ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 10 / UIF:8.2 / MODERNSCIENCE.UZ
892
THE EFFECT OF THE HEAT SOURCE ON THE AMBIENT DENSITY IN THE
PROCESSES OF NON-LINEAR HEAT PROPAGATION IN MULTIDIMENSIONAL
FIELDS
J.M.Toshtemirov
University of Science and Technologies (Tashkent, Uzbekistan)
jaloliddintoshtemirov0@gmail.com
M.A.Asadullayeva
University of Science and Technologies (Tashkent, Uzbekistan)
https://doi.org/10.5281/zenodo.10052972
Abstract.
In nonlinear heat propagation processes in the article in multidimensional fields
devoted to the study of the effect of the heat source on the density of the environment.
In the course of work, a self-similar solution was built to solve the linear heat diffusion
equation according to the characteristics of the environment density and heat source, reaction-
diffusion processes were observed, and theorems were proved. The following results were obtained
from this work: a preliminary estimate for the double linear heat diffusion equation, the
localization process was observed, the finite velocity was approximated, new effects were
observed, and an algorithm based on the obtained self-similarity was constructed. solution, the
program code was created in the programming language and the process was modeled. All results
were compared.
Keywords:
Reaction-diffusion,
parabolic
equation,
asymptotic,
finite
speed,
approximation, conductivity of convective migration, self-similar, Multidimensional areas, system,
numerical solution.
ВЛИЯНИЕ ИСТОЧНИКА ТЕПЛА НА ПЛОТНОСТЬ СРЕДЫ В ПРОЦЕССАХ
НЕЛИНЕЙНОГО РАСПРОСТРАНЕНИЯ ТЕПЛА В МНОГОМЕРНЫХ ПОЛЯХ
Аннотация.
В нелинейных процессах распространения тепла статья в
многомерных областях посвящена изучению влияния источника тепла на плотность
среды.
В ходе работы было построено автомодельное решение для решения линейного
уравнения диффузии тепла по характеристикам плотности среды и источника тепла,
наблюдались реакционно-диффузионные процессы и доказаны теоремы. В результате
работы были получены следующие результаты: проведена предварительная оценка для
двойного линейного уравнения теплодиффузии, обнаружен процесс локализации,
аппроксимирована конечная скорость, обнаружены новые эффекты и построен алгоритм
на основе полученного самоподобия. Для решения задачи был создан программный код на
языке программирования и смоделирован процесс. Все результаты сравнивались.
Ключевые слова:
Реакция-диффузия, параболическое уравнение, асимптотика,
конечная
скорость,
аппроксимация,
проводимость
конвективной
миграции,
автомодельность, Многомерные области, система, численное решение.
Introduction
At present, one of the most frequent processes of quenching is the process of heat
dissipation, when the main phenomenon in this process depends on what area the heat is
distributed. The philosophy of Life shows that the spread of heat in nature can be either in a
multidimensional area or in a one-dimensional area. Of course, in this process, we must draw up
the evolutionary equation of the process, taking into account the density of the environment, the
positive or negative of the heat source, the heat capacity, the dependence of the environment on
the coefficient of heat transfer. As we know, the processes of heat dissipation in multidimensional
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 10 / UIF:8.2 / MODERNSCIENCE.UZ
893
areas are in a non-linear state. In such a process, the main attention is paid to heat capacity. For
the process of heat dissipation without any scratches, the initial temperature (Cauchy condition)
must be given. For non-linear heat dissipation, quasi-linear diffusion models, the spatial
localization property and the thermal dissipation effect with limited speed have been studied by
many scientists over the years. It was also used in nonlinear processes encountered in
multidimensional areas of the studied mathematical models and found new effects not
characteristic of linear equations. The scientific research devoted to nonlinear systems for cases of
varying density, conductibility capacity of the environment, conductibility of convective
migration, which leads to parabolic equations with distortion, has begun to be viewed as the main
problems of the present day. In order to impress the process of heat dissipation without lines in
multidimensional areas, we see the following equation in
R
x
R
t
x
t
Q
,
:
,
area:
u
u
v
u
u
u
x
u
x
Au
l
p
k
m
t
2
1
2
1
(1)
initial (Cauchy condition)
,
0
,
0
0
x
u
x
u
(2)
let condition given.
