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About Dirichlet boundary value problem for the heat equation in the infinite angular domain
Boundary Value Problems volume 2014, Article number: 213 (2014)
In this paper it is established that in an infinite angular domain for Dirichlet problem of the heat conduction equation the unique (up to a constant factor) non-trivial solution exists, which does not belong to the class of summable functions with the found weight. It is shown that for the adjoint boundary value problem the unique (up to a constant factor) non-trivial solution exists, which belongs to the class of essentially bounded functions with the weight found in the work. It is proved that the operator of a boundary value problem of heat conductivity in an infinite angular domain in a class of growing functions is Noetherian with an index which is equal to minus one.
MSC: 35D05, 35K20, 45D05.
Different kinds of processes of mass and heat transfer lead to solving boundary value problems for parabolic equations in a domain with a moving in time boundary (non-cylindrical domain). These processes are the most important factor that affects, for example, the reliability of various contact systems. Due to the increased speed-in-action of the electrical contacts, that is, because of the short duration of the process, it is experimentally impossible to determine accurately the temperature field of the contact system and the dynamics of its change in time. Therefore, the study of boundary value problems of heat conduction in domains with moving boundary and the degeneracy at the initial time is actual. Consideration of a wide range of issues of mathematical physics , , in particular, the solving of boundary value problems in the heat equation degenerating domains leads to the need to study the singular integral equations of Volterra type when the norm of a integral operator is equal to unit. These problems have a direct connection with the theory of loaded equations , . It turned out that these issues have a close connection with the problem of establishing the classes of uniqueness from –, which have been further developed in – and other works.
2 On classes of uniqueness
Let us give a brief overview of some works on uniqueness classes for parabolic equations. In the domain for the boundary value problem
where A is the linear elliptic operator of orders 2p, the following classes of uniqueness are established:
(Ladyzhenskaya  for one equation with coefficients depending only on t).
3 VP Mihajlov’s example on the existence of non-trivial solution for the homogeneous Dirichlet problem in the degenerate domain
Let be the domain bounded by a closed curve Γ: , passing through the origin of coordinates and the point and symmetrical with respect to the axis 0t. The boundary value problem 
has a non-trivial solution
We note that although , the following inclusions hold:
4 Statement of the boundary value problem L
In the domain it is required to find a solution to the heat conduction equation
satisfying the boundary conditions
where has to belong to the class
We note that the functions
Boundary value problems of the form (1)-(2) arise in the mathematical modeling of thermophysical processes in high-current electric arc of the disconnecting device. Tool for describing the physics of the processes in the arc is the heat equation, in which the influence of the heat sources in the arc and the effect of the contraction of the axial section of the arc in the cathode region to a contact spot are taken into account. The diameter of the contact spot is several orders smaller than the diameter of the developed section of the arc column and this fact gives the chance to consider the contact spot as a mathematical point. The solution domain changes over time according to the law which is defined by the conditions of the bridging contact. At the fixed terminal time the contacts close and the solution domain of the problem degenerate. From a mathematical point of view, problematic nature of the problem consists in the presence of a moving boundary and degeneracy of the solution domain at the fixed terminal time. It should be emphasized that the boundary value problems for parabolic equations in a domain with a moving boundary are fundamentally different from the classical problems. Due to the size of the domain depending on the time for this type of problems methods of separating the variables and integral transformations not be applied, as remaining within the classical methods of mathematical physics you cannot conform the solution of the heat equation to the motion of the border line of the heat transport domain.
Excluding from system (5) , we find
We introduce the following notation:
Here we have used substitution in integral and the known equality , p.321, 3.325]
Substituting the value of the integral into (8) we have
We note that the kernel has the following properties:
and is continuous on ;
Indeed, properties (2) and (3) follow from the following equality:
where , t satisfy a condition .
To prove (3) we note that the first summand of this equality tends to zero when , since for large for the exponent we have the inequality:
Here we have used the asymptotic formula , p.718, Formula 3].
6 Investigation of the integral equation (10)
In the homogeneous integral equation (10) we transform its kernel. Using the relations
It is well known that in order to solve (10) it is sufficient to find a solution of the following equation:
To solve (13) we introduce the following substitutions:
As a result, (13) takes the form
From the latter equation using the next substitution of the required function,
Applying to (14) the Laplace transform, we obtain
that is, we have
The general solution of differential equation (15) is determined by the following formula:
To find the original of this function, we rewrite it in the form of a series
Applying the inverse Laplace transform to (17) we have
By inverse substitutions , and recalling that equality (18) takes the form
that is, for (10) a non-trivial solution is given by
For violation of condition (3), it is sufficient to show it for the solution (4), corresponding to the first term of the sum (20) which is constant. Violation of condition (3) really takes place, the homogeneous boundary value problem L (1)-(2) in class (3) has only the trivial solution.
