DAI HQCQuac GIATHANHPHa HOcHI MINII
TRUONG DAI HQC KHOA HQC TV NHIEN
:;::, ~?
NGUYEN DINH HIEN
L!P TRINH TINH TOAN HINH THUC
TRONG PHUONG PHAP PHAN T\J HOU HA.N
GIAI M(rr s6 nA.I TOAN
CO HQC MOl TRUONG LIEN T{)C
Chuycn nganh: CO HQC V~T RAN BIEN D~NG
Ma 86: 1.02.21
T6M TA'l' LU~N AN TlEN SY ToAN LY
Thanh ph6 1-16CHi MINH
- 2003
LPr
(~l), Ie;
NCr~ H
10-03
tong tr}nh duQc hoan thal1h t~i Khoa Toan Til1 , TrU<1Iigf)~i
hQc khoa hQc tlf nhien, D~i hQc q'u6c gia th1inh ph6 H6 chi Minh.
Nguoi huang dfinkhoa hQc:
1./ pas. TS. NGG THANH PHONG
TrW!118D~tih()(,'Khoa h()(,'TV l1hiel1Tp.HCM
2./ TS. NGUYEN DONG
Vi~n Co h()(,'Ong dfl/18. Vi~n KH&CN Vi~t Nmn
Phan bi~n I: PGS.TSKH. NGUYEN VAN GIA
Vi~l1Co h(J(,'0118dfwg. Vifl1 KH&CN Viiit Nan?
Phan bi~n 2: PGS.TSKH. CHtJ VI~T C00NG
P!uln vi~n G3ng I1gM TMng tin c BQP
Phan bi~n 3: POSTS. NGG KIEtJ NHI
D~lih()(,'BelchK!lOaTp.HCM
Lu~n an se duQcbaa V9trltac H0i dOngchtm lu~n an dp

Liinh vlfe E)~i s0 m,iy llnh khac vdi nhii'ng ehu'dng ldnh hi~n hii'll dti~1e
hlnh lhanh lren ncSnLang Hnh loan s0. Nhii'ng M lh6ng d~i sO'may Hnh
co lh6 thao lac lren nhii'ng d6i lu<;1ngloan hQe hlnh thuc hen e~nh nhii'ng
phep tinh'tren d~i lu<;1ngsO'hQe, Th~t v~y, vi Ilguyell tdc, bitt ky toall tii
toall h{IC 1l{1OClI ca,l trllC a(1i st{ crill!: ClI tld t/r(t'e I';fll (111{/etrell /rf d<,;
sfI may tillh.
M(lc dich cda 11Iq.1lall Ilay la Ilghiell cli'll cae gidi thuq.1 eda D(1i
,'iflmay lillh, ktt IU/p voi gidi thuij.t s{;'truyill tho"g Ilhdm xtiy dlfllg mQI
hf c/llidllg trillh t{llh toall hlllh tlllt'c gidi cac bai loall cd h{lc.
Cling dn n6i them Iii I~p lrlnh llnh lmin hlnh lhue khong e6 nghia Iii
phil nMn Hnh luetn s6. Hflll hc't cae lru'ong h~jp, tinh luein s01a gicli pluip
duy nhi\t di de'n Wi giiii eu6i cling. Vi~c tlm du'<;1enghi~m giai lich chinh
xae cUa me>tbai loan ed la ri\l hic'm.
Kef quit cda luij.II all La dii xtiy dlfllg qua trillh lillh toall cac ma Irij.II
pha'lI lit cila pIUMII!:pllap pJlli.Il I,t Ju1uh(lll d't'(n d(1llg cae toall t,~ d(1i
so: qua do, dii ell Ihi Iq.ptrillh tillh loall IIlllh tlui'c, Lam cd si'Jtie" lai
2
ml)t moi tn/dllg tif dl)lIg /rOlllljp trill/r (allto-codillg). Nc'u xiiy dt!ng giao
tiC'pt6t, cae cbudng trlnb Hnh lmln hlnb thue sc kc'l hl.lpvdi ehu(lng tdnh
tinh loan s6 truyen th6ng (Fortran) l~p thanb h~ chudng trinh tinh loan
mQtldp cac bai loan.
CHt!ONG 1
MOT SO KY HItU, DJNH NGHIA VA CAC KHAI NItM CO BAN
Chudng nay trlnh bay de ky hi~u t()(lnh{JC.Hldl,ln/{tron/{luc;invan, cac
dinh n/{hfa,cac c()n/{thue bie'n ddi tEch1'1/(1/1va cac djnh Iy c(1biin cua
gidi tEchham, d~c bi<$tkhai ni<$mv~ d~l1lgye'u (d~ng bic'n phan) cua bai
loan bien va cac xffp xi lam cd sa clIo pht/(Ing phdp pl/(1ntii hilu h~m.
Cae djnh ly v6 .qtl/(}i tl,l,t(1nt~linghi~m va cac ddnh /{id sai sf} cling
duQc nh~c l~i. Nh~m ung d1,lngphuong pIlar ph§n tu huu h1:J,ntrong cac
bai loan bien cua cd h<;>c,Ben cac khai ni~m cd hiln Clla If thuyet dcln

