Finite difference method in the Fourier equation With Newton’s boundary conditionsdirect formulas

(1)

FINITE DIFFERENCE METHOD IN THE FOURIER EQUATION WITH NEWTON’S BOUNDARY CONDITIONS

DIRECT FORMULAS

Grzegorz Biernat, Justyna Mazur

Institute of Mathematics, Czestochowa University of Technology, Poland e-mail: imi@imi.pcz.pl

Abstract. In the paper we give the direct FDM formulas for the solutions of the Fourier equation with the Newton boundary condition in the

^x,^t ^case.

1. Formulation of the problem

In limited spatial solid (centres) approximate to the one-dimensional case the temperature distribution ^T

^x^,^t with the Newton boundary conditions and deter- mined initial conditions (without the internal heat source) is defined by the equations

t t x c T x

t x T

w w w

w , ,

2

2 U

O ⁽¹⁾

(heat conduction in the centre-Fourier’s equation)

environment

_border

border

, ,

w

w _x

x

T t x n T

t x

T D

O ⁽²⁾

(heat exchange on the border-Newton’s equation)

x,0 Tinitial

T (3)

where the positive coefficients O,U,c ^andDwe receive as a constant.

Classical difference methods lead to the linear systems equations

t T cT x

T T

T_i _l _il _i _l _il _il '

'

1

2 1

1 2

U

O ⁽⁴⁾

where 1didm and lt1(internal case)

(2)

°°

¯

°°®

'

t T cT x

T T T

T x T

T T

l l l

1 0 0 2

1 0 env

env 0 env 1

2 2

U O

D O

(5)

where lt1and T_env !T₀_l !T₁_l (border case)

ini

0 T

T_i (6)

where 0didm (initial case)

2. Solution of the problem

From the linear system (5) we calculate only T₀_l with the limitation

DO DO _x_' 2

1 (7)

In fact, the first equation from the system (5) gives

l

l T xT T

T₁ _env 2 _env ₀

'

O

D ₍₈₎

so

¸

¹

¨ ·

©

§ '

'

2 2 1

0 env 0

env env

0 1

0 T T T xT T T T x

T_l _l _l _l _l

OD

OD ₍₉₎

and because T_env T₀l !0 and T₀l T₁l !0, that is

2 1

! O 'x

D _or

D O

!2

'x (10)

And now the linear system (5) gives

1 0 2 env

2 1 2 0

env 1

0

2

2 2

'

¸¹

¨ ·

©

§

' '

¸¹

¨ ·

©

§

'

l l

l l l

tT T c

T x T x

t x c

x T xT

T

U O

O O

U

D O D O

(11)

(3)

with the determinant condition

2 0 2

1 2 2

2

2 2

2

!

'

' '

'

' '

'

x x t c x x

t x c

x O U D O

O O

U D O

(12)

It only needs to point out that the determinant on the right side

2 2

2 2 2

2 1

t x c x x

t x

c '

'

' '

D U O

O U D

(13)

is positive, when

2 0 2 2

2 ¸!

¹

¨ ·

©

§ ' ' '

'

D O O D

x x x

x ⁽¹⁴⁾

so

!0 'O D

x or

DO

'x (15)

The solution T₀_l of linear system (5) has then the next intermediate form

env 1

0

0 2

2

2 T

x x t c

x T x

x x t c

t c

T _l _l

¸¹

¨ ·

©

§

'

' '

¸¹

¨ ·

©

§

'

¸¹

¨ ·

©

§

'

' '

'

O D U

O D O D

U

(16)

For the internal linear system (4) and from symmetry condition T_m₁_l T_m₁_l we receive

2 1 2 1

1 2 1

2 1 2 2

1 2 2 3

2 2 2 1

1 1 2 0

2 2 2 1

2 2

2

. ...

...

2 2

¸ '

¹

¨ ·

©

§

'

' '

¸

¹

¨ ·

©

§

'

' '

¸

¹

¨ ·

©

§

'

' '

'

¸¹

¨ ·

©

§

' '

ml ml

l m

l m ml

l m l

m

l l

tT T c

t x T c

x

tT T c

T x t x

T c x

tT T c T x

t x T c

x

tT T c T x

T x t x

c

U O

O U O

U O

O U O

U O

O O

U

(17)

(4)

with the positive determinant (see [1, 2])

m

x m

t c x

x x

t c x

x x

t c x x

x t c x

x t x

c

' u

'

' ' '

'

' ' '

'

' ' '

'

' '

2 2

2

2 2

2 0 2

0 . . .

.

0 2 . . .

.

2 0 .

. .

.

. .

. . .

.

. .

. . .

.

. .

. . .

.

. .

2 .