Here
t
x
u
,
-the temperature of the heat,
Q
c
t
v
-the speed of the environment,
q
n
x
x
x
x
2
1
,
-the density of the environment, which is a continuous function
-
parameter,
u
- represents the positive (
1
) or negative (
1
) power of the source of heat.
(1) the equation represents a number of physical processes: the reaction diffusion process
in a non-linear environment, the heat dissipation process in a non-linear environment, the filtration
of liquid and gas in a non-linear environment, they represent the existence of the law of polytropy
and other non-linear displacements.
(1) the Cauchy issue and boundary value issues for the equation were observed by many
authors in one-dimensional and multi-dimensional cases [1-5].
(1) in the processes represented by the equation, the phenomenon of finite distribution of
temperature occurs [4]. In the presence of an absorption coefficient, the phenomenon of the “rear”
front can occur, that is, the Left front can stop after a certain time and move along the medium.
Solution method
There are many ways to find a solution to the above (1) equation: in functional analysis, it
is possible to solve with the help of spaces, in mathematical physics and differential equations, to
build differential operators, and in practical mathematics, mainly by methods of constructing an
self-similar solution.
(1) we write the equation for N=2 case as follows:
2
2
1
1
1
1
2
2
2
1
2
2
1
1
1
2
1
1
1
u
u
t
v
u
u
u
x
t
u
x
u
u
t
v
u
u
u
x
t
u
x
i
l
p
k
m
i
q
n
l
p
k
m
i
q
n
(3)
here,
1
;
1
3
i
i
i
,
i
i
u
- the source of heat,
2
,
1
,
,
0
|
,
0
0
i
R
x
x
u
x
t
u
N
i
t
i
- the balance of conditions.
(3) system of equations to be present in solution in
1
0
,
1
,
0
,
i
l
n
k
m
position[3]
1
1
,
2
1
,
1
1
1
p
m
p
m
p
p
m
p
i
i
i
i
(3) in the system of equations, we perform the following substitution
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VOLUME 2 / ISSUE 10 / UIF:8.2 / MODERNSCIENCE.UZ
894
;
2
1
1
2
N
i
i
x
r
x
here is the size of the space. This replacement is called a radial-symmetrical replacement.
From the above substitution (3) the system of equations changes as follows:
2
2
1
1
1
2
2
1
1
2
2
2
1
2
1
1
2
1
1
1
1
1
2
1
1
1
1
1
u
r
u
r
r
t
v
r
r
u
r
u
u
r
r
r
t
u
r
u
r
u
r
r
t
v
r
r
u
r
u
u
r
r
r
t
u
r
N
N
l
p
k
m
N
q
N
n
N
N
l
p
k
m
N
q
N
n
(4)
(4) in order to solve the system of equations, we initially look for the solution as follows:
2
,
1
;
;
,
i
v
r
w
v
r
t
u
i
i
i
(5)
here
i
- the number being searched,
i
w
- an unknown function,
- parameter,
.
const
(5) provisions (1) we apply to the system of equations:
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
N
N
i
N
n
l
i
p
k
i
m
i
N
q
m
l
p
k
q
p
N
k
i
p
k
i
m
i
N
q
n
i
i
i
n
n
i
i
i
n
n
i
n
w
v
u
d
dw
d
d
v
r
u
r
r
v
r
d
dw
d
dw
w
d
d
v
r
u
r
u
u
r
r
r
i
w
w
v
v
w
v
v
w
v
v
v
t
u
r
3
3
1
2
1
1
1
2
1
1
1
2
1
2
1
1
1
1
1
3
2
,
1
;
v
-we set the following conditions for the function:
0
1
3
3
1
0
0
1
t
T
A
t
v
dt
dv
v
v
v
v
i
i
i
i
i
i
n
n
it is here
0
1
0
0
3
0
;
0
A
n
i
i
i
. If:
0
0
,
const
c
t
c
t
v
1
1
:
As a result, we will have a new equality
i
i
i
i
i
l
p
k
m
p
q
3
2
1
1
2
(6)
(6) from the equation, the following system of linear equations is formed:
1
2
1
2
1
2
2
2
2
2
1
1
2
1
1
1
p
q
l
p
k
m
(7)
(7) we use the Kramer method to solve a system of equations:
2
1
1
1
1
2
1
1
1
2
1
1
1
2
2
2
1
2
2
2
2
1
2
2
2
2
1
1
1
1
1
0
2
2
p
q
l
p
k
m
l
p
k
m
p
q
p
q
l
p
k
m
p
q
l
p
k
m
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895
.
0
1
2
2
1
1
1
2
3
3
2
1
2
1
2
1
3
2
2
2
2
2
2
1
i
i
i
i
i
i
N
s
n
i
i
N
s
N
s
N
s
n
l
i
p
k
i
m
i
i
N
s
i
N
s
l
i
p
k
i
m
i
f
f
s
N
d
df
s
N
s
N
s
N
s
s
N
s
N
d
df
d
df
f
s
d
df
s
N
s
N
d
f
d
s
N
d
df
d
df
f
d
d
1
2
2
1
1
0
1
1
2
p
l
p
k
m
k
l
s
b
const
a
i
p
p
i
(1) - the following theorem and lemma were obtained from the solution of the problem
using the linear separation algorithm.
Lemma 1. Let
)
(
1
0
0
R
C
u
. Then there are constant
0
2
a
,
))
1
(
1
exp(
S
T
0
ln
)))
1
(
1
(
1
(
2
)
1
(
T
S
S
s
such that the function
)
;
)
(
(
)
ln(
)
,
(
1
2
)
1
(
a
f
t
T
t
T
t
u
s
s
,
where
2
1
1
)
(
T
s
,
is the upper solution of problem (1)-(2).
Proof. It is sufficient to show that, under these constraints, function
)
);
(
(
2
a
f
Q
is
the
upper
solution
of
problem
(13)-(14)
(i.e.,
0
)
(
Q
D
)
everywhere
in
)
(
,
ln
2
a
T
.
Substituting function 1 in (13) we get.
s
s
s
s
A
d
F
F
d
a
d
A
DQ
2
)
1
(
0
)
1
(
2
)
1
(
2
1
2
2
1
1
0
)
(
)
(
1
1
2
, (15)
where
,
ln
,
,
0
1
2
2
2
2
2
T
a
s
a
d
4
)
1
(
2
)
1
(
1
2
1
)
2
)
1
(
(
4
1
1
)
(
1
s
s
s
s
s
s
s
s
F
,
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International scientific journal
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VOLUME 2 / ISSUE 10 / UIF:8.2 / MODERNSCIENCE.UZ
896
2
)
1
(
2
)
1
(
1
)
(
2
s
s
s
s
s
s
F
.
To fulfill inequality
0
DQ
in (10), it is necessary to fulfill condition
0
)
(
1
F
for all
T
ln
, which takes place under the imposed conditions on
T
and
s
, specified in the conditions
of the lemma. Then, due to the uniform boundedness of
functions
)
(
1
F
and
)
(
2
F
, for all
sufficiently large
2
2
a
, the inequality will be satisfied
1
1
2
2
2
2
)
1
(
)
1
(
2
)
1
(
0
2
)
(
)
(
2
1
)
(
F
F
d
a
A
F
s
s
s
s
,
which provides the implementation of inequality
0
DQ
. In turn, for large
2
a
, the
estimate is also valid
)
;
)
(ln
(
)
;
(ln
)
(
2
0
a
T
f
T
Q
Q
,
and hence
2
;
ln
a
T
f
is the upper solution of problem (13)-(14) (Lemma 1.).
