Solution of one initial-boundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method
Bohdan Kopytko
,Zhanneta Tsapovska
Journal of Applied Mathematics and Computational Mechanics |
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SOLUTION OF ONE INITIAL-BOUNDARY WENTZEL PROBLEM FOR A PARABOLIC EQUATION WITH DISCONTINUOUS COEFFICIENTS BY THE BOUNDARY INTEGRAL EQUATION METHOD
Bohdan Kopytko 1, Zhanneta Tsapovska 2
1 Institute
of Mathematics, Czestochowa University of Technology
Częstochowa, Poland
2 Ivan Franko Lviv National University
Lviv, Ukraine
bohdan.kopytko@gmail.com, tzhannet@yahoo.com
Received: 11 January 2017; accepted:
10 March 2017
Abstract. In this article we consider the question of existence in the Holder
class of the solution of the initial-boundary problem for a linear parabolic
second-degree equation with discontinuous coefficients in noncylindrical
domain. This domain is bounded of the smooth elementary surfaces of the Holder
class .
The boundary conditions and the conjugation condition of the Wentzel type are
given to external and internal boundaries of
a domain respectively. We use the potential method to
solve this problem.
MSC 2010: 35K20
Keywords: linear parabolic second-degree operator, potential method, conjugation condition of Wentzel type
1. Introduction
The theory of potentials is very important in the study of the Cauchy problem, the boundary-value problem, the conjugation problems for the heat equations and the general second-degree parabolic equations as well. The potential method is used to thoroughly examine the initial-boundary problems for the uniformly parabolic equations when the order of the differential boundary operators is less than the order of the equation in the domain [1-8]. We can encounter the initial-boundary problems which contain derivatives of the second and higher orders. The Wentzel problem is a vivid example of this type [9]. This is the initial-boundary problem for the parabolic equation with the boundary condition which has the form of a parabolic operator on the tangential variables. That problem arises, in particular, into the theory of Markov processes in the construction of a diffusion process in a domain on predetermined the diffusion coefficients and the boundary conditions.
The parabolic initial-boundary Wentzel problem (in a cylindrical domain) was investigated in the works [10-12] by the methods of functional analysis. In the papers [13, 14] (in cylindrical and noncylindrical domains) this problem was studied by the boundary integral equation method using a simple-layer potential. As for the parabolic problem with Wentzel conjugation conditions, this problem, for the case of a cylindrical domain, is studied in the most general formulation in the papers [15, 16].
In this article, we consider one of the problems in the assumption that the boundaries of the domains are the elementary noncylindrical surfaces of the Holder class.
2. Problem statement and its solution
In a
layer where
is fixed,
is the
-dimensional Euclidian space of the points
we consider the domain
with the smooth
boundary
where
,
We
assume, that the surface
subdivides the
domain
into two domains
and
with
the boundaries
and
Let
in particular by
we denote
at
By
we denote the
unit normal vector at the point
to the surface
which is in the section
and the vectors
and
directed
inwards to the domains
and
respectively.
is the value of the function
on the surface
i.e.
The differential operators with respect to and
we denote by
and
(
) is a tangent differential operator on
i.e.
where
is the Kronecker symbol. If
then
Let and
be any domains,
and
are the closure of them,
and
are some numbers,
is an integer and
Similar to [3, ch. I, § 1] is the class
surface,
and
are the
corresponding Holder spaces with the norms
and
which are defined on
and
respectively.
is the subspace
of functions from
that together with admissible derivative with respect to the time
variable, vanishes at
are positive constants independent of
We
are not interested in their specific value.
In a layer let us consider
two second-order uniformly parabolic equations
![]() | (1) |
Assume that the coefficients of the operator are
defined in
and the following assumptions are true:
(А1)![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Assumptions (A1), (A2) guarantee the
existence of a fundamental solution (f.s.) for each equation from (1) (see [3,
ch. IV, § 11]) which we will denote by (
),
Let us consider the integrals - the parabolic simple-layer potentials:
![]() ![]() ![]() | (2) |
![]() ![]() ![]() | (3) |
where the functions and
defined, bounded
and continuous on surfaces
and
respectively.
We note some properties of the potential (2),
(3) (see [1-4]). The functions (2) and (3) satisfy the equation (1) at each point and
respectively,
and they also satisfy the initial condition
At the points of the surfaces and
let the conormal vectors
and
where
be defined. The important property of
the simple-layer potential reflected in the boundary relations for the conormal
derivative of this potential (see [2, ch. V, § 2], [3, ch. IV, § 15], [4, 5]).
Using a f.s. we can identify
and explore in the unbounded
domain the properties of two integrals connected with the operator
![]() ![]() | (4) |
![]() ![]() | (5) |
The first integral is called the volumetric
potential and the second one is called the Poisson potential. If specified
functions and
are bounded and
continuous,
satisfies the Holder condition for variable
uniformly
relative to
then the function
satisfies the equation
![]() ![]() | (6) |
in the
domain with a zero initial condition
and the
function
satisfies equation (1)
in the same domain with the initial condition
![]() ![]() ![]() | (7) |
Considering this, we can give the general
classical solution of the Cauchy problem
(6), (7) as the sum of potentials (4), (5). And
if
then the potentials (4),
(5) and therefore
the solution of the prob-
lem (6), (7) belongs to the Holder class
We will consider the following conjugation
problem: we have to find the function based on the
conditions
![]() ![]() ![]() | (8) |
![]() ![]() ![]() | (9) |
![]() ![]() | (10) |
![]() | (11) |
![]() | (12) |
![]() | (13) |
where
We assume
that for the coefficients of the Wentzel type operators and
the
following conditions hold:
(В1) ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
(В2)
![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
Also we assume, that
![]() ![]() ![]() | (14) |
![]() ![]() ![]() |
![]() ![]() ![]() ![]() | (15) |
We will assume that for the function from
(8)-(13) the agreement conditions hold at
and
these conditions are determined by
a given the boundary conditions (12), (13) and the conjugation condition (11).
