a4 bai 7 - nguyen quoc lan
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BO MON TOAN NG DUNG - HBK-------------------------------------------------------------------------------------
TOAN 4
CHUOI VA PHNG TRNH VI PHAN BAI 7: PHNG TRNH AO HAM RIENG
TS. NGUYEN QUOC LAN (5/2006)
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NOI DUNG--------------------------------------------------------------------------------------------------------------------------------
1 TONG QUAN
2 PHNG TRNH PARABOLIC (TRUYEN NHIET)
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TONG QUAN (SGK, TRANG 248)-------------------------------------------------------------------------------------------------------------------------------------
P/trnh ao ham rieng: ham an nhieu bien, ao ham rieng
VD: Phng trnh Laplace (cap 2, tuyen tnh) 022
2
2
=
+
yu
x
u
PTHR & K bien, K au Ton tai & Duy nhat nghiem
( )
=
+
2
2
2
2
2
,
0
Ryxyu
x
u
( ) ( ) = ssgsu ,
Elliptic (tnh static): ( )yxfyu
x
u,2
2
2
2
=
+
Parabolic (truyen nhiet): 022
2=
x
ua
tu
Hyperbolic (truyen song): 022
22
2=
x
ua
t
u
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TONG QUAN -------------------------------------------------------------------------------------------------------------------------------------------
Tm nghiem bai toan bien sau ay: ( ) yeyuyu
x
u 38,0,4 =
=
Bc 1: Tach bien: ( ) ( ) ( )yYxXyxu =,
Bc 2: Thay vao ptrnh va phan ly bien so ( )( )( )( )yY
yYxXxX '4' =
2 ve phai cung bang hang so ( )( )( )( ) kyY
yYxXxX 4'4' ==
( )( )
=
=
0'
04'kYyY
kXxX ( ) ( ) kykx BeyYAexX == ,4 ( )yxkCeu += 4
Tm C: K bien ( ) ( ) yxyky eyxukC
eCeyu 3123 8,38
8,0 +=
=
=
==
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TONG QUAN PHNG PHAP TACH BIEN------------------------------------------------------------------------------------------------------------------------------------
Phng trnh ao ham rieng dang n gian, an u = u(t, x)
( )*0,,,,,,,2
2
2
=
txu
tu
x
u
x
uutxF K
Bc 1: Tach bien: u(t, x) = T(t).X(x) Thay vao (*)
Bc 2: Bien oi (*) [ X(x) ] = [ T(t) ]: Cung = k = const!
Bc 3: (*) [ X(x) ] = k: Pt vi phan thng theo x. Giai X
[ T(t) ] = k: Pt vi phan thng theo t. Giai T
Bc 4: Nghiem c s ucs = T(t). X(x). Neu PT thuan nhat
= cstq.tn uu ieu kien au (bien) Gia tr C
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BAI TOAN TRUYEN NHIET: PT PARABOLIC -------------------------------------------------------------------------------------------------------------------------------------------
x
t
>== ttutu lBienK
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V DU (SGK, trang 253) -----------------------------------------------------------------------------------------------------------------------------------
Giai phng trnh parabolic (bai toan truyen nhiet) sau
( ) ( ) ( )
===
>
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TONG KET PT PARABOLIC (BAI TOAN TRUYEN NHIET) -------------------------------------------------------------------------------------------------------------------------------------------
PT PARABOLIC 1 chieu & KB thuan nhat & K au
)()0,(: xxu =auK
( ) 0,0,0,122
2 >== ttutu lBienK
PTrnh (thuan nhat):
3/ Nghiem sau cung: ( )
=
=
0
2
222
sin,n
tan
n exnbtxu ll
pipi
4/ Tnh he so bn: ( )
=
l
ll 0
sin2 dxxnxbnpi
1/ Tach bien u(x, t) = X(x).T(t) BT bien (PTVP thng)
( ) tan
nn exnbtxu 2
222
sin, ll
pipi
=2/ GT rieng n Nghiem c s: