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Asgarov S. (Azerbaijan), Alakbarov M. (Azerbaijan), Aliev Z. (Azerbaijan), Babayev N. (Uzbekistan), Chiladze G. (Georgia), Datskovsky I. (Israel), Garbuz I. (Moldova), Gleizer S. (Germany), Ershina A. (Kazakhstan), Kobzev D. (Switzerland), Kohl O. (Germany), Ktshanyan M. (Armenia), Lande D. (Ukraine), Ledvanov M. (Russia), Makats V. (Ukraine), Miletic L. (Serbia), Moskovkin V. (Ukraine), Murzagaliyeva A. (Kazakhstan), Novikov A. (Ukraine), Rahimov R. (Uzbekistan), Romanchuk A. (Ukraine), Shamshiev B. (Kyrgyzstan), Usheva M. (Bulgaria), Vasileva M. (Bulgar).
Phisics and Mathematics
The theory of the differential equations with the deviating argument belongs to number of the relatively young and roughly developing sections of the theory of the ordinary differential equations. There is a number of monographs, entirely or partially devoted to various aspects of this theory. We will specify, first of all, Myshkis A.D. monographs. [1], Elsgoltsa L.E. [2; 3], Krasovsky N. N. [4], Pinni E. [5], Bellman R. and Cook K.L. [6], Norkina of Page B. [7]. The equations with the late argument appear, for example, every time when in the considered physical or technical task force operating on a material point depends on speed or the provision of this point not only at present, but also at some moment preceding this.
For the equation with the deviating argument the considerable number of mathematical works is devoted to creation of the theory of boundary tasks in recent years. Now one of the directions in this theory is developed by Azbelevy N. V. and its pupils [14].
Problem definition. Let  the square limited to pieces:
Through we will designate a set of functions twice continuously differentiable on and once continuously differentiable on in area . The border of area is understood as set of pieces
We will consider in Hilbert space the mixed task for the heat conductivity equation:
.
. 
(1) (2) (3) 
where
To find Fourier decomposition of the solution of the mixed task (1)(3).
The purpose  to receive Fourier submissions of solutions of a task (1)(3).
II. MATERIAL AND METHODS
The regular solution of a task (1)(3) we will call the function turning into identity the equation (1) and regional conditions (2)(3).
We will call function the strong solution of a task if there is a sequence of functions and meeting regional statements of the problem such and , as meets in respectively to and at .
The regional task (1)(3) is called strongly solvable if the strong solution of a task exists for any right part and only.[15]
In work methods of the complex analysis, theory of operators, theories of the differential equations, the spectral theory of differential operators and the theory of regular expansions are used.
III. RESULTS
Through we will designate the operator determined by a formula
In space , it is obvious that the selfconjugate and unitary operator meeting a condition , where  the single operator.
Affecting with the operator both members of equation (1), we have
.
Now we investigate spectral properties of the operator . For this purpose we will consider a spectral task:
или . 
(4) 
We look for the solution of this task in a look:
.
Having substituted this expression in the equation (4), we will receive
,
,
.
Therefore,
,
.
from where and .
For functions we will receive an infinite series of spectral tasks:
.
Thus, at everyone fixed values it is necessary to solve a spectral problem:

(5) 
We will consider more general task:

(6) 
where  any complex (generally speaking) constant, and spectral parameter.
Having differentiated the equation and having used a boundary condition (6), we will receive a problem of StormLiouville:
,
If function is own function of a regional task (6), it is also own function for a problem of StormLiouville:

(7) (8) (9) 
Now we will assume the return, i.e. let function be own function of a problem of StormLiouville (7)(9), then whether there will be it own function for a task (6)?
We will find own functions of a regional task (7)(9). From the equation (7) we have
Believing , we will receive
which common decision has an appearance:
where  any constants. Having substituted this expression in boundary conditions (8)(9), we have
,
Therefore,
.
As, that , then , i.e. own values of a regional task
are squares of roots of the equation:
,
and own functions have an appearance:
.
Having substituted this expression in the equation (7), we have
.
Having reduced by both parts of equality, we have
.
We will transform the right part of this equality:
Therefore,
,
from where
, ,
, ,
.
Thus, if function is own function of a problem of Storm Liouville (7)(9), it is own function of also regional task (6), where
, .
We proved the following lemma.
Lemma 1 Function is own function of a regional task (6) in only case when it is own function of a regional problem of StormLiouville (7)(9).
If the material size, a regional task (6) is selfconjugate therefore the problem of StormLiouville also is selfconjugate and therefore has no the attached functions, so rated own functions of a regional task (6) form orthonormalized basis of space .
Lemma 2 If the material constant, rated own functions of a regional task (6) form orthonormalized basis of space .
We will designate own values of a regional task (5) through , and own functions corresponding to them through , and own functions of a regional task (4) through , then equality takes place:
Lemma 3 Rated own functions of a regional task:
, . 
(10) (11) 
form orthonormalized basis of space .
Proof. Orthogonality of own functions of a regional task (10)(11) follows from symmetry of the operator therefore it is enough to prove completeness of system of own functions.
Let at some functions equality take place:
Then owing to Fubini's theorem it is had
almost everywhere in area , as was to be shown.
In our case and , therefore we can formulate the following lemma.
Lemma 4 Regional task (5) has an infinite set of own values:
where  roots of the equations

(12) 
and corresponding to them own functions
where  rated coefficients, and .
We will assume that , then also equality takes place:
,
where . Therefore,
.
Then
i.e. the operator we will turn. We will find the return operator .
As , that of the last equality follows quite the operator's continuity .
Now we will return to our initial task. The solution of our task has an appearance:
where
 roots of the equations (12),  normalizing coefficients.
IV. CONCLUSIONS
As a result of research the following theorem is proved.
Theorem
(a) The mixed task (1)(3) for the equation of heat conductivity is strongly solvable in space ;
(b) The return operator is quite continuous on this space and Voltaire;
(c) "Spectral" decomposition takes place:
where
,
 roots of the equations (12),  normalizing coefficients,
 orthonormalized basis of space .
Thus, spectral properties of the indignant operator of heat conductivity are investigated; Fourier submission of solutions of the mixed task for the heat conductivity equation is brought.
Results of article are an essential contribution to development of the general spectral theory of regional tasks for the differential equations.
The received results can be applied in further researches of regional tasks to the differential equations and theories of operators.
2. Elsgolts L.E., 1955, Qualitative methods in the mathematical analysis. GTTI, Moscow, 137151.
3. Elsgolts L.E., and Norkin S. B., 1971, Introduction to the theory of the differential equations with the deviating argument. Science Publ., Moscow, 116158.
4. Krasovsky N. N., 1959, Some tasks of the theory of stability of the movement. Fizmatlit Publ., Moscow.
5. Pinni E., 1961, Ordinary differentialdifference equations. SILT Publ., Moscow, 88107.
6. Bellman R., and Cook K.L., 1967, Differentialdifference equations. World Publ., Moscow, 271327.
7. Norkin S. B., 1965, The differential equations of the second order with the late argument. Some questions of the theory of fluctuations of systems with delay. Science Publ., Moscow, 301318.
Rustemova K., Zhunisbekova D., Ashirbaev Kh., Dzhumagalieva A. ABOUT FOURIER OF REPRESENTATIONS THE SOLUTION OF THE MIXED TASK FOR THE HEAT CONDUCTIVITY EQUATION. International Journal Of Applied And Fundamental Research. – 2016. – № 5 –
URL: www.sciencesd.com/46725050 (06.06.2023).