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Chuyennganh: loAN GJArItCH
\13 s6 01.01.01
II
TOM T£(TLu.~N AN
Fh6 Tie'nSIKhoahQcToan Ly
Thanh ph6116 ChI Minh
II

II
Cd Quan nJU)Hxet :
II'
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III
Ii! =
II
Lu~n ~n se du'<;fcbaa v~ tq,iH9i D6ng Chill Lu4n an Nh;) Nll'OChqp
tq,iTru'CJngDq.ihQcKhoa h9CTv 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?llMinh
aID
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D/\I HQC Quc5c CIA THANH PH6 HO CHi MINH
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NGUYEN C(1NG TAM
cHiNn HOAj\U)Tso nAI ToAN NGU<;jC
TRONG KI-IO;\H()C (fNG D\JNG
Chuycn nganh: ToAN GIAr TfCH
Mil s6 : 01.01.01
, ,I .,

Trong lu~n an, chung t6i khao sat mQt s6 bili loan Cauchy coo phu'ong trinh
Poisson va Laplace trong doc mi~n khac nhau ct'Ia R2 va RJ. Nhil'ng bili loan nay
co 9nghia qua" tn,mg trong ling d\Ing, chAng h;;tnnhu' trong V~t 19Dja du, VI
nghi9m ctta cac hili loan nay se du'<,1cxac dinh khi ta hic't di~u kic$nDirichlet (hay
Neumann) tren loan bien mi~n khao sat nen vic$cgiai hili loan Cauchy coo
phuong trInh Poisson hay Laplace du'<,1ccoi nhu bili loan tlm du kic$nd~u vao v la
di6u ki9n hien Dirichlet (hay Neumann) khi bic't du kic$nd~u ra F la di0u ki~n
bien CalJ<.:h'y(trcn mQt ph~n bien) va hc$thi')ng A chino la phut1ng trInh Poisson
hay Laplace tuUng (tng, hong D,a V~( Iy, hili loan nilYc6 9 nghia th\l'cto vlugu'\fi
ta thuong khang (ht; do d;;tc gia tri tntong trl)ng lifc, tr9ng 1\l'caj thll'ong hay
gradient dw n6 tren loan bien m't chi c6 th6 do tren mOtph~n bien ma thai,
- I -
!-)u vilO
I
H th6ng
!-)u ra
v
F
A
Phdn I chung t&i xet 3 ba.i toin Cauchy cho phtiong trmb Poisson trong
dl3 troll don vi Dc R2.trong nU'am~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'<;1ccho tren mQt ph~D bien cua mien.
CI}th~. chung t3i Ian lu'<;1tchuy~n dc ba.i toan kh3.o sit ve vi~c giai mQt
phtiong trmh rich phh Fredholm lo~i mQt :
Av=F
(1)
trong d6 A la. mQt loan tU'A:H
~ Hi. v(JiH. Hila bai kh8ng gian Hilbert. Trong

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'anghi~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 dod."eF£va dU'ki~n ehinh xac F la &
,nghlala
~F,-FII< Ii
(4)
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 hay
[l{~)r;(0<&<1)
nghlala IIv£-vll < c,fS (5)
II v vll < c[tr{~)r
hay
(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:

!!.U= f trong D (9)
u E C2(D)nC1(15)
va di\!u kic$nCauchy du'<1ccho tren mQt phh bien ct1a D nhu' sau :
U(CO50,5inO) = uo(O)
~:(COSO,5inO)=Ul(0) O«}<a (10)
vdi f cho tru'dctrong D; Uo . III cho tru'dc tren r = {e'o:0 < 0 < a} .a
t3u
cho tru'dc Q< a < 21f kj hic$u
- la d~o ham theo hu'dng pMp tuyen
. t3n
n
=(c~O ,sinO) tren t3D hu'dngra ngoai d5i vdi D
2. Thitt l/jp phr/dnll lrinb deb phlin
iJu. . t3u
Ch9n v(O)
=- (cosO ,5mO) ,Q ~ 0 ~ 21f 13.an ham, () d~y - la
t3n t3n
d~o ham theo hu'dng pMp tuyen ngoai tren vong troD ddn vi 13D.
- 6 -
B~ng phu'dng phap Green chung toi du'a bai toaD (9), (10) v~
phu'dngtrlnh tich phan Fredholm lo~i mQtd6i vdi /inham vnhu'sau :
2r
I
I - B
I
J v(/)ln2sinTdl::: F(O)
a
vdi F(O) :::1T[U(O)- uo(O)] -
Ilit (I) In21sin I ~ 0 Idl
+ ~ Iff(~, 1}){21n[(cosO - ~)2 + (sinB - 1})2] -In(~ 2 + 1}2))d~d1}

eho tru'de x~t bai loan : T1m
P(\'p,ffJ)+<AVp,Af!J> = <F.AffJ> ,Vf!JEI!(a,2f'l)
(15)
trong do ( . , .) va <.,. > Ih lu'~t la tieh vo hu'dng trong L2(a.2TC) va
L2(O.a). Chung ta ky hi~u cae ehuin tu'dngU'ngla 11.11 va 11.11 .Ta co ke't
H HI
qua:
Dillh Iv 1.1: Vdi m6i P > 0 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 ella phu'dng trlnh
Avo =Fo
thoa di~u ki~n : T6.ri t~i v E L2 (O.a) sao cho
(vo.ffJ)=<v.AffJ> , VffJEL2(a.27r)
(16)
(17)
Killd6 ta co
Dinh It 1.2:
GiasltF.FoEe(O,a) thoallF-Foll <Ii vavo thoa
HI
(16) ,(17). G<;>iv, lil nghi~m da phu'dng trlnh bie'n phan (15) rl'ng vdi
p =£ thl ta co daub gia
livE - VO~H < Mii
M=C+I~U~.r
(18)
trong do (19)
5. Phlidnf!. IJhti~ s(J:'
XtSt phu'dng trlnh bie'n phan (Ii > 0) :
£(v.,ffJ)+<Av AffJ> = <F.AffJ> ,VffJEL2(a.27r)
(20)

T
(m-I) -
12
VB - VB m - , '0"
v~O) E L2 (a 27t) tily y
Taco
(25)
,,~m)= (/- PEl v~m-l) - pA *(A 'J~m-I)- F)
v8i fJ nhu'trong Djnh Iy 1.3
(26)
Mellh d~ 1.2:
Gia sarv~ thoa (16), (17). Khi do sai s6 giii'a v~m) va vo 13
Il
vlllll - v
II
< (' kill + M r;
I> 0 /}ra 21t) I> v'<'-
(27)
- 9 -
IITv.(0)-". (0)11,
d da C = L(a,2") ,
(
28
)
Y . I-k '
k - h~ s8 co eua anh x~ co T ; (0 < k < I) va M de djnhb"l (19)
Ml!nh dO1.3:
, ~f)
ChQns6tVnhl~nm. > Ink
Bat v = v (M,) khid6

- 10 -
~ 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 p1uh1nlltrinh 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)
v<'1i
Jv(~)~- ~Id~= F(x)
J
F(x) = 1T(Uo(x) - u,.,)-
Ju1 (~) lntx- .;Id.;
-I
- ~ If 1(';",) I {(x- .;)2 + ,,2 ]d~d"
(36)
(35)
3. Khdo sat phJif1n6, trinh tic" phlin
i)
ii)
Hi)
Gi:lthie't:
UO,UI E L2(J)
.1=(-1.1) (37)
f ~ L~(P')
={rI[II'« ,.)f'«, .)<1<}. <w} (38)
vdi 1l(.;",)=(I+I.;I+I"r2l1; () >0 chotru'&
v EL~(J) = {v:[ p(~)v2(.;)d.; < cO}
voi P(~)=(1+1.;~2

hay Ia
vC1i
liV. +A'Av. =A'F
£v. +A'Av. -A'F = 0
v, =v, - P(EV, +A'Av. -A'F)
/3 > 0 se ch<;msau
(41)
(42)
- 12 -
V~y Vs =T jIB VOlloan tifT du<jcdinh nghTanhu sail:
2 2
T:Lp(J)~;Lp(J)
T,,=v-p(Asv-A"F)
i'1day
As
=E.ld + A" A
va Id - loan tu don vi trong L~( J)
Dillh Ii 1.4:
VoiP= E 2 thi T 1aphep co trong L~(J)
(H36)
(43)
(44)
He Qua1.2:
\::IE>0, \::IFE L2(J) cho tru'oc phuong trlnh (41) co nghi~m duy nhKt
2
vsELp(J)
Giii su rnng phuong trlnh
Avo =Fo
co nghi~m chinh xac Va san cho t6n t':li v E L2 (I) thoa
(vO,q»L~(J)= (It; Aq»Ll(/) \::Iq>EL~(J)

jJ
::: thl
. (6+36)2
r- 62 1
v;" L i. (E ~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~'lqua Clla lIJlle !Jay se ch(yc caug b6 trou!?,131
.
(48)
Ill. BA.ITOA.NCAUCHY CHO PHUONGTR1NHPOISSON TRONG NUA
KHONGGIANTR.t:N:
Llld{.O!I.T1':'
f)~t
R;::: {(x,y,z): -00< x,y< OO,Z > a}
J{3::: {(x,y,z): -00< x,y<oo,z z a}
Q::: {(x,y): X2 + y2 < I}
T1m ham u E C2(R;1)nC(R;), UzE C(R/) thoa
t>.u:::f trong R;
(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) E Q (50)
trong d6 f cho Iniac trong R; ; uo,u. cho tru'<'Jctrong Q ; IIz d~o ham rieng
nla u theo z .
II chlnh qui d v() rUng. nghia la
r 1
l~d ,,+~+~)lI(x,y,z)l
j
:::a
L z>o

Uo E L (0) (55)
Ut bj ch~n trong Q (56)
f EL: (R) 0 {fo Iff p( x,y ,z) f' (x,y,z )dxJydz < ro}
. vdi P(X,y,z)=(l+Jxz+i+zzY (57)
(Hi)
Bil di 1.5:
V(x,y) E 0 tich ph~n soy rQng
II
dl;d17
J(x,y) =
Q J(x-~)Z +(y- 17)Z
hQit~ va J(x,y)::;; 41f
Bil di 1.6:
- 15 -
Xet ham 'l/x.y:Q -) R+
(x,y) E 0 Ia tham s5
I
Ij/xr<I;, 17) = r: 2. z J 2 2' vII-ex-';) +(y-11) (x-';) +(Y-17)
fhl '(1',1E L2(Q). Hdn m1a
~
II
11
2 2- ~x +Y
VI'. ~ 21r In
x;, Lt(Q)
I
J2 2
- x +y
!1ff !li1.7;
Xet ham x.,,:RJ -)R+ ;(X,Y)EO lathams5

VJi roM 'EL:(Q) =!, II w«, "),' «, ")d91" < 00)
trong do W(';,17)= 1+ J,;2 + 172 (';,11) E Q
- 16 -
If
,,(;;.rt)d;;drt
J)~[ (;1,,)(x,y) =
Q ~(x-;;)2 +(y_rJ)2
Khi do A:L~.(Q) ~ L2(0.) lit loan ttYtllyc'n tinh lien t\le.llon nITa
(
16
)
1/2
11;1lis 1t 3 In -;-
4. Chlnh hOa nehiem:
Phuong trinh (53) co d~ng
Av= F
Leongdo FE L2(0.) cho tru'oc va loan ttYA xac dinh theo (58).
(58)
(59)
Xet phuong trinh bic'npMn LeongL~(Q)
'" '"
EVE+ A AVE =A F
Khido ve =T vevd'itminttYTdulfcdinh nghIanhusau:
2 2
. T:L (Q)~L (Q)
Tv= v-I3(Aev-A "'F)
P> 0 se xac djnh sail
'"
Ae =E./d+A A
/d Ja loan ttYdon vi tTOngL;. (Q). Chung Laco kc't qua