Respectively
)
,
(
t
w
is the upper solution of problem (4)-(5), and
)
,
(
x
t
u
is the problem (1)-
(2). The lemma is proved.
Theorem. For any function
)
(
1
0
0
R
C
u
, there are constants
0
a
,
0
a
, and
1
T
such that the solution of problem (1)-(2) in
1
)
,
(
R
T
satisfies the inequality
)
;
(
)
ln(
)
(
)
,
(
)
;
(
)
ln(
)
(
1
2
)
1
(
2
)
1
(
a
f
t
T
t
T
x
t
u
a
f
t
T
t
T
s
s
s
s
Proof. The validity of the theorem follows directly from Lemmas 1, where the existence of
lower and upper solutions is proved and the corresponding estimates are obtained. In this case, the
constant T is chosen taking into account the condition of Lemma 1 and requirement
0
)
0
,
(
T
u
, the constants
a
and
a
, for example, as follows:
1
a
a
,
T
s
a
a
ln
1
2
.
Then, under the conditions of the theorem, we obtain
)
;
(
)
,
(
;
a
f
Q
a
f
everywhere in
1
)
;
(ln
R
T
. Then the statement of the theorem follows taking into
account (13)-(14) and (4)-(5). The theorem is proved.
Results of numerical experiments and visualization
When solving the problem numerically, the equation is approximated on a grid using
an implicit scheme of variable directions (for the multidimensional case) in combination
with the balance method. Iterative processes were built on the basis of the Picard, Newton
method as well as special method.
A computational algorithm has been developed. When developing software that illustrates
the simulation process of solution behavior over time (visualization), the Visual Studio 2019 C#
environment was used with the inclusion of the Open GL graphics library.
ISSN:
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VOLUME 2 / ISSUE 10 / UIF:8.2 / MODERNSCIENCE.UZ
897
The results of computational experiments show that all iterative methods are suitable for
the constructed scheme. To achieve the same accuracy, the Newton method requires fewer
iterations than the Picard method. In some cases, the number of iterations is almost twice, and the
maximum number of iterations is 3-4 times less than in other methods. Since power-law
nonlinearity is present in the right-hand side of equation (1), naturally the special method gives
better results than the Picard method.
As a test example, we used solutions of equation (1) obtained by the methods of reference
equations and nonlinear splitting [2]. Figures 1-4 show the calculation results for various values
of parameters
,
,
,
,
,
,
l
k
m
n
p
and time.
Fig.3. p=2.1, n=1.2, m=1, k=1, l=1
Fig.4. p=2.3, n=1, m=1.2, k=1.3, l=1
Conclusion
The results of computational experiments show that all of the listed iterative methods
are effective for solving nonlinear problems and lead to nonlinear effects if we use self-similar
solutions constructed by the nonlinear splitting method and the standard equation method as the
initial approximation of the solution [4, 6].
As expected, in order to achieve an identical accuracy, the Newton method requires fewer
iterations than the methods of Picard and a special method because of the successful choice of the
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 10 / UIF:8.2 / MODERNSCIENCE.UZ
898
initial approximation. Note that in each of the cases considered, the Newton method has the best
convergence due to the choice of a good initial approximation.
In some cases, the total iteration amount is almost two times and the maximum iteration is
almost 4 times less than other methods.
The results of numerical calculations show the effect of the finite velocity of disturbance
propagation and the localization of the solution depends on the values of the numerical parameters.
All results of numerical experiments are presented in the form of visualizedanimation.
Acknowledgements
We are very grateful to experts for their appropriate and constructive suggestions to
improve this template.
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