eong khcldT trang hiti lOcin h(: nhi0u v~t 111;1.J.Willenhurg (1;11;1111:
~ [ ,,; [ m,;~ - i;) , ",;', [ J, ,;,- Ai:)] - 0
Voi cae bai loan Cd h()c stYdl,lng phu<lng phap phftn Ill' hull h1.ln,chung
toi t<;\mphilo chia thilnh hai dOi tu\lng :
II. Cac hili WclllxuJ't phcll tiYplllf<lngIrluh vi philn:
L(u)+ j;: =0 tren mien V
Voi :('
(
II
)
=
/
' , Iren bien 8
"
va ('
(
11
)
=-::
f
' , Iren bien 8
',. , ,. " , " I
dc;tOgbie'nphiln: W(u)= flfl[L(u)+}; ]dV=O trongdu VIj/ thuOcm(H
,.
khong gian ham xac dinh nilo d6, vii cae dieu ki(:n biellnhu Iren
4
2/.Cac bi\i loan dl,i'a lrcn m(H nguycn 19 bie'n pMn cua cd hQc,
chAngh~n lrong cac bi\i loan cd V~lrAn: gO (u) = 0 '
vdi: D(Il)=! JCi/k/l'ij(II)l'k/(II)dV- JIll; dV - JF;II;dS
2" r ,l"

dam tinh CI , c2
ciia xa'p xi cling e6 th~ du<;1eki~m chung d~ dang.
Cae ma tr~n ph§n ltl d~u du<;1exac djnh ireD may d d~ng bi~u thue lrude
khi ehuy~n d6i (tlf dQng) qua eae ngon ngu s6 truy~n th6ng. M",e dich
eu6i eung eua bai loan vfin la cae kC"tqua s6, vi~e I~p tdnh tinh loan
hlnh thue nh~m xay dlfng la't cii cae bi6u lhue giai lieh trung gian, vdi
5
ham xa'p xi va d<;lngphfin tli kha tiiy y ( 20, 32 nut ), tie'n de'n tt,i
di)ng boa xfiy d1,fngcae ehudng trlnh can. Cae ky thu~t Hip nip ma tr~n,
giai cae M phu'dng tr1nh d<;lis6 v~n lien hanh t~cae ngon ngu s6 truy~n
th6ng.
CHUONG 3
MQT HIE U DIEN CUA PHUONG PHAP I)HAN TD HUU H~N
CHO TiNH ToAN HINH THUC
Chuong nay xfiy thIng m(Hbi~u oi6n kluic eua phuong pluip ph~n tll hITu
h<;lnphii h<;1peho Hnh loan hlnh thue tren may Hnh.
3.1 PHUONG PHAP PHAN TO HUU H~N TONG QUAT
Xct m<)ibi) ba (K,P,L) trong do cae m6i lhanh phjn K, P vaL co cae th~
hi~n va quaDh~ nhu sau:
KIa mQtt~p con compact thuQc R", khae r6ng co bien lien t\,leLipschitz,
P la thong gian vector huu h<;ln(m) ehi~u , g6m cae ham xae dinh IreD
K, co gia tri tht,ie.T~p h<;1pL = {oi},i= I, ,m,eua cae di(~m G;E KIa P-
duy nhflt giai ne'u: '\fa; E R,i = I, ,m , 3!p E P sao eho
p(a;)~a;,'\fi = 1, ,111.Ph~n tll hITulu~n li'I IIl~Ji00 oa(K.P.~) sao eho
P-duy nha't giiii. Ham P; E PIa ham cd sd eua philo ta huu h<;ln,xae
dinh bhg : Pi (OJ)= OJ;' 1:$ i,.i:$ III
3.2 L!P TRiNH TINH TOANHINH THDc
M\,Iclieu la I~p tr)nh Hnh loan h1nh thlie clIOc:ic pllli<fng Ir1nh vi phfin va
cae o~ ng oie'n pIta n. IUt1ngung lit cae ma tr~n 111,.,k". C", F" (el11ow)'e /III)
ta cae 111atr(in nay cua cae pluin tu lulu "<In)

