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)

NOI DUNG--------------------------------------------------------------------------------------------------------------------------------

1 TONG QUAN

2 PHNG TRNH PARABOLIC (TRUYEN NHIET)

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

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 +=

=

=

==

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

BAI TOAN TRUYEN NHIET: PT PARABOLIC -------------------------------------------------------------------------------------------------------------------------------------------

x

t

>== ttutu lBienK

V DU (SGK, trang 253) -----------------------------------------------------------------------------------------------------------------------------------

Giai phng trnh parabolic (bai toan truyen nhiet) sau

( ) ( ) ( )

===

>

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: