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- BO GIAO ])t)C vA 1);\0 T~O
[11)1
H()C Qu6c cIArl..jANH PH6 H6 CHI MINH
TRUc)NCD~I HQC KHOAHQC H,! NHlt N
a
a
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"
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N
",1\..1:', ,,'_.Tn..,
D
CHiNH. UO,\ l\iQT 86 BAI ToAN NGtf
- -
~} I.';
)
"J, ' ~
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
l1li
.. 'II
II
II1II
II
Ng1f(Y]nhan xet 1 :
II
a
III
a III !I
II
..
~i1j hh1tnxet 2 :
CI
II
II'
Cd Quan nJU)Hxet : Ie
'III
III
Ii! =
II
Lu~n ~n se du'
- nO GIAo D~JC vA. BAo L'}O
D/\I HQC Quc5c CIA THANH PH6 HO CHi MINH
TRU
- Lu4n an n~y du
- MO 8AU
Trong Khoa hl)c ling d\Ing, nolI du khao sat hili loan nglic;1cdii xullt
hi~n tit' lau, Coo de'n nhung nam 60, d6ng thdi VOlvi~c phat tri6n cac c6ng C\l
loan hQC,cac hiii loan ngu'
- Phdn I chung t&i xet 3 ba.i toin Cauchy cho phtiong trmb Poisson trong
.
dl3 troll don vi Dc R2 trong nU'a m~t phAng tren p+ c R2. va. trong mta kh8ng
ih
glaD tIeD R\ ; v(Ji dfl ki~n Cauchy (u. c7v- d~o ham theo htidng phap tuye'n
ngoai tren bien cua mien) du'
- nay la m5 hlnh R3cua bai toan da:du'qc khao s.h (xem D.D.Ang. D.N.Thanh &
V.V.Thanh: Regularized Solutions of a Cauchy problem for the LAplaceequation
H
in an irregular strip ", :Tournal of integral equations and Applications, Vo1.5, N2.4,
(1993), p,p. 429_441),
B~ng phu'dng phap Green va ly thuye't the' vi, chung t51 da: du'a du'6ngg6p quan tn;mg khac trong Lu~n an Ia chUng t5i dii dauh gia du'
- 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'
- I"~
C[~;)r
1\ v.-vll <
trong d6 h~ng s6 C chi ph'} thuQc vao Ilv~lh'11)
Lie ke"lqua cbillh CIIa LlI~ll all (hi
- 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..
D =I(X,Y):X2+ l < I}
G~i
15 = I(x,y): X2 + y2 ~ I}
Tlm ham u = u(x,y) th
- B~ng phu'dng phap Green chung toi du'a bai toaD (9), (10) v~
phu'dngtrlnh tich phan Fredholm lo~i mQtd6i vdi /in ham v nhu'sau :
2r I- B
( 11)
a v(/)ln2sinTdl:::
J F(O)
I I
I lit (I) In21sin I ~ 0 Idl
vdi F(O) ::: 1T[U(O) uo(O)] -
-
Iff(~, (12)
-
+~ ~)2 + (sinB - 1})2] -In(~ 2 + 1}2))d~d1}
1}){21n[(cosO
D
3.Khiio sat phlidnf!. trinh tfch vhlin-,-
Rtf di 1.1.. Ne'u lIo,U1E L2(0,a) va f E L2(D) tIll FE L2(0,a) vdi
F xac dinh CI(12).
1- 0
Bil tfi 1.2:
hamt~ln2sin2
';jOE(O,a). thuQcL2(0,21T).
I I
2r I-O
.
(13)
v(/)ln2sinTdt
£)~t (Av)(O)::: J
a r j
thl ta c6:
Ml!nh tfi 1.1: Toan tU' A: L2(a ,21
- P>0 eho tru'de x~t bai loan : T1m
Vdi va FE L2(O.a)
vfl E [}(a,2TC) saoeho
(15)
P(\'p,ffJ)+ = ,Vf!JEI!(a,2f'l)
trong do ( . , .) va Ih lu'~t la tieh vo hu'dng trong L2(a.2TC) va
va 11.11HI . Ta co ke't
L2(O.a). Chung ta ky hi~u cae ehuin tu'dng U'ngla 11.11 H
qua:
P >0
Dillh Iv 1.1: Vdi m6i va FE L2(O.a) phu'dng trlnh (15) co cluy
nha't mQt nghi~m 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
= Fo (16)
Avo
v E L2 (O.a)
thoa di~u ki~n : T6.ri t~i sao cho
(17)
(vo.ffJ)= , VffJEL2(a.27r)
Kill d6 ta co
Dinh GiasltF.FoEe(O,a) thoallF-Foll 0) :
(20)
£(v.,ffJ)+ = ,VffJEL2(a.27r)
hay tu'dng du'dng
-8-
- 1>1' +;\*;\1' B =;\*r (2\ )
F.
ludo
( )
I'F. =\' F. -n. 1>1' +;\*;\1' F. -;\*r (22)
f' F.
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 :
(23)
Tv= v-p(ABv-A *F)
() day
, *
(24)
AB =;E.ld+A A,
vii ld - to0 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)
m 1, 2 '0"
- -
VB
VB
y (25)
v~O) E L2 (a..27t) tily
Taco
(26)
,,~m) = (/- PEl v~m-l) - pA *( A 'J~m-I) - F)
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
r;
+M
< ('
vlllll - v (27)
kill
Il II
I> 0 /}ra..21t) I> v'
- IITv.(0)-". (0)11,
( 28 )
d da C= L (a,2") ,
Y . I-k '
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
.
v, - < (1+ M)JE (30)
VoI,
ilL(a,l..)
Il
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 p+ = {(x,y): -oo
- ~ diiy Vu - gradient cua u.
. Uo,u.
f chotn10ctrong r tho tru'
- Ai! 1.3:
e >0
Voi chotniOc ,-1 0 se ch
- =T
V~y Vs VOlloan tifT du0, \::IF E L2(J) cho tru'oc phuong trlnh (41) co nghi~m duy nhKt
2
vsELp(J)
Giii su rnng phuong trlnh
Avo = Fo (45)
vE
co nghi~m chinh xac Va san cho t6n t':li L2 (I) thoa
(46)
(vO,q»L~(J)= (It; Aq»Ll(/) \::Iq>E L~(J)
Ditlh IV 1.5:
Giasu va thoa (45), (46) va !IF- Fo 1~1(/)< E khi do
F
lIvE- Vo II~(J) < M
day Va - nghi~m ciia phuong mnh bie'n phan (41), cfing ill di~m b1lt
i'1
1/2
2
. 1+ 111,111
L(l)
~? .
dQngcua T, M = 2
[ ]
5. PIll/dill!vM,} sri:
Ta tinh bang phuong phap xa'p xi lien tie'p
jIB
"
,(111-1) m-- I,-,...
Is - 7'
,(111)- IE
,,
2
(0)
E 1 (J) tHYY .
Vs 'p
- 13 -
- 6
thl
Chon :::
jJ
. (6+36)2
1
r- 62
(48)
v;" ~36)T 1';",.1)- (c +C36)2 '(Av~"-.) - F)
A
i. (E
L
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< OO,Z > a}
z a}
J{3::: {(x,y,z): -00< x,y.u::: f trong
vdi dil ki~n Cauchy duo
(ii) T3n t~i hhg s6du'dng C sao cho
l (52)
+zz dtl Wn
C v
- = uz(x,y,O);
Ch
- (x,y) E 0 Ia tham s5
Xet ham 'l/x.y:Q -) R+
I
J
Ij/' r
nguon tai.lieu . vn
