Solution of one initialboundary 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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@article{Kopytko_2017, doi = {10.17512/jamcm.2017.1.05}, url = {https://doi.org/10.17512/jamcm.2017.1.05}, year = 2017, publisher = {The Publishing Office of Czestochowa University of Technology}, volume = {16}, number = {1}, pages = {5162}, author = {Bohdan Kopytko and Zhanneta Tsapovska}, title = {Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method}, journal = {Journal of Applied Mathematics and Computational Mechanics} }
TY  JOUR DO  10.17512/jamcm.2017.1.05 UR  https://doi.org/10.17512/jamcm.2017.1.05 TI  Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method T2  Journal of Applied Mathematics and Computational Mechanics JA  J Appl Math Comput Mech AU  Kopytko, Bohdan AU  Tsapovska, Zhanneta PY  2017 PB  The Publishing Office of Czestochowa University of Technology SP  51 EP  62 IS  1 VL  16 SN  22999965 SN  23530588 ER 
Kopytko, B., & Tsapovska, Z. (2017). Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method. Journal of Applied Mathematics and Computational Mechanics, 16(1), 5162. doi:10.17512/jamcm.2017.1.05
Kopytko, B. & Tsapovska, Z., 2017. Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method. Journal of Applied Mathematics and Computational Mechanics, 16(1), pp.5162. Available at: https://doi.org/10.17512/jamcm.2017.1.05
[1]B. Kopytko and Z. Tsapovska, "Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 1, pp. 5162, 2017.
Kopytko, Bohdan, and Zhanneta Tsapovska. "Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method." Journal of Applied Mathematics and Computational Mechanics 16.1 (2017): 5162. CrossRef. Web.
1. Kopytko B, Tsapovska Z. Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(1):5162. Available from: https://doi.org/10.17512/jamcm.2017.1.05
Kopytko, Bohdan, and Zhanneta Tsapovska. "Solution of one initialboundary Wentzel problem for a parabolic equation with discontinuous coefficients by the boundary integral equation method." Journal of Applied Mathematics and Computational Mechanics 16, no. 1 (2017): 5162. doi:10.17512/jamcm.2017.1.05
SOLUTION OF ONE INITIALBOUNDARY 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 initialboundary problem for a linear parabolic seconddegree 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 seconddegree 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 boundaryvalue problem, the conjugation problems for the heat equations and the general seconddegree parabolic equations as well. The potential method is used to thoroughly examine the initialboundary 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 [18]. We can encounter the initialboundary problems which contain derivatives of the second and higher orders. The Wentzel problem is a vivid example of this type [9]. This is the initialboundary 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 initialboundary Wentzel problem (in a cylindrical domain) was investigated in the works [1012] 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 simplelayer 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 secondorder uniformly parabolic equations
(1) 
Assume that the coefficients of the operator are defined in and the following assumptions are true:
(А1); (А2) 
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 simplelayer 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 [14]). 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 simplelayer 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 righthand 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 simplelayer 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 simplelayer 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 righthand 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 righthand 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 righthand 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 integrodifferential operators similar to the operator that was introduced (see [4, 8, 1316]) in the study of the first boundaryvalue parabolic problem by the boundary integral equation method.
Let In this case, the integrodifferential 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 integrodifferential operators which are similar to the operator from (29). To this end on the righthand 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 approximations, 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 [1316]. The theorem is proved.
7. Conclusions
In the article, we investigated the question of the classic solvability of the parabolic initialboundary 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 simplelayer 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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