ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–4_ Мая –2025
444
2181-3187
INVERSE BOUNDARY VALUE PROBLEM
Muzaffar Sulaymonovich Azizov
Senior Lecturer at the Department o
f Mathematical Analysis and Differential Equations,
Fergana State University
PhD in Physics and Mathematics
E-mail:
muzaffar.azizov.1988@mail.ru
Odina Isroiljon qizi Ismoiljonova
3rd-year student of the Applied Mathematics program,
Group 22-08, Fergana State University
E-mail:
ismoiljonovaodina88@gmail.com
Gulnoza Abrorjon qizi Ibrohimova
3rd-year student of the Applied Mathematics program,
Group 22-09, Fergana State University
E-mail:
jabborovagulniza2004@gmail.com
Annotation:
This article discusses the steepest descent algorithm of the gradient
method used to solve systems of linear algebraic equations. The method for finding the
solution by minimizing a functional is explained step-by-step. Theoretical foundations
based on gradient and error vectors are presented. A practical example involving a
system of four equations is solved using Python, and the convergence rate and accuracy
of the algorithm are demonstrated. The results show that the gradient method is a
simple and efficient computational tool suitable for solving large-scale linear systems.
Key words:
Gradient method, steepest descent, iterative method, symmetric
matrix, positive definiteness, functional, optimization, Python, algorithm, error vector.
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–4_ Мая –2025
445
2181-3187
Аннотация:
В статье рассматривается обратная задача определения
граничного условия для уравнения теплопроводности. Неизвестная функция,
зависящая от времени, определяется на основе дополнительной информации о
решении. Существование и единственность решения доказываются с помощью
теорем 1 и 2. Кроме того, показано, что задача может быть сведена к
интегральному уравнению типа Вольтерра.
Ключевые слова:
обратная задача, уравнение теплопроводности,
граничное условие, интегральное уравнение, единственность, уравнение
Вольтерра
Annotatsiya:
Ushbu maqolada issiqlik tarqalish tenglamasi uchun chegaraviy
shartni aniqlashga oid teskari masala ko‘rib chiqiladi. Noma’lum bo‘lgan vaqtga
bog‘liq funksiyani aniqlash uchun yechim haqidagi qo‘shimcha ma’lumotlardan
foydalaniladi. Masalaning yechimining mavjudligi va yagonaligi 1- va 2-teoremalar
orqali asoslanadi. Bundan tashqari, bu masala Volterra turidagi integral tenglama orqali
yechim topishga keltirilishi ham ko‘rsatib beriladi.
Kalit so’zlar:
teskari masala, issiqlik tarqalish tenglamasi, chegaraviy shart,
integral tenglama, yagonalik, Volterra tenglamasi
Introduction:
Achieving fast and accurate solutions to systems of linear algebraic equations is
one of the key directions of modern computational mathematics. In particular, for
systems with symmetric and positive definite coefficient matrices, gradient-based
methods—especially the steepest descent algorithm—are highly effective. This
method operates on an iterative approach, moving in the direction of the greatest
decrease of the objective function at each step. This article presents the theoretical
foundations of the gradient method, the steps of the algorithm, and a practical example
implemented using the Python programming language.
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–4_ Мая –2025
446
2181-3187
Determination of Boundary Conditions.
We consider an inverse problem
that involves determining a time-dependent function included in the boundary
condition, using additional information about the solution of the boundary value
problem for the heat conduction equation. Let us assume that
( , )
u x t
the function
2
, 0
, 0
,
t
xx
u
a u
x
l
t
T
=
(1)
(0, )
( ),
u
t
t
=
0
,
t
T
(2)
( , )
( ),
x
u l t
t
=
0
,
t
T
(3)
( , 0)
0,
u x
=
0
,
x
l
(4)
boundary issues of the solution , let it be.
Assume let,
( )
t
given the function,
( )
t
of unknown function, and (1)-(4)
of problem solution about
( , )
( ),
u l t
g t
=
0
,
t
T
(5)
additional information is,
( )
t
the function to determine required will be, this
earth
( )
g t
- given function.
Let us consider the issue of the uniqueness of the solution to the given inverse
problem
.
The uniqueness of the solution to this problem has been studied in a more
general form by a number of authors. Let us present one of the results obtained in this
direction, as provided in [51], and apply it to the case of the heat conduction equation
(1) with constant coefficients.
We introduce the following notation:
{( , ) : 0
, 0
}.
lT
Q
x t
x
l
t
T
=
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–4_ Мая –2025
447
2181-3187
1-teorema.
Assume let me, the function
2,1
( , )
(
)
lT
u x t
C
Q
and (1) equations of
the
lT
Q
area will build. It without , you
0
t
T
for
( , )
( , )
0
x
u l t
u l t
=
=
be,
lT
Q
at
( , )
0
u x t
=
will.
This teorema from given
( )
t
and
( )
g t
the function for (1)-(5) from
( )
t
the
function identifying issues of the singleness come out can. Fact is,
2,1
1
2
( , ),
( , )
(
)
lT
u x t u x t
C
Q
lT
Q
in the area of (1) the equation may build and
1
1
(0, )
( ),
u
t
t
=
2
2
(0, )
( ),
u
t
t
=
0
,
t
T
1
2
( , )
( , )
( ),
u l t
u l t
g t
=
=
1
2
( , )
( , )
( ),
du
du
l t
l t
t
dx
dx
=
=
0
t
T
which is.