. .

. 2 0

DET

O U O

O O

U O

O O

U O O

O U O

O O

U

(18)

4 2

DET

' m

m D

D Ox (19)

where

2 ...

2 2 2

1 2 1

8 4 4 2 4

2 2 2

2

¸ '

¹

¨ ·

©

§

'

¸¸¹ '

¨¨© ·

§

¸ '

¹

¨ ·

©

§

'

¸¸¹ '

¨¨© ·

§

¸¹

¨ ·

©

§

' '

x t x

j c x t x

j c t x

D c

j j

j

j U O U O O U O O

(20) Finally, we can write

m

x m

t c x

x t x

c x

x t x

c x x

t x c x

x x

t c

' u

'

' '

'

' ' '

'

' ' '

'

' '

2 2

2

2 2

2 0 2

0 . . .

.

0 2 . . .

.

2 0 .

. .

.

. .

. . .

.

. .

. . .

.

. .

. . .

.

. .

2 .

. .

. 2 0

DET

O U O

O O

U O

O O

U O O

O U O

O O

U

(21)

(5)

¸¸

¸

¹

·

¨¨

¨

©

§

'

¸¹

¨ ·

©

§

' '

'

¸¹

¨ ·

©

§

' '

'

¸¹

¨ ·

©

§

' '

¸ '

¹

¨ ·

©

§

' '

¸

¹

¨ ·

©

§

' '

2 ...

! 4

5 6 7

2

! 3

4 5

2

! 2

3 2

2

16 8 8 2

12 6 6 2

8 4 4 2 4

2 2 2

DET 2

x x t c m m m

x t x

c m m

x t x

c m x t x

c

x m t c

m m

m

O O U

O O U O

O U

O

U (22)

Next, the algebraic complements of the matrix

m

x m

t c x

x t x

c x

x t x

c x x

x t c x

x t x

c

u

»»

»

¼ º

««

«

¬ ª

' ' '

'

' ' '

'

' ' '

'

' ' '

'

' '

2 2

2

2 2

2 0 2

0 . . .

.

0 2 . . .

.

2 0 .

. .

.

. .

. . .

.

. .

. . .

.

. .

. . .

.

. .

2 .

. .

. 2 0

O U O

O O

U O

O O

U O O

O U O

O O

U

(23)

there are the simple induction formes (see [2])

1

2 1

1 1 2 2

1 1

4 2 2 2 1

2

1 1

for 1

2 1 2

1 0

for 1

d

¸ d

¹

¨ ·

©

§

'

¸¹

¨ ·

©

§

'

¸¹

¨ ·

©

§

'

d

¸¸¹ d

¨¨© ·

§

'

¸¹

¨ ·

©

§

'

m mm

k k m k

m mk

k k m m

k km

p k m p

k m k p p

k p

kk

D A

m k x D

A

x D x

t A c

k m p x D

D x D

A

O

O O U

O O

(24)

according to the formula for D_j.

Also, the direct solutions of the linear system (17) are given

DET

1 1

1 1 1

1 1

2 0l l i m l m i ml mi

il

A tT A c

tT A c

tT T c T x

'

¸

¹

¨ ·

©

§

' '

U U

U

O

(25)

for 1didm2

(6)

DET

1 1 1

1 1 1 1

1 1 1 2 0

1

'

¸

¹

¨ ·

©

§

'

' ^l ^l ^m ^m ^l ^m ^m ^ml ^mm

l m

A tT A c

tT A c

tT T c T x

U U

U

O

(26)

DET

1 1

1 1 1

1 1

2 0l l m m l m m ml mm

ml

A tT A c

tT A c

tT T c T x

'

¸

¹

¨ ·

©

§

' '

U U

U

O

(27)

according to the formulas for A_pq.

References

[1] Mostowski A., Stark M., ElemenW\DOJHEU\Z\ĪV]HM, PWN, Warszawa 1970.

[2] %LHUQDW*%RU\Ğ-&DáXVLĔVND,6XUPD$The three-band matrices (to appear).

[3] Majchrzak E., 0HWRGDHOHPHQWyZEU]HJRZ\FKZSU]HSá\ZLHFLHSáD, Wydawnictwo Politechniki

&]ĊVWRFKRZVNLHM&]ĊVWRFKRZD

[4] Mochnacki B., Suchy J.S., 0RGHORZDQLH L V\PXODFMD NU]HSQLĊFLD RGOHZyZ, WN PWN, War- szawa 1993.

[5] Majchrzak E., Mochnacki B., Podstawy teoretyczne, aspekty praktyczne i algorytmy, Wydaw- nicWZR3ROLWHFKQLNLĝOąVNLHM*OLZLFH