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ILMIY METODIK JURNAL
V.I.Romanovskiy nomidagi Matematika
instituti Fizika-matematika fanlari
doktori S.Z.Djamalov tаqrizi ostidа
Kamoldinov Muhammadsodiq Baxtiyor o‘g‘li
Oziq-ovqаt texnologiyаsi vа muhаndisligi
xаlqаro instituti аssistent o‘qituvchisi
Telefon rаqаmi: +998 94 992 51 52
Orcid:
https://orcid.org/0009-0009-3416-121X
UCH O‘LCHOVLI FAZODA ISSIQLIK TARQALISH TENGLAMASI UCHUN IKKI
NUQTALI TESKARI MASALANING KORREKLIGI HAQIDA
Annotatsiya:
Ushbu maqolada uch o‘lchovli fazoda issiqlik tarqalishi tenglamasi uchun nolakal
chegaraviy to‘g‘ri va teskari masalalarning umumlashgan yechimini Sobolev fazosida mavjud va
yagonaligi o‘rganilgan.
Kalit so‘zlar:
Parallelepiped, funksiya, differensial tenglamalar, matritsa, sohada, hosila.
О КОРРЕКТНОСТИ ДВУХТОЧЕЧНОЙ ОБРАТНОЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ
РАСПРЕДЕЛЕНИЯ ТЕПЛА В ТРЕХМЕРНОМ ПРОСТРАНСТВЕ
Аннотация:
В данной статье исследуются существование и единственность обобщенного
решения нелокальных краевых прямых и обратных задач для уравнения диффузии тепла в
трехмерном пространстве в пространстве Соболева.
Ключевые слова:
Параллелепипед, функция, дифференциальные уравнения, матрица,
область определения, производная.
ON THE CORRECTNESS OF THE TWO-POINT INVERSE PROBLEM FOR THE
EQUATION OF HEAT DISTRIBUTION IN THREE- DIMENSIONAL SPACE
Annotation:
In this article, the existence and uniqueness of the generalized solution of nonlocal
boundary value direct and inverse problems for the equation of heat diffusion in three-dimensional
space in Sobolev space is studied.
Keywords:
Parallelepiped, function, differential equations, matrix,
in the field, derivative.
Introduction:
It is worth mentioning that in many scientific works, we emphasize that the
boundary conditions have a local character and the methods of solving the equations are classical
methods. However, the modern problems of natural science create the need to formulate and study
qualitatively new problems, a clear example of which is the research of non-linear boundary
problems. The relations connecting the value of the solution and its derivatives at the boundary
and interior points of the sphere are called nonlocal problems. There are many open and
understudied problems in the theory of nonlocal boundary value problems for differential
equations of both theoretical and practical interest. Among them, for example, we note a wide
category of questions related to the uniqueness, existence and stability (stationarity) of solutions.
In recent decades, nonlocal problems for differential equations have been actively studied by
many mathematicians. During the study of non-local issues, it was found that they are closely
related to the opposite issues (management issues). In the theory of differential equations, such
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ILMIY METODIK JURNAL
problems are called inverse problems, in which the problem of determining the coefficients or the
right side (external force) of the equation is understood along with determining the solution of the
differential equation. The interest in studying inverse problems for mathematical physics
equations is explained by the importance of their application in various fields of mechanics,
seismology, medical tomography, and geophysics.
Analysis of literature on the topic of research
: The main content and results of the research
work at the conference “theoretical foundations and practical issues of modern mathematics” of
Andijan State University on March 25-26, 2022. On April 21-22, 2022, the scientific and
Practical Conference of young scientists on the topic “Mathematics, Mechanics and intellectual
technologies” of the National University of Uzbekistan, named after. On April 29-30, 2022, in the
city of Dushanbe, it was expressed in the conference called "Modern problems of the number
theory of mathematical analysis". The general approach to solving incorrect problems was
developed by A. N. Tikhonov, and A. N. Tikhonov, J. Lyons, Lattes, M. M. Lavrentiev, A. S.
Alekseev, V. A. Morozov, V. Ya Arsenin, V. G. Romanov, S. I. Kabanikhin, B. A. Bubnova, A.
Kojanov, K. S. Developed by Fayazova, D. Durdiev and others. Two-point inverse problems for
mathematical physics equations have been studied very little.
Let's look at the following equation of heat
distribution in a field in the form of a parallelepiped.
(3.2.1)
Here
and
- given functions. We consider the
coefficients of equation (3.2.1) as sufficiently smooth functions. Two-point linear inverse
problem. equation (3.2.1) in the field and the following boundary conditions
(3.2.2)
(3.2.3)
(3.2.4)
And additional conditions
(3.2.5, 3.2.6)
Satisfactory and belongs to the following class
find functions. We introduce the following definitions
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elements through
we
define the second, ordered square matrix consisting of functions, i.e
,
we define its determinant by
Theorem. We assume that the following conditions are met for the coefficients of equation
(1).
optional
for s, here
and
we
assume
let
the
inequality
be
fulfilled,
here
Then it's optional
and
the following conditions
the solution of the problem (3.2.1)-(3.2.5) is given for satisfying functions
is available and unique in the class.
The unknown functions are defined as follows.
here
functions
is found from the semi-indefinite problem set in the following system of
loaded infinite parabolic equations in the field.
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ILMIY METODIK JURNAL
(3.2.7)
(3.2.8)
(3.2.9)
here
(parabolic operators)
We
prove
the
theorem
in
several
stages:
First
of
all
function
(3.2.5, 3.2.6) we show that it satisfies additional conditions.
For this we assume the opposite. Suppose such a function
exists, let the following
condition be relevant for it
, i.e
Then it's a new function
for,
using problems (3.2.7)-
(3.2.9) in the field, solve both sides of the equation (3.2.7)
from 1 by multiplying by
summing
up,
we
get
the
following
loaded
equation.
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From formula (3.2.10) to make it easier for the first time
let's look at the situation, taking
into account the designations, we create the following formula from the formula (3.2.10).
Then it's a new function
because
using the problems
(3.2.7)-(3.2.9) in the field, we form the following equation from the formula (3.2.11).
(3.2.12)
from equality (3.2.12) and conditions (3.2.8), (3.2.9) to the given function in the theorem
Based on the given conditions, we form the following problem.
(3.2.13)
(3.2.14)
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(3.2.15)
Our task now is to show that problem (3.2.13)-(3.2.15) has a unique solution. For this, the
following equation
Integrating by pieces, conditions of the theorem, limit
(3.2.14),( 3.2.15) conditions and
taking into account, we form the following inequality
from here
Thus, since the problem (3.2.13)-(3.2.15) has a unique
solution
originates. Hence the condition of problem (3.2.5).
done. Also, condition (3.2.6) of the problem, i.e
execution is shown. Our task
now is to prove that the solution of the problem (3.2.7)-(3.2.9) has a unique solution. For this
we use successive approximations, Galerkin and a priori estimation methods for the system of heat
diffusion equations loaded in the field.
(3.2.16)
(3.2.17)
(3.2.18)
To investigate this problem (3.2.16)-(3.2.18) and prove the theorem, we need the following
definitions, spaces and auxiliary lemmas.
It has the following finite norm
(C)
we introduce the space of vector functions
.
It can be seen that finite (C) has a norm
the space will be a Banach space.
according to the definition, the following relationship is appropriate.
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Now
through (3.2.17),(3.2.18) satisfying the conditions
we introduce
the space of vector functions.
Description. (3.2.16)-(3.2.18) as the solution of the problem, satisfying the equation (3.2.16),
we say to the vector-function.
Lemma 1.
Let's assume that all conditions of the lemma are fulfilled. In that case, the
following a priori estimates are appropriate for solving the problem (3.2.16) - (3.2.18).
I)
II)
here
Proof. We prove the first a priori estimate:
For this we consider the following equation:
(3.2.19)
Integrating equality (3.2.19) piecewise, using the conditions of the theorem and boundary
conditions (3.2.17), (3.2.18) and Cauchy's inequality, we form the first estimate.
Now we prove the second a priori estimate:
For this we consider the following equation:
(3.2.20)
Integrating equality (3.2.20) piecewise, using the conditions of the theorem and boundary
conditions (3.2.17), (3.2.18) and Cauchy's inequality, we form the second estimate.
Now
we define a new function in space:
Lemma 2. Suppose that all the conditions of theorem and theorem 1 are fulfilled, in that case
the following a priori estimates for the functions are appropriate.
III)
IV)
, here
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Proof. From problems (3.2.16)–( 3.2.18).
in the field
we form the following problems
for the functions.
(3.2.21)
(3.2.22)
(3.2.23)
since the conditions of lemma 1 are reasonable for the functions, we repeat the methods of
obtaining the first and second values
we generate a priori estimates for functions III) and
IV).
Now
radius in space
was
we get a ball.
, here
some constant number.
Lemma 3. Suppose that all conditions of lemmas 1 and 2 are fulfilled, then (3.2.16)–(3.2.18) is
the problem
will have a unique solution on the sphere.
Proof.
we define the following operator (reflection) on the sphere:
Since the conditions of Lemma 1 and Lemma 2 are valid for vector functions:
1)
– operator
reflects the sphere to itself with one value.
2)
– there will be a compression reflection.
Based on the principle of compressive reflections, we note that problem (3.2.6) - (3.2.8)
will have a unique solution on the sphere.
in
using Parseval-Steklov equality for functions (3.2.1)-(3.2.3)
we get the
solution from the class. The theorem is proved.
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Result-1. If given
if we consider the function as a solution to the following problem
(3.2.24)
(3.2.25)
(3.2.26)
In that case
here
and
the proof of the theorem can be much simplified.
Result-2. For equation (3.2.1), replace condition (3.2.2) in problem Koshi
condition, we can get the same result.
Summary:
we conclude that in this work, the correctness (having a unique solution) of the
generalized solutions of Cauchy and nonlocal problems for the string vibration equation in
Sobolev spaces was studied using modern methods, methods of a priori estimates.
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4. Bitsadze AV, Samarskii AA. (1969) On some simple generalizations of linear elliptic boundary
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5. Vragov VN. (1983) Boundary problems for non-classical equations of mathematical physics.
Novosibirsk, 216pp.
6. Glazatov SN. (1985) Nonlocal boundary problems for mixed type equations in a rectangle.
Siberian Math. J., 26(6):162-164.
7. Djamalov SZ. (1989) On correctness of nonlocal boundary problems for many-dimensional
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