Application of finite difference methods for solving the twodimensional equation of heat conductivity

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Navruzov, D. (2022). Application of finite difference methods for solving the twodimensional equation of heat conductivity. Результаты научных исследований в условиях пандемии (COVID-19), 1(05), 98–102. извлечено от https://inlibrary.uz/index.php/scientific-research-covid-19/article/view/8467
Dilshod Navruzov, M.V. Lomonosov Institute. Urazbaeva AN ruz

PHD I year of study in the laboratory "Mechanics of fluid and gas"

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Аннотация

A comparison is made of finite-difference schemes with the exact solution of a parabolic partial differential equation. A stability analysis has also been carried out. To solve the parabolic equation, one-step and twostep finite-difference methods are used.


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References

:

1.

Pabis A

.,

Rawle R.J

.,

Kasson P.M

. Influenza hemagglutinin drives viral

entry via two sequential intramembrane mechanisms//

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S A.

2020 Mar 18. pii: 201914188. doi: 10.1073/pnas.1914188117. [Epub

ahead of print]

2.

Faravonova T.E., Olenina L.V., Kuzmina T.I., Sobolev B.N., Kuraeva T.E.,

Kolesanova E.F., Archakov A.I. Identification of glycosaminoglycan-binding
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3.

Wang T., Palese P. Universal epitopes of influenza virus

hemagglutinins// Nature structural & molecular biology. – 2009. – N2. – C.1-
2.

4.

4.Khamidov D.Kh., Lim A.V., Salikhov R.S. et al. Immunoaffinity

fractionation of neutralizing antibodies against nerve growth factor//
Chemistry of natural compounds. – 1991. – N6. – P.828-832 [in Russian].

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203 [in Russian].




Navruzov Dilshod Primqulovich, PHD I year of study in the laboratory

"Mechanics of fluid and gas" Institute MISS them. Urazboeva AN zUz

,Tashkent ,Uzbekistan

APPLICATION OF FINITE DIFFERENCE METHODS FOR SOLVING THE TWO-

DIMENSIONAL EQUATION OF HEAT CONDUCTIVITY.

Navruzov D

Abstract: A comparison is made of finite-difference schemes with the

exact solution of a parabolic partial differential equation. A stability analysis
has also been carried out. To solve the parabolic equation, one-step and two-
step finite-difference methods are used.

Keywords: two-step method, nonlinear equations, implicit scheme, heat

equation, finite difference scheme, parabolic.


Introduction: In this article, various finite-difference schemes have

been studied in detail, with the help of which it is possible to solve the
simplest model equations of heat conduction. We restrict ourselves to
considering the diffusion equation. Difference schemes with a first order of
accuracy are considered. For the numerical solution of the heat equation, the


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99

Kranko-Nicholson method and implicit methods of variable directions are
used [1].

The two-dimensional heat equation is a parabolic partial differential

equation that describes the process of heat propagation or diffusion [1-2].

)

(

2

2

2

2

y

U

x

U

t

U



















(1)

This equation is the simplest model equation for parabolic equations.

heat propagation rate.

Consider the problem of temperature distribution in a pipe. In this case,

equation (1) takes the following form

)

(

2

2

2

2

y

T

x

T

t

T



















(2)

We now turn to the study of finite-difference schemes for solving the

two-dimensional heat equation.


Numerical method

Application of model methods for solving

)

(

2

2

2

2

y

U

x

U

t

U



















the heat

equation.

1.

The Kranko-Nicholson method for the two-dimensional heat

equation.

































)

)(

(

2

,

1

,

2

2

,

1

,

n

j

i

n

j

i

y

x

n

j

i

n

j

i

T

T

t

T

T



(3)


To shorten the notation, two-dimensional central-difference operators

are introduced here

2

x



n

j

i

T

,

and

2

у



n

j

i

T

,

defined by the relations

2

,

2

2

,

1

,

,

1

,

2

)

(

)

(

2

х

T

х

T

T

T

T

n

j

i

x

n

j

i

n

j

i

n

j

i

n

j

i

x





















(4)

2

,

2

2

1

,

,

1

,

,

2

)

(

)

(

2

y

T

y

T

T

T

T

n

j

i

y

n

j

i

n

j

i

n

j

i

n

j

i

y





















2.

Th

e implicit method of variable directions.


