions to such problems. Such as methods: V.P. Maslov’s school, A.B. Vasilieva’s–V.F. Butuzov’s–N.N. Nefedov’s school and others. One of such methods is the method of regularization S.A. Lomov’s school. This method allows you to build a solution over the entire region of integration. A “simple” pivot point means that the eigenvalues of the limit operator are isolated, and one eigenvalue at separate points of
t vanishes. Problems with such features were considered in the works [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]. The Cauchy problem for a singularly perturbed parabolic equation with a “simple” rational turning point was considered in [
12].
When solving such problems, functions arise that describe the irregular dependence of the solution on
. The identification and description of these functions is the main problem of the regularization method. Under conditions of a stable spectrum of the limit operator, i.e. under conditions
, the eigenvalues of the limit operator
,
,
,
, the singularities are:
In this paper, we construct regularizing functions related to the singularities of the boundary value singularly perturbed problem.
2. Nomenclature.
All quantities in the article are dimensionless:
1. -variables;
2. ;
3. -small parameter varying within ;
4. ;
;
- functions;
5. -operators;
6. - constants;
7. - problem solution area;
8. -integers.
3. Formulation of the Problem. Construction of an Asymptotic Solution
And conditions are met
- (1)
, , ;
- (2)
, ;
- (3)
, , .
To prove an estimate for the remainder of the asymptotic series of a solution, additional assumptions are required
- (4)
;
- (5)
;
- (6)
.
Singularly perturbed problems arise in the case when the domain of definition of the initial operator depending on
, at
, does not coincide with the domain of definition of the limit operator at
. Under the stability condition for the spectrum of the limit operator, essentially singular singularities are described using exponentials of the form
,
,
, where
are the limit operator’s eigenvalues,
are smooth (in general, complex) functions of a real variable
t. If the stability conditions for the limit operator are violated for at least one point of the spectrum of the limit operator, then new singularities arise in the solution of the inhomogeneous equation. When studying problems with a “simple” turning point, we are faced with a problem when the range of the original operator does not coincide with the range of the limit operator [
12] (p. 1).
To regularize the problem, we introduce an additional variable:
and the extended function
, which when narrowed
gives an exact solution to the problem (
1).
Let us calculate the derivatives for the extended function
:
In what follows, ∼ will be omitted. Then the problem (
1) takes the form
The regularizing functions of the problem (
2) have the form
Here
.
The properties of the operator
are described in the application. The operator
transfers any smooth function
to the solution of the problem:
Note that
for
after restricting to
is essentially special. Following the regularization method, the solution to problem (
2) is sought in the form
where the functions
—are smooth in
and depending on
.
Substituting (
4) in (
2), we get a problem with respect to functions
,
,
,
:
Based on the distribution of the parameter
in system (
5), it follows that the functions
,
,
decompose in whole powers
, or
by powers
:
Note that if , then the term with this index is equal to zero.
Since does not satisfy the theorem on the pointwise solvability of the equation , then the expansion of the solution begins with .
. System (
6) in this case has the form
Hence , , , is an arbitrary function.
From the initial and boundary conditions, we have
To determine the unknown functions, consider the following iterative problem.
. System (
6) in this case has the form
here
.
As
, then we get an equation for
or:
From here
where
arbitrary function.
From the properties of the operator
F we obtain for
. Then
. Putting
, we get
Solution (
7) has the form
The function
at this iteration step is arbitrary. To determine
, consider the system (
6) at step zero.
. System (
6) looks like:
Subordinate the right side of the equation for
to the conditions of the point solvability theorem. To do this, expand the right-hand side of the equation in the Taylor-Maclaurin formula.
Then
where
—whole and fractional parts.
After
is determined, the
solution is found at the “
” iteration step:
or
Using the property of the operator
,
can be represented as
The solution of the equations of the system (
8) have the form
Let’s solve the equation for
. Since
, then
Solution (
11) has the form:
As , then
.
From the initial conditions for
we have
Thus, at the step “
”
is defined. Taking into account the properties of the operator
, the solution
can be represented in the form
The function
is arbitrary at this iteration step. To determine
, it is necessary to consider the system at the “1” th iteration step; to determine
, it is necessary to consider the system at the step “
”. At the zero iteration step,
has the form
Submit to the boundary condition:
Solution (
12) has the form
To determine
, consider the equation at step
for
where
is determined from the relation
Subordinate the right side of the Equation (
14) to the conditions of solvability. For this we expand
by the Taylor-Maclaurin formula.
Thus the terms for the solution at the zero iteration step have been finally found:
Thus, at this iteration step, after restricting the solution to
, we obtain the main term of the regularized asymptotics of the solution to the boundary value problem on the semiaxis for the parabolic equation:
Using the properties of the operator
(Lemma 3) can be represented as