' < M J6
, L,«I)
( 11111'11£,(0) )1\ "
v<Ji /0.,1=l ; ) va v. la diem b:it dQng cua anh x'!-co T dng v<1iG
(t{(c v, Iii nghi~m daphlfdng trlnh bi€n phan (60»
5. Ph"dn~s{l:
B€ til1hdi€m bit dQl1gv. ctia anh x'!-co T chung ta dung phu'ong
phap x;i'pXl lien ti€p
v(",)= T\,(",-I) , m = 1,2"
. .
v;o)E L~(Q) tuy Y
s
Chon
fJ
=-, thl
.
i 16
)
2
l£ +311'21n~
2
(m)_
1
S (",-I) s
A
'
(A
(m-I)
F
'

ux(x,y,9'{x,y))= f(x,y) ,
uA x,y,qj(x,y)) = g(x,y)
uA x,y,qS(x,y)) = h(x,y)
u (x,y,qj(x,y)) = u1(x,y)
vdiJ. g . h . Ul cho tru'dc trong R2.
(67)
(68)
i'
Bay Ia bai toaD Cauchy cho phu'dng trlnh Laplace va nhu' chung ta
da: bitt 111.bai toan kh8ng chInh.
2. THANH LAP PH(j(JNG TRlNH TicH PHAN
Ch9n v(x,y) = u(x.,y,O)lam lin ham (-oo<x,y< 00)
X6t ham Green cua bai toaDDirichlet cho phu'dngtrlnh Laplace
trong nua khong gian tren
G(x,y,z;~,17,0 = r(x,y,z;~,t'/,0-r(x,y,z;~,t'/,-0
- 19 -
v~i f(x,y,z;.;, 1],0 = 1 ~' 1 la nghi~m cd ban
4Jr (x-,;y +(y_1])2+(z-02
crlaph1.fdngtrlnhLaplacetrongkhonggianR3; (69)
Gia thi8t :
i)
:(x,y)= :(x,y) = 0;
v~i r = ~X2+y2 dll l~n
(70)
ii)j{x,y) , g(x,y) , h(x,y) , /lb,y) d4n v~ 0 dll nhanh, nh1.f
- ~ khi ~X2 +y2 ~ ro (71)
\jX2 + l
ill) JI+x2 + y2 .v(x,y) EL2(R2)
(72)
iv) u chinh qui <"vo cung, nghla la t8n t~i h~ng s6 d1.fdngA thoa

-G,(x"v,z;.;: 1],,p(.;,tJ»)- ,p(t;,tJ)
at;
0
-G~(x,y,z;1; ,1], ,p(.; ,1]»)- ,p(1;,tJ) (77)
01]
Cho z ~ ,,(1;,1]) trong (75) chung ta nMndu'c!c phu'dng trlnh tich
phan vdi ftn ham v(x,y)
(76)
~
JJ
,p(.;,1])v(,;,tJ)d.;dtJ -~u x
[
2 2 2
]
~ - I ( ,Y)
211:-<r>-CD(x - t;) + (y - tJ) +; (';,1]) 2
- J J G(x,y, ;(x,y);.;, 1];;(t;, 1]»)f (1;,17)d.;d1]
-aHX>
+
J JG1(x,y, ;(x,y);.;, 1];(J(I;, 1]))UI(1;,17)dl;d1]
i'
3. THANH LAP PHUONG TRiNH TicH PHAN CHAP
1
JJ
zv(~,17)dl;dtJ
Ham H(x,y,z)
=-
]
1/ di8u boa trong
2