. Dinh dS}.nghinh hQc cua cac pIlau to' cIlia .
. XAydtfng cac ham nQi sur N; (x) tu'dng dng tren m6i phdn to'.
Chu v: Cae dii11l 11(J;,fUy klu)ng I1I/(ttthilt chlla diim mit hll1h 11{)e,ma
can C()thi cd cae diim bell trol1g 17'1£111tt~.
Tuy nhien, xAy dtfng cac xa'p xi tren cae pIlau to' thtfe, thi ma tr~n
nQisur N se phI) thuQc vao tQa dQcae di~m nut cua phftn to', nghia la
phl! tIllIQCvao d{lllg hlllh h{)c, va cIlllllg kluic llhall vai miJi plUtll tit.
Ne'u xa'p xi lren du(je lh1!ehi~n lren cae phall tii Iham chie'" sao eho ma
tr~n ham N fa dl)c lijp vai d{lllg hlllh hvc c,;aplUtll Iii Ih~lc,thi cae ham
nay co lh€ xu d\,lngeho mQiphfintu e6 ph5n lu tham ehie'ugi6ng nhau
(Cae ph~n lu lh1!ese e6 clIngphfinlu tham chie'ukhi chung gi6ng nhau
v~: [o(ti hinl1 d(l11g, s(j' nut IIll1h h()c )1([,W;' l1ut 11(J;suy). Sau d6 la xay
d1!ng phep bic'n d6i tu(ing dU(ing (hinh hQc) giii'a ph~n tu lh1!c va ph~n tu
7
tham ehie'u. GQiphep bie'n d6i hlnh hQe giua ph~n tu tht1cva phfin tu
tham ehie'u la:
r:f ~ x(f) = N(f)(xm) trong do: ~=(~,17,(), x=(x,y,z)
Yi~c tinh loan ham xa'p XlN du'<;1etie'n hanh blnh thu'ong nhu' cae xa'p Xl
phfin tu hUll h<;1nquell thuQe vdi cae tQa dQ nut ~ eua phfln tu tham
ehi6u H\ua niC't, qua cae nu'de :
. C/u!n da thlic ("(1,Wlcua xap xl :
.
TEnhma trrJnnut
P(~)=(PI P2 Pm)
1'"=(PI(~»),i,j= I, ,m
. Tinh N(~) theoc(jngthlic : N(?)=/'(?)I',,-I
Phep bie'n d6i hlnh hQe tren eho phep ta ehuy~n cae Lichphan eua mOt
ham f tren ph~n tii'th\fc thanh tich phan don ghln bon, tren mQLph~n tii'
tham ehie'u. -
3.3 XAI»xi TRftN PIIAN 'I'D TIIAM Cillfiu