Then the following statement is true.
Theorem. Let for the coefficients of the
operators and
conditions (А1), (А2) and
(В1), (В2) hold, respectively, for the surfaces
and
the functions
from
the right-hand side (8)-(13) the conditions (14) and (15) hold. Then the
problem (8)-(13) has a unique solution
![]() ![]() | (16) |
in the performance the appropriate agreement conditions and the estimation
![]() | (17) |
is true.
Proof. We will look for the solution of the problem (8)-(13) in the form
![]() ![]() ![]() | (18) |
of the sum of the simple-layer potentials (2), (3) with the unknown densities and the potentials (4), (5) with the known functions
Using the properties of these potentials, we will find the unknown
functions
and
so that for
the conditions (10)-(13) have been met.
Let us consider a priori that unknown
densities and
satisfy the conditions
![]() ![]() ![]() | (19) |
Now we pass to investigating the conjugation condition by Wentzel type (11). First, we transform this equality by separating its tangential and conormal components in the expressions that contain the derivatives of the first order in space variables using ratio
![]() ![]() | (20) |
where is a
tangent differential operator on
Then using (20) and the relationship from the theorem on the jump of the conormal derivative from the simple-layer potential (see [4]), we can write the condition (11) as
![]() | (21) |
where
![]() | (22) |
For the kernel in the first and second integral from (22)
the estimations (
)
![]() | (23) |
are true.
Now, consider (21) as the autonomic parabolic
equation on for the function
For its solution we introduce the
following transformation of the variables:
![]() ![]() ![]() ![]() ![]() |
The conjugation condition (21) in new variables will take the form:
![]() | (24) |
where
|

If follows from the conditions of Theorem,
the additional assumption (19), formulae (25) and properties of the potentials
that the coefficients and the function on the right-hand side of this equation
belong to the space It is known that, the unique
solution
of the equation (24) which satisfies
the initial condition
can
be represented by the formula
![]() | (26) |
where (
) is a f.s. of the uniformly para-
bolic equation
Returning to the variables
we can write equality (26) as
![]() | (27) |
where
The function
belongs
to class
as well.
Thus, we have two representations for values
of function on
relation (18), where one should put
and relation
(27). Then, comparing the right-hand sides of equalities (18) and (27) and
taking into account (25), we obtain the first integral equation for the unknown
functions
and
Using the
equality (27) and the conjugation condition (10), we find the second equation
for these functions. The third and fourth equations of the required system for
and
we obtain from
the boundary conditions (12) and (13) similar to the way in which we found the
first equation. After appropriate transformations, an obtained system of four
equations for the unknown functions
and
can be represented as
![]() | (28) |
where
![]() ![]() |
![]() |
![]() |
![]() |
and
are the f.s. of the parabolic equations
which we obtained after transformation of the boundary conditions (12) and (13)
using the scheme to obtain
the equation (21). The kernels
are expressed by the functions which
have a „weaker” singularity than the function
at
So, we have a system of four integral
Volterra equations of the first kind (28) for and
The functions
and
from right-hand side of equations of this system belong to the Holder classes
and
respectively.
In order to transform each of the equations of this
system, we introduce the special integro-differential operators similar to the
operator that was introduced (see [4, 8, 13-16]) in the study of the first
boundary-value parabolic problem by the boundary integral equation method.
Let In
this case, the integro-differential operators (denote them by
) which will be used to transform the first two equations of system
(28), can be defined by the formula
![]() | (29) |
Here the function (
),
is a f.s. of
the uniformly parabolic operator, which is a trace of the operator
on
To transform the third and fourth
equations of the system (28), we use the integro-differential operators
which are similar to the operator
from (29). To this end on the right-hand side (29) we should
replace the function
and integrate over the surface
to the function
and
integrate over the surface
respectively. Here
(
) is a f.s. of the parabolic operator, which is a trace of the
operator
on
Applying and
to both sides of the corresponding
equations of the system (28), we transform this system into the equivalent system of the integral
Volterra equations of the second kind
|

where
![]() ![]() |
![]() ![]() |
And for the kernels the inequality (23) is true.
Solving the system of equations (30) by the
method of subsequent approxima-tions, we find One can additionally verify that
and
satisfy the condition (19).
We obtained the solution of the problem (8)-(13) by formulas (18), (30). To complete the proof of Theorem, we have to only check that this solution satisfies the condition (16) and the estimate (17). We have to also verify the statement of the Theorem on the uniqueness of this solution.
In this regard, we note that the strict proof of these facts practically repeats the similar statements in the papers [13-16]. The theorem is proved.
7. Conclusions
In the article, we investigated the question
of the classic solvability of the parabolic initial-boundary problem with the
boundary conditions and one Wentzel
conjugation condition in the assumption that the boundaries of the domains are
the elementary noncylindrical surfaces of Holder class The
solution
is obtained by the usual parabolic simple-layer potentials by using the
boundary
integral equation method. The proposed approach can be used to solve a similar
conjugation problem in the noncylindrical domain of the more general type.
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