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100


Step 1































)

(

2

/

,

2

2

/

1

,

2

,

2

/

1

,

n

j

i

у

n

j

i

x

n

j

i

n

j

i

T

T

t

T

T




(5)

Step 2



































)

(

2

/

1

,

2

2

/

1

,

2

2

/

1

,

1

,

n

j

i

у

n

j

i

x

n

j

i

n

j

i

T

T

t

T

T



2

x



2

/

1

,



n

j

i

T

,

2

у



n

j

i

T

,

and

2

у



1

,



n

j

i

T

defined by the relations

2

2

/

1

,

2

2

2

/

1

,

1

2

/

1

,

2

/

1

,

1

2

/

1

,

2

)

(

)

(

2

х

T

х

T

T

T

T

n

j

i

x

n

j

i

n

j

i

n

j

i

n

j

i

x































2

,

2

2

1

,

,

1

,

,

2

)

(

)

(

2

y

T

y

T

T

T

T

n

j

i

y

n

j

i

n

j

i

n

j

i

n

j

i

y





















(6)

2

1

,

2

2

1

1

,

1

,

1

1

,

1

,

2

)

(

)

(

2

y

T

y

T

T

T

T

n

j

i

y

n

j

i

n

j

i

n

j

i

n

j

i

y































Calculation results.
Here are some specific examples illustrating the briefly described

models above. The results of calculations are comparable to those of
calculations [1].

In Fig. 1. The results of the temperature distribution by the Cranco-

Nicholson method are derived.

1.

Cranco-Nicholson Method.


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Scientific research results in pandemic conditions (COVID-19)

101

Fig. 1.
2. The implicit method of variable directions.

Fig. 2.


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102

Conclusion: The calculation results are compared. It is shown that these

finite-difference schemes give very close calculated results for the exact
solution of the parabolic equation.


References:
1. Anderson D, Computational hydromechanics and heat transfer //

Moscow "Mir" 1990, 382 p.

2. Spalart P. R., Allmaras S.R. A One-Equation Turbulence Model for

Aerodynamic Flows. AIAA-92-0439.

3.A. Faysman, Professional programming in Turbo Pascal. 1992.
4. Shur M., Strelets M., Zaikov L., Gulyaev A., Kozlov V., Secundov A.

Comparative numerical testing of one and two leveling turbulence models
for flows with separation and addition, "AIAA Paper 95-0863, January 1995
year

5. Shur M., Strelets M., Zaikov L., Gulyaev A., Kozlov V., Secundov A.

Comparative numerical testing of one and two leveling

models of turbulence for flows with separation and accession, "AIAA

Paper 95-0863, January 1995

6. L. G. Loytsyansky, Mechanics of liquid and gas, M .: Nauka, 1970. 904

s .; L. G. Loitsyansky, Fluid and Gas Mechanics, Moscow, Nauka, 1970, 904
pp. (In Russian)

7. Vladimirov V.S. Equations of mathematical physics. - M .: Nauka,

1988.512 s




Bahodir Ibragimov, assistant of the Department of Obstetrics and

Gynecology, Samarkand State Medical Institute, Uzbekistan

THE RELATIONSHIP OF METABOLIC DISORDERS WITH POLYCYSTIC

OVARIAN SYNDROME IN YOUNG WOMEN

B. Ibragimov


Annotation. The article presents some pathogenetic mechanisms of the

development of metabolic disorders in young women with polycystic ovary
syndrome. In recent years, polycystic ovary syndrome is considered as part
of metabolic disorders [1]. Metabolic disorders are manifested by disorders
of carbohydrate and lipid metabolism, abdominal obesity, hypertension,
followed by the development of type 2 diabetes mellitus and cardiovascular
diseases.

Keywords: metabolic disorders, polycystic ovary syndrome, insulin

resistance, hyperinsulinemia, dyslipidemia

Библиографические ссылки

Anderson D, Computational hydromechanics and heat transfer // Moscow "Mir" 1990, 382 p.

Spalart P. R., Allmaras S.R. A One-Equation Turbulence Model for Aerodynamic Flows. AIAA-92-0439.

A. Faysman, Professional programming in Turbo Pascal. 1992.

Shur M., Strelets M., Zaikov L, Gulyaev A., Kozlov V., Secundov A. Comparative numerical testing of one and two leveling turbulence models for flows with separation and addition, "A1AA Paper 95-0863, January 1995 year

Shur M., Strelets M., Zaikov L, Gulyaev A., Kozlov V., Secundov A. Comparative numerical testing of one and two leveling

models of turbulence for flows with separation and accession, "A1AA Paper 95-0863, January 1995

L. G. Loytsyansky, Mechanics of liquid and gas, M .: Nauka, 1970. 904 s .; L. G. Loitsyansky, Fluid and Gas Mechanics, Moscow, Nauka, 1970, 904 pp. (In Russian)

Vladimirov V.S. Equations of mathematical physics. - M .: Nauka, 1988.512 s

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