About Dirichlet boundary value problem for the heat equation in the infinite angular domain
© Jenaliyev et al.; licensee Springer 2014
Received: 4 May 2014
Accepted: 4 September 2014
Published: 25 September 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
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
4 Statement of the boundary value problem L
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.
and is continuous on ;
where , t satisfy a condition .
Here we have used the asymptotic formula , p.718, Formula 3].
6 Investigation of the integral equation (10)
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
and is continuous on ;
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.
10 Solving the characteristic equation
Assuming that the right side of (34) is known, we will find its solution, that is, the solution to the characteristic equation.
11 Reducing (33) to the Abelian equation
12 Solving the Abelian equation (43)
and the function is determined according to (45).
14 Estimate of non-trivial solution (49)
where we use the substitution .
The following is established.
15 The main result. Classes of uniqueness
(The main result)
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.
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’).
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