BO GIAO ])t)C vA 1);\0 T~O
[11)1H()C Qu6c cIArl..jANH PH6 H6 CHI MINH
TRUc)NCD~I HQC KHOAHQC H,! NHlt N
a
a
~~I:&'~~
N
C TrY'~
",1\..1:',
D
. ,.
N
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( 'r'
)l \T(i T AM
,,'_.Tn..,
CHiNH. UO,\ l\iQT 86 BAI ToAN NGtf<1C
TRONe KHOA HOC rfNG Dt)NG
Chuyennganh: loAN GJArItCH
\13 s6
01.01.01
II
TOM T£(TLu.~N AN
Fh6 Tie'n SIKhoa hQcToan Ly
II
Thanh ph6116 ChI Minh
- 1996-
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)
"J, '
-
I.';
~
Lu~n ~n n1iydu'Qchean thanh t~i Khoa Toh - Tin h9c
II
Tru'CJngBl}i hQc &boa hQc TV lJl'Mn Thanh pho' hI6 Chi Minh
IIiIo
Irii
'II
Netti1i
hu'OIH~ d1in :
"
II
iii
GS TS J:)~NG DINH ANG
rI/lI
..
II1II
l1li
'II
Ng1f(Y]nhan xet 1 :
II
II
II
a
III
a
II
III !I
..
~i1j
hh1tnxet 2 :
CI
II
II'
Cd Quan nJU)Hxet :
Ie
'III
III
Ii!
=
II
Lu~n ~n se du'<;fc
baa v~ tq,iH9i D6ng Chill Lu4n an Nh;) Nll'OChqp
tq,iTru'CJngDq.ihQc Khoa h9C Tv Nhien Thanh pho' H6 Chi Minh VaG
hk~
giCJ ~
ngay
--
thclng
~
Ham
1~96.
'
III
..
III
IJI
C6th!
tlm hilu Lutjn dn tQi cdc tllltvifl1 :
!:I
-' Tnto/'lg Dqi h9C Khan h9C T~(Nhien Thc'rl1hpluJ' H6 Chi Minh
- Khaa H9c nfng fir]) Thanh ph/f H6 C?ll Minh
a ID
nO GIAo D~JC vA. BAo L'}O
D/\I HQC Quc5c CIA THANH PH6 HO CHi MINH
TRUHI
Trong phan II chung toi xet bai toan Cauchy cho phu'ong trmh Laplace
trong t~ng g6 gh8 cua R3nhu'san
D = {(x,y,z):- 4>(x,y) ,
V(x.y)
E R2
Sd dl}ng phu'dng phap chlnh h6a Tikhonov (xem A.N.Tikhonov and
V.Y.Arsenin : Solutions of ill-posed problems. Winston. Willey, New York,
(1977», chung toi xiy dlfng mQt phu'dng trlnh bie'n pMn (phu'dng trlnh chInh
h6a) (00) :
~ Ve = Fe
(3)
Trong d6 bai toaD too nghi~m v=v" da phu'dng trlnh (3) la bai "toaD
chlnh, nghia la
i) T8n t~i duy nha't v"thoa (3)
ii) v" phI} thuQc lien tl}c vao Fe
E>6ngg6p quan tn;mg khac trong Lu~n an Ia chUng t5i dii dauh gia du'<1c
5ai 56 giU'a nghi~m chlnh h6a v" neu teen so voi nghi~m chinh xac v cua phu'dng
trlnh (1)
-3 -
Cl}the la ne'u sai s6 giiia dii ki~n do d."e F£ va dU'ki~n ehinh xac F la
&
, nghlala
(4)
~F,-FII< Ii
thl eh11ngt8i eh1fng t6 du'<;1ela sai s6 giiia nghi~m ehlnh h6a v£ va nghi~m chinh
xacv (Vdi~iathie'ttrdnthichh<;JP)C6b~C,fS
nghlala
IIv£-vll < c,fS
hay
[l{~)r;(0<&<1)
(5)
hay
II v.-vll < c[tr{~)r
(6)
trong d6 h!ing s6 du'dng C kh6ng phl} thuQc S Ta chuiln 11.lIl1y trong cae kh6ng
gian tu'dng 1fng
.
Hdn the' niia, ehl1ng t8i thi~t l~p dU<;1f;
thu~t roan Giai tlch s8. Cl} the; nhu'
sau:
a) £>6ivdi cac bar roan khao sat trong phh I, chung toichd'ng Minh
du<;1e
rhg v.chinh la diem b1t dQngduy nh:lt cua mQtroan ttl'co thieh h<;Jp.Do d6
de dang dy dvng mQtthu~t roan l?p M tinh xa'p xl v£ . O9i v£(rn)la budc l~p thd'
m .Chung toi da dua ra dU<;1cdaRb gia sai s6
Iv,("> -vl<
C,k'" +C~
(7)
C£ 13.h!ing s8. phl} thuqc s. kh8ng phl} thuQc m . k E (0.1) 13.h~ s8 co. Hdn niia
ne'u chQn budc l~p t6i thieu m=m. tIll chung t8i thu du'<;JcdaRb gia sai so'
~v}",)
- vii < (1 + C)J;:
(8)
b) £>6ivdi bai roan trong phh II, chung toi du'a ra du'<;1ccong th1fc tu'Clnp,
minh tinh v. theo dU'ki~n do d~c F£ thong qua bie'n d6i Fourier (hai chi~u) thu~II
va ngu<;1C.Vdi gia thi6t v du trdn (v E Hl(R2» chung t8i thu du'<;JcdaRb gia sai s6
-4-
I"~
1\ v.-vll
<
C[~;)r
trong d6 h~ng s6 C chi ph'} thuQc vao Ilv~lh'11)
Lie ke"lqua cbillh CIIa LlI~ll all (hi<,lccong b6 trong
c!til/c cong b6"lrong [:\J.[4 J.[S J"
tJ
-5 -
[1] .[2] va se
PJIANM6r
cAc sAI loAN CAUCHY
CHO PHUONG TRINH POISSON
I. BM roAN CAUCHY CHO PHtJdNG TRINH POISSON TRONG HINH
TRON BdN VI :
1.Bdi loan..
G~i
D =I(X,Y):X2+ l < I}
15 = I(x,y): X2 + y2 ~ I}
Tlm ham u = u(x,y) th0
vfl E [}(a,2TC)
eho tru'de x~t bai loan : T1m
va FE L2(O.a)
saoeho
P(\'p,ffJ)+
=
trong do ( . , .) va <.,.
(15)
,Vf!JEI!(a,2f'l)
> Ih lu'~t la tieh vo hu'dng trong L2(a.2TC) va
L2(O.a). Chung ta ky hi~u cae ehuin tu'dng U'ngla
11.11
H
va 11.11HI . Ta co ke't
qua:
Dillh Iv 1.1:
Vdi m6i
nha't mQt nghi~m
P >0
va FE L2(O.a) phu'dng trlnh (15) co cluy
vp E L2(a.27r)
, hdn m1a vp phV thuQc lien t1}c vao
FE L2(O.a).
Ghl sU' Vo13.ngill~m chinh xac ell a phu'dng trlnh
Avo
thoa di~u ki~n : T6.ri t~i
= Fo
v E L2 (O.a)
(vo.ffJ)=
(16)
sao cho
,
(17)
VffJEL2(a.27r)
Kill d6 ta co
Dinh
It 1.2:
GiasltF.FoEe(O,a)
thoallF-Foll
HI
iv, lil nghi~m da phu'dng trlnh bie'n phan (15) rl'ng vdi
p =£ thl ta co daub gia
livE - VO~H <
trong do
Mii
(18)
r
(19)
M=C+I~U~.
5. Phlidnf!. IJhti~ s(J:'
XtSt phu'dng trlnh bie'n phan (Ii > 0) :
£(v.,ffJ)+
=
,VffJEL2(a.27r)
hay tu'dng du'dng
- 8 -
(20)
1>1'F. +;\*;\1'
B =;\*r
(2\ )
ludo
I'F. =\'
(
F. -n.f' 1>1'F. +;\*;\1'
F. -;\*r
)
(22)
v(fj fJ > 0 sc ch9n sau,
=
V~y vI> T vI>v8i T: L2 (a.,27t) ~ L2 (a.,27t)
du'(jc xac djnh nhu' sau :
Tv= v-p(ABv-A
() day
,
AB =;E.ld+A
(23)
*F)
*
(24)
A ,
vii ld - to
Dillh Ii 1.3:
V8i P = (E+
He Qua1.1:
IIA If
Y
thl T Iii phep co trong L2 (a.,27t)
'liE:>0 cho tmac, phu'dngtrlnh (20) ho~c (21) co nghi~m duy
nha'l VBE L2 (a.,27t)
Ta linh VI>bAng phu'dng phap xa'p xi lien tie'p
(m) T (m-I)
VB
VB
m 1, 2 '0"
--
--
v~O) E L2 (a..27t)
y
tily
(25)
Taco
,,~m)= (/- PEl v~m-l) - pA *( A 'J~m-I) - F)
(26)
v8i fJ nhu'trong Djnh Iy 1.3
Mellh d~ 1.2:
Gia sar v~ thoa (16), (17). Khi do sai s6 giii'a v~m) va vo 13
Il
vlllll - v
I>
0
< ('
II
/}ra..21t)
kill
I>
-9-
+M
r;
v'<'-
(27)
d da
C =
Y
IITv.(0)-". (0)11,
.
I-k
L (a,2")
,
'
( 28 )
k - h~ s8 co eua anh x~ co T ; (0 < k < I) va M de djnhb"l (19)
Ml!nh dO 1.3:
~f)
,
(29)
ChQns6 tVnhl~n m. > Ink
Bat
. v = v (M,) khid6
..
Il
v, -
VoI,
ilL(a,l..)
< (1+M)JE
(30)
Cilu tb.ie!!.;.
m.la s8 bd&:l~p t8i thi~u di ta e6 danh gia teen.
MQt ph~n ket qua eua mvc nay dii ddcjcc6ng b6 trong [I] va[2].
II. BAI ToAN CAUCHY CHO PHVcJNGTRINH POISSON TRONG NUA
MAT PHANGTRtN:
1. Biz; loan:
GQi
Tlm
p+ = {(x,y):
-ooO}
]5+= {(x,y):
-ooy~O}
U E Cl(P+)nC2(]5+)
Au = f
.
thoa
C(P+)
trong p+
Uy E
(31)
u(x,O) = uo(x)
"Ix E 1 = (-1,1)
Uy(x,O) = UI(x)
(32)
u ehinh qui d v8 eung. nghla Iii.t6n t~i h~ng s8 dddng B sao el1o
lim sup u(x,y) = U'"
R-++a> x'+,,'-R'
y>o
B
IVu(x,y)\~
Xl +
l
'V(x,y) E P+ vhl
+l
- 10 -
duMn
(34)
~ diiy Vu - gradient cua u.
f chotn10ctrongr
. Uo,u.
tho tru'<'1ctrong (-1.1) ; Uy - d~o ham rieng cua
u theo y .
2. Thitt llip p1uh1nll trinh tlch phlin
Ch9n v(x)
..
=Uy(x,O) , x E J=R\I=
{x:~1 ~ I} lam in ham.
B~ng phu'dng ph3.p Green, chung tBi du'a bal to~n (31 ).(32). (33),
(34) v8 phu'dng trlnh tich phan Fredholm lo~i mQt sau dAy d6i v<'1ilin ham
v(x) ..
JJ v(~)~-
v<'1i
(35)
~Id~= F(x)
F(x) = 1T(Uo
(x) - u,.,)-
- ~ If 1(';",)
J u1 (~) lntx - .;Id.;
-I
I..{(x- .;)2 + ,,2 ]d~d"
(36)
3. Khdo sat phJif1n6, trinh tic" phlin..
Gi:l thie't:
i)
UO,UI E L2(J)
ii)
f ~ L~(P')
vdi
Hi)
.1=(-1.1)
={rI[II'«
(37)
,.)f'«,
.)<1<
}. 0 chotru'&
v EL~(J) = {v:[ p(~)v2(.;)d.; < cO}
voi
(39)
P(~)=(1+1.;~2
- 11 -
Ai! 1.3:
Voi
e >0
chotniOc
,-1 0 xet phu'dngtrlnh bi~n philn
(41)
liV. +A'Av.
=A'F
tu'dng du'dng
£v. +A'Av.
-A'F
hay Ia
vC1i
v, =v, - P(EV, +A'Av.
=0
-A'F)
/3 > 0 se ch<;msau
- 12 -
(42)
V~y Vs
=T
jIB
VOlloan tifT du0, \::IFE L2(J) cho tru'oc phuong trlnh (41) co nghi~m duy nhKt
2
vsELp(J)
Giii su rnng phuong trlnh
(45)
Avo = Fo
co nghi~m chinh xac Va san cho t6n t':li
vE
(vO,q»L~(J)= (It; Aq»Ll(/)
L2 (I) thoa
(46)
\::Iq>EL~(J)
Ditlh IV1.5:
Giasu va thoa (45), (46) va !IF- Fo 1~1(/)< E khi do
lIvE- Vo II~(J) < M
i'1
F
day Va - nghi~m ciia phuong mnh bie'n phan (41), cfing ill di~m b1lt
~?
1/2
2
1+ 111,111
.
L(l)
.
dQngcua T, M =
2
[
]
5. PIll/dill!vM,} sri:
Ta tinh
jIB
bang phuong phap xa'p xi lien tie'p
,(111)-- 7'
Is
(0)
Vs
,(111-1)
IE
, ,
2
E 1'p (J) tHYY .
- 13 -
"
m-- I,-,...
6
Chon
.
r-
v;"
L
(6+36)2
1
62
i. (E
thl
:::
jJ
(48)
~36)T 1';",.1)- (c +C36)2A'(Av~"-.) - F)
Khi d6 ta co hai mc:nh M (I ') v~ 1.6) IIMnp,ht vai hai mt$nh d~ 1.2 va 1.3 d
Inl)CI. MQI phau k~'l qua Clla lIJlle !Jay se ch(yc caug b6 trou!?,131 .
Ill. BA.ITOA.NCAUCHY CHO PHUONG TR1NH POISSON TRONG NUA
KHONGGIAN TR.t:N:
Llld{.O!I.T1':'
f)~t
R;::: {(x,y,z):
-00<
x,y<
J{3::: {(x,y,z):
-00<
x,y.u::: f
OO,Z > a}
trong
Uz E C(R/)
R;
thoa
(49)
vdi dil ki~n Cauchy du<;1crho tnf(1c tren dla Iron ddn vj et1a m~t phang z==O
u(x,y,a)::: uo(x,y)
11,('1",)',0):::U,(x,y)
V(x,y)
EQ
(50)
trong d6 f cho Iniac trong R; ; uo,u. cho tru'<'Jctrong Q ; IIz d~o ham rieng
z.
nla u theo
II chlnh qui d v() rUng. nghia la
r
(i)
l~d ,,+~+~)lI(x,y,z)l
L
(ii)
I
1
z>o
:::a
(51)
j
T3n t~i hhg s6du'dng C sao cho
vu(x,y,z)I:- Xem thêm -