Please cite this article as:
Grzegorz Biernat, Justyna Mazur, Finite difference method in the Fourier equation with Newton's boundary conditions
direct formulas, Scientific Research of the Institute of Mathematics and Computer Science, 2008, Volume 7, Issue 2,
pages 123-129.
The website: http://www.amcm.pcz.pl/
Scientific Research of the Institute of Mathematics and Computer Science
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: [email protected]
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 determined initial conditions (without the internal heat source) is defined by the equations
O
w 2 T x, t wx 2
Uc
wT x, t wt
(1)
(heat conduction in the centre-Fourier’s equation)
O
wT x, t wn xborder
D T x, t Tenvironment xborder
(2)
(heat exchange on the border-Newton’s equation)
T x,0 Tinitial
(3)
where the positive coefficients O , U , c and D we receive as a constant.
Classical difference methods lead to the linear systems equations
O
Ti 1l 2Til Ti 1l
'x
2
where 1 d i d m and l t 1 (internal case)
Uc
Til Til 1
't
(4)
Finite Difference Method in the Fourier equation with Newton boundary conditions …
125
­ T1l Tenv
D T0l Tenv °°O 2'x
®
°O Tenv 2T0l T1l Uc T0l T0l 1
°¯
't
'x 2
(5)
where l t 1 and Tenv ! T0l ! T1l (border case)
Ti 0
(6)
Tini
where 0 d i d m (initial case)
2. Solution of the problem
From the linear system (5) we calculate only T0l with the limitation
1O
O
'x 2D
D
(7)
In fact, the first equation from the system (5) gives
Tenv T1l
2D
O
'xTenv T0l (8)
so
T0l T1l
T0l
Tenv 2D
O
Tenv T0l §¨ 2D 'x 1·¸
'xTenv T0l © O
¹
(9)
and because Tenv T0l ! 0 and T0l T1l ! 0 , that is
2D
O
'x ! 1 or 'x !
O
2D
(10)
And now the linear system (5) gives
DT0l O
2'x
T1l
O ·
§
¨D ¸Tenv
2'x ¹
©
O
§ Uc 2O ·
2 ¸T0l 2 T1l
¨
'x
© 't 'x ¹
O
'x 2
Tenv Uc
't
(11)
T0l 1
G. Biernat, J. Mazur
126
with the determinant condition
D
Uc
't
O
2O
'x 2
2'x
O
'x 2
O
Uc
D
2O
2'x 't 2
'x
1
2 !0
'x
(12)
It only needs to point out that the determinant on the right side
Uc
D
2O
't 'x 2
1
2
'x
2D Uc 2O
'x 't 'x 2
(13)
is positive, when
2D
2O
2
'x 'x
2 § O
·
D ¸ ! 0
¨
'x © 'x
¹
(14)
so
O
'x
D ! 0 or 'x O
D
(15)
The solution T0l of linear system (5) has then the next intermediate form
Uc
't
T0l 1
Uc 2 § O
·
D
¸
¨
't 'x © 'x
¹
T0l
2 § O
·
D ¸
¨
'x © 'x
¹ T
env
Uc 2 § O
·
D
¸
¨
't 'x © 'x
¹
For the internal linear system (4) and from symmetry condition Tm 1l
receive
O
O
Uc
§ Uc 2O ·
T1l 1
T 2 ¸T1l 2 T2l
¨
2 0l
t
't
'
'x
'x
'x ¹
©
O
O
Uc
§ Uc 2O ·
T2l 1
2 T1l ¨
2 ¸T2l 2 T3l
't
'x
'x
© 't 'x ¹
.......................................................................
O
§ Uc 2O ·
Tm 2 l ¨
2 ¸Tm 1l 2 Tml
'x
'x
© 't 'x ¹
Uc
2O
§ Uc 2O ·
Tml 1
2 Tm 1l ¨
2 ¸Tml
't
'x
© 't 'x ¹
O
2
Uc
't
(16)
Tm 1l we
(17)
Tm 1l 1
Finite Difference Method in the Fourier equation with Newton boundary conditions …
127
with the positive determinant (see [1, 2])
Uc
't
DET
2O
'x 2
O
0
'x 2
Uc 2O
't 'x 2
.
O
'x 2
.
O
'x 2
.
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0
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0
DET
Dm O
Uc
2O
't 'x 2
'x 2
0
'x 2
Uc 2O
't 'x 2
2O
2
'x
O
'x 2
0
O
O
O
'x 2
Uc 2O
't 'x 2
mum
(19)
Dm 2
'x 4
(18)
where
j
§ j 1·§ Uc 2O ·
§ Uc 2O ·
¸¸¨
2 ¸ ¨¨
2¸
¨
© 't 'x ¹
© 1 ¹© 't 'x ¹
Dj
j 2
j 4
§ j 2 ·§ Uc 2O ·
O4
¨
¸
...
¸
¨
'x 4 ¨© 2 ¸¹© 't 'x 2 ¹
'x 8
(20)
O2
Finally, we can write
Uc
't
DET
2O
'x 2
O
'x 2
.
O
'x 2
Uc 2O
't 'x 2
.
0
O
'x 2
.
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0
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0
O
Uc
'x 2
't
2O
'x 2
O
'x 2
0
O
'x 2
Uc 2O
't 'x 2
2O
2
'x
(21)
0
O
'x 2
Uc 2O
't 'x 2
mum
G. Biernat, J. Mazur
128
§ § Uc 2O · m2 O2 m 3 § Uc 2O · m4 O4 ·
¨¨ ¸
¸
¨ ¸
¨ © 't 'x 2 ¹
2! © 't 'x 2 ¹
'x 8 ¸
'x 4
¨
¸
m
m 6
O6
¨ m 5m 4 § Uc 2O ·
¸
§ Uc 2O ·
¨ 2¸
¨ 2 ¸ m¨
¸
12
'
'
t
3
!
t
'
'
'
x
x
x
¹
©
¹
©
¨
¸
¨ m 7 m 6 m 5 § Uc 2O · m8 O8
¸
...
¨
¸
¨ 2¸
16
¨
¸
'
4
!
t
'
'
x
x
¹
©
©
¹
DET
(22)
Next, the algebraic complements of the matrix
ª Uc 2O
« 't 'x 2
«
« O
«
'x 2
«
.
«
.
«
«
.
«
.
«
«
«
.
«
«
.
«
¬
O
'x 2
Uc 2O
't 'x 2
.
.
0
O
'x 2
.
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0
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0
O
Uc
'x 2
't
.
2O
'x 2
O
'x 2
0
.
O
'x 2
Uc 2O
't 'x 2
2O
2
'x
º
»
»
»
.
»
»
.
»
.
»
»
.
»
0
»
»
O »
2
'x »
Uc 2O »
»
't 'x 2 ¼ mum
.
(23)
there are the simple induction formes (see [2])
p
§
·
O2
¨
D
D
Dm k p 2 ¸¸ for 0 d p d m k 1
¸
k 1 ¨ m k p
2
4
'x
© 'x ¹
©
¹
O ·
Akk p
12k p §¨ Akm 1
1k m 1 §¨ Uc © 't
O ·
¨ 2 ¸
x ¹
'
©
mk §
Amk
2 1
Amm
Dm 1
2O ·§ O ·
¸
¸¨ 'x 2 ¹© 'x 2 ¹
m k 1
Dk 1
(24)
mk
Dk 1 for 1 d k d m 1
according to the formula for D j .
Also, the direct solutions of the linear system (17) are given
Til
Uc
Uc
Uc
§ O
·
T1l 1 ¸ A1i Tm 1l 1 Am 1i Tml 1 Ami
¨ 2 T0l t
t
'
'
't
© 'x
¹
DET
for 1 d i d m 2
(25)
Finite Difference Method in the Fourier equation with Newton boundary conditions …
Tm 1l
Tml
Uc
Uc
Uc
§ O
·
T1l 1 ¸ A1m 1 Tm 1l 1 Am 1m 1 Tml 1 Amm 1
¨ 2 T0l 't
't
't
© 'x
¹
DET
Uc
Uc
Uc
·
§ O
T1l 1 ¸ A1m Tm 1l 1 Am 1m Tml 1 Amm
¨ 2 T0l 't
't
't
¹
© 'x
DET
129
(26)
(27)
according to the formulas for Apq .
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, Warszawa 1993.
[5] Majchrzak E., Mochnacki B., Podstawy teoretyczne, aspekty praktyczne i algorytmy, WydawnicWZR3ROLWHFKQLNLĝOąVNLHM*OLZLFH