hQc(x )cua cac di~m nut cua philn hi tht/c Vc.
Nhll I'QY ta C() thi xiiy d1!lIg C£lC11£111111; cita plu!p billl ddi hillh h{JCtheo
euIIg 17l{Jte£lch tlllle nhll x£iy d1!ng c£le 11£1111nOi suy Ni (~), I'a lllu yrling
ham N (C; ) lil d(jc lq.p vui d(l1lg hillh h{Jc cLiaphall tLi:tlll.jC V. Do vQY
eae ham nay C()thi xii'dl:lngclIO /1/{Jiplul11Iii'C(}pluln Iii' thWll ehilu diJe
tn(llg qua: Hinh d(lIIg , ,ffJ'mit hillh h{Jc, S(J'nut n{Ji .wy :
Xa'p xi lren ph~n It!' lham ehie'u :
uex(~) = u(~) = U; = (NI(~)N2(~) NII, (?;)XUI U2" . u"J1'
-
{
o khi i oF j
Nj(l;;)=
I
kl
" " .
11 1= J
Xa'p xi lren ph~n It!'lham ehie'u ph~lid~lmbao Hnhlien ll,lelren phh It!',
va Hnh lien ll,legiifa de phdn It!'
Ta s~ biegudi~n II(~) lrcn ph~n It!'lh~\lnehic'u du'di d~ng mOll6 h~p .
luye'n Hnh eua eae ham s6 dOe I~p dii ch9n lru'de PI(;f) ,P2(;f), ,
lhOng lhu'Clngnha'll1\ eae d(/n tlulC dOc I~p. Vi~e eh<;>nIt!a de ham p;(;f)
la mOLva'n d~ ed ban eua phu'dng pMp phftn HIhifu h<.tn:
u(:[) = (PI(:[)P2 (:[) Pml (:[)XOI 02" .a"d Y = P(:[)(a,,)
T~p h~p de ham lrang I'(;f) L<.t°Den cd .'111cLia xap xi . SII sInu;l1lg ciia
110phdi blillg .'IIIbif/" mit hay .'IIIbq.c t{t dO"d trIa phih, tit .
Tom tift cac blluc xiiy d{tllg ham (11latrq.1lhillll) N(~):
» Ch{JIlda tl"ie C(f,ffl I' (;f )
» Tinh17latrQnnut Pn=(Pj(';») ,i,j=I, ,nJ
» Ngh;ch ddo ma trQII nut p"
» Tinh N(:[) theocvngthue N('t)=P('t)Pn-1

thuoc :
ff(x)/ (x)/'(x), dxdydz = f f(x(~) )/(x(~) )(("(~) ),.ldcl(.J)ld~d!!£It;
1" 1"
voi J la ma lr~nJacobicua phcp bie'no6i
Ngoai ra chudng nay con oua ra mOLsCSofnh nghla cua chuii'ncua sai sCS,
nh~m sii'dt;lllgcae Hnheha't eua ham xa'p xi dS ti€n hanh Hnhloan hlnh
lhlie eho cae sai s6 nay lren may linh.
CHUaNG 4
L~P TRINH TINH TOAN HINH THDC CHO BAI TOAN CO HQC.
CluJ'cJngnay Irlnh bay de ed sa eua cae tinh loan hlnh lhlic (symbolic)
tren may Hnh,dl!a tren n~n tang eua cae tinh loan hlnh thlie tren eac da
thue. Cae khai ni<$mdin ban eua cae ca'uIrue o!,lis6 oil ou~1e06 c~r
nh~m d~n dAtde'n cae linh loan phuc I~p h<.1neho cae ma lr~n. cae loan
tii' d~o ham, Heh philo, bi€n d6i Laplace,
10
Vi~c I~p trlnh Hnh loan hlnh thuc cho phlidng phap ph~n tii' hUll h~n, chu
ye'u t~p lrung vao vii;c dlia biLitoe", biell vi d(l1lg bitll p"lill (d~ng ye'u)
ho~c sii'dl,lng cac nguyen Iy bie'n phfin (Lagrange chdng h~n) va till"
toe", d(l1lggidi tic" cae 11latrQ-llp"OIl tll (ma lr~n cung, ma tr~n can
ho~c ma tr~n khoi Ili<;1ng)sau khi xa'p xi ph~n tu hUll h~n .
Sd d5 qua trlnh nay the hi~n nhli sd d6 phia sau :
SO DO LAP TRINH TfNH ToAN HINH THUC CHO PP.PTHH
. ,
Bil i tm! n cd hQc
Lll =f
Nguyen 1:9
com! khii di
ChQn philo tl't- xac t1inh philo tl'fthalli chiC'lI
Xac djnh ham lien tI,IcdC'n dip mil'y'!
ChQn da thuc xa p xi p(:f)

In J<.<m!ra file CJ clan
N6i ke't vOi cae ehu'dng tr'lnh con kh:lc: nh~p s6
liell. IllIh ilia IrQII KillIheo s{) liC;lIlIh~l'. 1:11'nip
ma lr~n loanel;1e-,glMI!~ KU=F
In ma tr~n U (U Iii loi giiii sO'eua ke'tqua din
tlm)
Chu'dng nily cling d6 c~p d6n ghli thu~t hlnh thuc cho phu'dng plulp phitn
Iii'hull h<.1nugh nhicn d~ giai cae bili to:ln bien co ehua dc tham 86
hay qua trlnh ngau nhicn, trong do viOc xflp xi c~tc da thuc X~ICc.1inh
tu'dng ung nhu' trang ph§n Iii' hull h<.1nxae c.1inhc.1u'<;1ethay bang cae da
thue chaos thong qua khai tri6n Karhuncn-Locvcn, cae c.1alhue nily
trang tru'ong h<;1pt6ng quat co th~ lien hanh tu'dng It! nhu' giiii thu~t
Gram-Smith, da thuc chaos clip n co d<.1ng:
(
0 ,
(
"
)
2)-1)' I n~'1 fI~"
rpk ""'~I)= ,." n(I", J,P'I
.
/.,11
I , ~(-Ir-I"
fg / n ~I
)
I'(Ji11Ie
L L . \,-,.,1 I
"-I< .(1,., '.) I. I
trang do n-(.) Iii phep hoan vi (nghia Iii t6ng thlfe hi~n ln~n cae phep
hmin vD, cae d<.1nge\l th~ se du'<;1ctrlnh bay trong ehu'dng sau,

Khi Unh loan tren may, ta chuy€n d6i cho cac ph~n t\1thalli chiC'u Qr
(tttdng ling vai ph~n t\1 th\fc). NC'u cac ph~n t\1th\fc lil phh t\1kh6i sau
m~t, ph~n t\1thalli chiC'u co d<;\ng:
Ch<;>ncd sa Xa'Pxi : P =(1 ; 11
~ ;11 11~ ;~ ;11~)
G<;>iJ lil ma tr~n Jacobi cua phep
bie'n d6i nily
D~t :
12
trong Q x (O,T)
Q=r'
13
HI" =((:) (~) (~)r; H{ =((~~)(~) (:)f
va v = ((u,,) (VII) (w,,))
Khi d6 ta c6 .:
111,.= jN'Nldct(J)ldO, 'k" = fN'NVQlJ{ldct(.J)lclH,
n, ",
k'2 =()" IN'Nldetl(J)dO, 'k., +k., = JB/Q'DQB~ldet(J)ldO,
0, 0,
d
'
(
I' 0
Iron 0:
k,.,= vfJIN'N.J,dl;d77' g f) = () II
I", 0 0
I
[
)
'

.
. u"
. " ~
. mHO
. "."
: ::::::
. "
. ""
. "."
. "o~.
. ,
. """
. .n
.
. "n
/~
.
-
.
~~
.
'
.
~'
.
'
.
'
.
'

.~
.~
.~

.~
.~
.~

-~
-~
.~

.~
-~

.~
-~
.~
.0-
.M-


huy ).
Tli'de thu vi~n chuyen d\lng (Packages) co th€ lie'n t\f dOngboa
l~p trlnh (t<,\oma ngu6n cho cac chudng trlnh giai so).
Ke't qua bili loan Ihay ddi Ihco de vi lrf ngu6n, v~n 16egio. V~
dinh Hnhnh~n Ihily hoiln loan h<;lply. Co Ih~ md rOngvi dl,llhanh mOl
th\fe nghi~m IreDmay Hnh.
5.2 Bai toan 2: Phudng phap phftn hi hii'u h~n giai bai toan bien
voi v~t Ii~u dan nhut
-diing huang -ddng nhi~t
15
5.2.1/ Nguyen 19tu'ongling-Bai tmin daD h6i ke't hop:
Thea nguyen ly luong ling, nghii;m cua hai loan hi0n dan nhdl
tuye'n Hnhco lh€ thu du<;1ctit nghi~m cua bai loan bien dan h6i, lrong do
cac hhg s6 dan h6i du<;1clhay bling cac loan Iii'ham ph", thuOclhai gian
(modun chung ling sua't ho~c ham chay cMm).
Qua bie'n d6i Laplace ta lhu du'<;1cbai loan dan h6i k61h<jp:
aO" if + Fi == 0 ; e,; == ~
(
au, + auJ
]
ox;
- 2 ax; ax;
- -
ai; =Cijk}EkJ ' trong do : CijAi==p' C'ikl
p: anh cua bie'n thai gian lh1fct qua ph6p bi6n d6i Laplace.
Cijk: la iinh ciia ~;kJ qua phep bie'n d6i Laplace.
Va bie'n d6i cac di~u ki~n bien:
- -
U
I

x./v
Nhao xet (Xem hloh) : Chuy€n dicit u' ,v' tl.lidi€m ireD Idp song song
va each m~t giil'a mOLkhm\lIg z , ll,i thili didm I - do s1futIli colig lfi'm
tl.lOra - co dl.lng :
82w
u'==-zsin(a(t)):::::: -ztg(a(t)) ==-z-
8x8t
16
a2w
v'= -zsin(a(l)) ~ -zlg(a(l)) =-z-
ayal
I OW Ow
Laplace(u) =-zp-, Laplace(v') =-zp-
OX oy
Ti:rnMn xet teen ke't h<;lpquaIl Mbie'n u<.1ng-ehuy~nvi cho La:
- 2- 2-
- 02 W - 0 W - 0 W
Ii =-z
p
-, Ii =-z
p
-, Ii =-z
p
-
rx OX2)J' oy2'.'. oxoy
Ti:rquaIl Mung sua't-bie'n di;\ng cho La:
(

' m1lX . 111ry 16q
qmn = 2 sll1-sll1-dxdy= L
ab 0 0 a b ,,2"111
- 16qo ~ ~ 1 . 11111X . I17ry
11'=-= ~ ~ 5In-5In-
1f6 D - -
(
2 2
)
2 a b
m-I,J,3 "-I,U, 111 11
/11/1 -+-
a2 b2
Ta chi dn tinh w khi m, n lil nhung s6le. Vol tai ngoili philn b6 o~u,
m~t giua khi u6n phai d6i xung, cae s6 h<.1ngm, n chii'lltuong ung yoi dO
yong khong d6i xung Hen chung phili b~ng khong.
Thea nguyen 19Luongung chung ta se thu du<;lcll1igiiii cua bili loan oiln
nhot ti:rWigiiii cua bili loan oiln h6i ke't h<;lpb~ng phep bie'n deSi
Laplace ngu'<;Ic.
w = Laplace-I(;)
17
1 . //I l1X . mcv
](
//I2~
)
2SII1-;-'Sll1h
//III +
a2 b2
[
16 '" '"

IV" - , (, L, L,
h :r ",'1.'.' " ",.
W= WIW()
I . IIITTX. n:rv
,5\11 5\11 '-
(
1111 /11
)
II h
111/1 '1-
a2 b'
Giii sii'v~lli~u dltn nhdl mo tii nhu'sau:
v
=consl; E = Eo(l-~e+'} lhco loi gidi teen ta c6:
(
'I' /
)
trong d6: W = 2e 2" cosh !L-
'I 2
_
[
12(I-V2) 16q" ~ ~ 1 . 1Il11X . nny
IV,,- J 6 L, L 2 25111 5111-
E"h 71: m~I.Un~I.U
(
Ill /12
)
a b
1Il/1 02 + b2
W = W" W"" '

ling 0 2 ve':
" ,,",
I Jr.'R;C'RJ:Uou,/dV = I fTe'S;j;,dV+ I JTe'S;frdi
e=1v, ('=1", e=1 r,
B\e'n 06i Laplace ngu<;Jcbi~u thlic nay, chu yr~ng chi co (;', u,/,fv,fr
11\chlia bie'n p:
(ky hi~u Laplace -I (f) la bie'n d6i laplace ngu<;Jcham cua hamf)
tT:[}R;(Lapla" -,(C''u" ))R.dV] T,,U"= ~ t' S;/,dV + ~ ,f T: S;fcdr
Ky hi~u :
Ke = JR;(Laplace-I (C' . U"))RedV 11\ma tr~n 09 cling ph~ntU'.
v.
19
K=~1'f.K .1" F=~
J
T')l
j
'dV+
~
I
r'S'
t
',/I'
L ,e " L ,,',V L "~,Ie
<=1 e=1 "0 e=1 ro
ta dj d611hi011
thuc sau: K .U0 =F
Ghli M phU'dng tdnh d~i tuye'n tren ta Om dU'Qc Uo Hi cac chuyen vi clan
h6i t~i cae nut. Do d6: nghil$mGan nhdt la :u (I) = U 0 . U" (t)
D~ Hm ham u1J(t) ta 100y de'n gia thie't Uola nghi~m dan hai, tit do
.my ra 10:11latrQIl K fJIld; 10:11latrQIl /zlillg dfil ya; t. f){iYchinh la ma tr~n

E(1) =y x Eo
I: :

! :
TIHli giall t
B6 thi bie'n d~ng nhat W,/(t) .
1
-: ;; __d -
.:: '-

I
20
theo thai gian vii h~ s61']
'L=do Z
i
" I
,
.
Thui gian t
Du'ai day Iii bi~u d6 bi~u di€n st! thay d6i chuy~n vi cua Him theo thai
gian. Biii loan du'<;1ckhilo sat vai cae gi,i tri cua 1']=0.2, 0.5, 0.7, 1.0 ,
trang l6m l~t chi th~ hi~n tru'ong h<;1p1']=0.2
0 I4T QuA BAI ToAN: TfNH BIEN DA.NGTAM, VAT LI~U DAN NHOT
BANG PHUONG PHAp PTHH
D. Ihl<"", ,,, lheo Iholglo. ,
g." ,
NUl
",." '
. .""."
.

khue eo dtp dil de sinh fa khong gian ham Ihii'.(de ddn gian each vie'lla
sc bo qua ky hi~lIe ) :
N
L u;[L(x)+ a(x)R(x)]gj(x) = lex)
;=1
Nhan 2 vfSeho gj(x) vii lich phan Iren Illi~n D s :
t[I~L(X)g;eX»)g/X)dx+ Ia(x)[R(X)g;(X)]g/X)dx}; =yex)g;(X)dx
Khai trien a(x,O) lheo dIng cd s() nhtt u(x,O) , khi do vC'Inti se baa g6m
mOt ma lr~o co de philo Iii'Il(dng qual1.
I)~ roi n.IC haa mOL qua Ir1nh ngftu nhien, la ap u~lng kiwi IriOn
AI
a(x) = I AII~II(/II(X)
Kafhllnen-Loeve rho a(x,O) :
II ~ I
Anvii an(x) Iii cae gi;, Iri ricng va vector ricng coa ham llt't1ngquan coa
ham a(x, e) , va ~"fa tijJ1cae bie'i, IIgiillll!riell tqlc ciao. ~" fa cae bie'll
trollg cac da tltuc cltaos (ltall lo{lll). Cac da tltuc lIay dii du{lc xay d1!llg
dum d{lllg gidi licIt qua 11lf!tcltUdllg trillit tillIt toall Itillit thuc.Nhit do,
vi~c riti r{lc 11lf!tqua trillit Ilgau Illtiell dii th(lC lti?1l dU{lc.
N
[
M
]
" K +"J:K.(."j 1I.=
(
.;j=l, ,N,
L '1 L ,"'1 ' . 1
;= I ,,~I
, K;j = HL(x).!:;(x)Fj(x)dx; K;~") = ja,,(x)[L(x)g;(X)Fj(x)dx
tfong do : I) /J

m~t dQ <p(x,y,z,t) cunfj tnti'lng ng/iu nhien.
Roi r1).cmi~n xac dinh va xflp xi trcn tung ph§n tll :
cp(x,y,z,t,B) = Ne(x,y,z)<P,,(t,B) voi <D,,(t,B)= ~<D(t,B)
chUng ta tho du'<;1cM phu'ong trlnh ma nghi~m la gia trj cua cp t1).inut
. .
ciia cac ph§n Iii': M<D(t,B) +K<D(t,B) = F(t,e)
Khai tri~n Karhunen-Loeve clIo d1).iIU(1ngngfiu nhien v~n to'c gio:
- - - AI . - (*)
V(x,y,,",/,B) - V(x,y,z,/) + Lc;k (B)JAk(/).fk(X,y,-,t)
kd
Ap dl,lngphu'ong phcip phfln IIIhull h~\l1ngiill lIllicH,ta co ph6 cua <bet)
co d1).ng:
[
AI
]
-1
[
AI
]
-"
S<I><I>«(») = J + t;;kR~k' SI'I,«(v) J + h;k Ril)
(Chi sO'H th~ hi9n bie'n d6i Hermite)
ho~c la sii'd\lI1gkhai tri~n Neumann clIo to<intll ngu'<;1c, ta co:
if) if)
S'f>Ij'(lV)= II(-I);+i[;kR~k)r SI",(OJ)[;kR~k)r
;=1}=I
23
5.3.3/ LSp-~t~IJJgJJnIUvii nJ,;h"Lb.iIi Iwi II_Y.UI ni 11biOn dilll~\{t1LyO IliOu nJ
cae tham s6 ng~u nhi6n- VIdu bili loan t[tlll:
Xct mQttffmmong eo cae di~u ki~n bien phu h<;1p.Gia sii'modul dan

ed hQc.