5027720109

Modelowanie Matematyczne w Fizyce i Technice

APPLICATION OF THE SYMMETRIC POLYNOMIALS IN THE THREE - DIMENSIONMAL FOURIER EQUATION

Grzegorz Biernat, Sylwia Lara-Dziembek, Edyta Pawlak

Institute of Mathematics, Częstochowa University of Technology, Częstochowa, Poland

[e-mail], [e-mail], [e-mail]

In this paper we return to the development of the determinant of the błock matrix in the 3D case. We express this determinant by the symmetric polynomials. The considered problem concerns the effective formulas expressing the symmetric polynomials of three groups of variables by the symmetric polynomials due to each of these groups.

The heat flow in the 3D domain is described by the Fourier eąuation

f d²T(x, y, z, t) ^ d²T(x, y, z, t) 5²T(x, y, z, t)^ dT(x, y, z, t) ^

I s7²    + sT ⁺    s7- )^pc st ⁽¹⁾

where X is a thermal conductivity, c is a specific heat, p is a mass density and 7', x, y, z, t denote the temperaturę, geometrical co-ordinates and time.

The Finite Difference Method leads to the internal system of eąuations

X _    21

(Ax)    (4x)

(Ay)^;

A

(Az)²

tT,,-

2A

pyf

2 2

(Az)²

I. i.*..

W

A

"W

A

(Az)²

^Ti+l,j,k,l ⁺

r^j+u,.⁺

(2)

_pę.

in each time step 1.

The main determinant of the system of differential eąuations obtained from the Fourier eąuation takes the following form

det A=(A, -X₂    - W    (3)

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