Now
1
2
( , )
( , )
( , )
u x t
u x t
u x t
=
−
the function we do not look at. This function is 1-
teorema terms and conditions will be content. Therefore,
lT
Q
in the field ,
( , )
0
u x t
=
ie
[0, ]
t
T
for
1
2
( )
( ).
t
t
=
Attention give, 1-teorema in
( , )
u x t
the function for the initial conditions, it is
not given.
(1)-(4) boundary issues for other reverse the issue , we do not look at.
Assume let,
( )
t
given the function,
( )
t
unknown following
0
( , )
( ),
u x t
g t
=
0
t
T
(6)
Wholesale of view (1)-(4) of problem solution about additional information
known if it is,
( )
t
the function to identify necessary, this earth on
( )
g t
a given
function,
0
[0, ].
x
l
The uniqueness of the solution to this inverse problem can also be
established using Theorem 1.
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–4_ Мая –2025
448
2181-3187
2-teorema.
If
2,1
1
2
( , ),
( , )
(
)
lT
u x t u x t
C
Q
the function of the
lT
Q
area in (1) in
the equation and (3), (4), (6) terms and conditions does it build,
[0, ]
t
T
for
(0, )
( ),
1, 2,
i
i
u
t
t i
=
=
is. It without
[0, ]
t
T
though
1
2
( )
( )
t
t
=
it will be.
Proof.
We consider the function
1
2
( , )
( , )
( , )
u x t
u x t
u x t
=
−
where
( , )
u x t
2
0
,
, 0
,
t
xx
u
a u
x
x
l
t
T
=
0
( , )
0,
u x t
=
0
,
t
T
( , )
0,
x
u l t
=
0
,
t
T
( , 0)
0,
u x
=
0
,
,
x
x
l
boundary of the issue the solution is. Optional
0
0
[ , ],
x
x l
[0, ]
t
T
for
( , )
0
u x t
=
this is that we show. For this, we multiply the equation by a function
( , )
u x t
integrate, and
0
0
2
0
0
( , ) ( , )
( , ) ( , )
l t
l t
t
xx
x
x
u
u
d d
a
u
u
d d
=
ensure you will. Integrated with many pieces in the calculation, the initial and
boundary conditions we will use, the result
[0, ]
t
T
for the following equality able
you will be:
0
0
2
2
2
0
1
( ( , ))
(
( , ))
2
l
l t
x
x
x
u
d
a
u
d d
+
That come out,
0
[ , ],
x
x l
[0, ]
t
T
to
( , )
0
u x t
=
. It without
0
( , )
0.
u x t
=
So done,
0
[0,
],
x
x
[0, ]
t
T
when
( , )
u x t
the function (1) in the equation and
[0, ]
t
T
for
0
0
( , )
( , )
0
u x t
u x t
=
=
terms and conditions will build. We apply Theorem 1 on the
rectangle
0
0
, 0
x
x
t
T
, for
[0, ]
t
T
, and thus
1
2
(0, )
(0, )
u
t
u
t
=
or
1
2
( )
( )
t
t
=
ollows. The proof of Theorem 2 is completed..
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–4_ Мая –2025
449
2181-3187
Let us present an example of reducing the problem of determining a
boundary condition to the solution of a first-kind integral equation.
We consider the boundary value problem for the heat conduction equation on the
half-line:
, 0
, 0
,
t
xx
u
u
x
t
T
=
(7)
(0, )
( ),
u
t
t
=
0
,
t
T
(8)
( , 0)
0,
u x
=
0
.
x
(9)
(7)-(9) problem solution about
0
( , )
( ),
u x t
g t
=
0
,
t
T
0
0,
x
(10)
Additional information is given,
( )
t
the function to identify required is.
(7)-(9) the issue of the solution
2
3 2
0
( , )
exp
( )
4(
)
2
(
)
t
x
x
u x t
d
t
t
=
−
−
−
has the following form. Thus, in this case, the inverse problem
0
( , ) ( )
( ),
t
K t
d
g t
=
0
,
t
T
It is reduced to a Volterra integral equation of the first kind, where the kernel
is...
2
0
0
3 2
( , )
exp
.
4(
)
2
(
)
x
x
K t
t
t
=
−
−
−
General Conclusion:
In this paper, an inverse problem related to the determination of a boundary
condition for the heat conduction equation was investigated. The problem involved
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–4_ Мая –2025
450
2181-3187
determining an unknown time-dependent function using additional information about
the solution. Based on Theorems 1 and 2, the uniqueness of the solution was proven.
Furthermore, a practical method for obtaining the solution was demonstrated by
reducing
the
problem
to
a
Volterra-type
integral
equation.
The results of this study can be applied in modeling and controlling physical processes
and contribute to the theoretical and practical understanding of inverse problems.
References:
1.
Lavrentyev, M.M., Romanov, V.G., Yanyushkin, S.P. –
Obratnye
zadachi matematicheskoy fiziki
. Moscow: Nauka, 1986.
2.
Tikhonov, A.N., Arsenin, V.Y. –
Reshenie nekorrektnykh zadach
.
Moscow: Nauka, 1979.
3.
Isakov, V. –
Inverse Problems for Partial Differential Equations
.
Springer, 2006.
4.
Cannon, J.M. –
The One-Dimensional Heat Equation
. Addison-
Wesley, 1984.
5.
Aliev, N.A. –
Introduction to the Theory of Inverse Problems
.
Tashkent: Fan, 1998.
6.
Mukhamedov, A.K. –
Equations of Mathematical Physics and Their
Inverse Problems
. Fergana: FSU Publishing, 2022.
7.
Rasskazov, A.V. –
Theoretical Foundations of Methods for Solving
Inverse Problems
. Novosibirsk, 2001.