Thus we have established the following.
7 Statement of the adjoint boundary value problem
In the domain we consider the following problem: it is required to find a non-trivial solution to the heat conduction equation
satisfying the boundary conditions:
where has to belong to the class
We note that the functions
Excluding from the system (25) , we find
Here we have used the substitution in the integral and the known equality , p.321, 3.325]
Substituting the value of the integral into (28), we have
We note that the kernel has the following properties:
and is continuous on ;
Indeed, taking into account that we have
From the last equality, the validity of properties (2), (3) of the function follows.
9 Investigation of the integral equation (30)
An important feature of (30) given by property (3) of the kernel is expressed by the fact that the corresponding non-homogeneous equation cannot be solved by the method of successive approximations.
In the homogeneous integral equation (30) we transform its kernel. Using the relations
It is well known that to solve (30) it is sufficient to find a solution of the following equation:
10 Solving the characteristic equation
To study the integral equation (33) we allocate its characteristic part, namely
Assuming that the right side of (34) is known, we will find its solution, that is, the solution to the characteristic equation.
where (see , p.86, according to the (1.2.56)] for )
11 Reducing (33) to the Abelian equation
We now calculate the inner integral in (41). We have
We calculate the integrals and . Making the substitution we find
For the integral using the equality , p.321, 3.325]
So, we have
Substituting the value in (42), we obtain
Thus, (41) takes the form
12 Solving the Abelian equation (43)
It remains to transform the first sum in (48):
since we have the decomposition , p.320 3.321 (1)]
The second sum of (48) is equal to
Now we can find the solution of the homogeneous boundary value problem for the heat conductivity equation in an infinite angular domain,
which has a nonzero solution , defined by the formula
and the function is determined according to (45).
14 Estimate of non-trivial solution (49)
To find the class of the non-trivial solution (49) we find the accurate estimate on its order of growth
We estimate the second summand of (50). We have for
where we use the substitution . Further,
Since , we have for the function
where we use the substitution . Hence it follows
Thus we have
The first two inequalities can be replaced by the following one:
Further, for the first summand in the function (50) we have
We express the function as the sum
where we use the substitution . Further, since , we have
Here we use following substitutions:
Thus we obtain
Using (53), we estimate the following summands of solutions and :
where we use the substitution .
Further we have
Thus we have
, , i.e.,
The following is established.
15 The main result. Classes of uniqueness
(The main result)
Classes of uniqueness are
Analyzing the previous expression for the function we obtain:
To summarize, the following is established.
In an infinite angular domain for the homogeneous Dirichlet problem for the heat conduction equation the existence of a unique (up to a constant factor) non-trivial solution, which, however, does not belong to the class of summable functions with the weight found in the work is proved.
For the boundary value problem adjoint to the Dirichlet problem, the existence of a unique (up to a constant factor) non-trivial solutions, which belongs to the class of essentially bounded functions with the weight is found in the work is established.
In the weight class of summable functions it is shown that the index of the Dirichlet problem is equal to minus one.
Weight classes of uniqueness for the boundary value problem considered in the work are found.
Problems in non-cylindrical domains, similar to those considered in this paper are highly relevant not only for modeling the processes of electrical contact apparatuses but also in the related field of the designing plasma torches. Similar problems arise in creating the new technologies, the production of crystals, laser technology and in the other branches. Mathematical modeling these processes allows one to carry out the optimal choice of parameters and operating modes of technological equipment and maximize economic and environmental benefits. Finally, we note that the results can be developed for non-homogeneous boundary value problems of heat conduction, when the data are selected from the corresponding weight classes.
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This study was financially supported by Committee of Science of the Ministry of Education and Scinces (Grant 0112 RK 00619/GF on priority ‘Intellectual potential of the country’).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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he Jenaliyev, M., Amangaliyeva, M., Kosmakova, M. et al. About Dirichlet boundary value problem for the heat equation in the infinite angular domain. Bound Value Probl 2014, 213 (2014). https://doi.org/10.1186/s13661-014-0213-4
- unique classes
- heat conductivity
- angular domain
- boundary value problem
- non-trivial solution
- Volterra integral equation