chỉnh hóa một số bài toán Moment 10

luận văn này, khảo sát bài toán moment và tìm cách chỉnh hóa một số bài toán moment cụ thể.

Chuang II: MQt s6 b~liloan quy v~ b~liloan moment

CmJONG II: MOT SO BA.I TOAN QUY VE BA.I TOAN MOMENT

2.1 Bai toaD bie'n d6i Laplace ngu'o'c :

Xet bai loan: TIm ham u khi bie't bie'n d6i Laplace cua no t~i cac di~m nguyen :

B~t t = e-x.Khi do x=-lilt

r u(x).e-kxdx = ru(x).e-x e-kx+xdx

= ru(x).e-x e-(k-,)xdx

= !w(t).tkdt Voi wet)=t lie-lilt)

M~t khac do x ~ 0 => 0 < e-x ::; 1 => 0 < t::; 1

V~y (2.1) dU<;1Cvie't l~i thanh: lw(t).tkdt= f-lk+1 k = 0,1,2,

Bay la bai loan moment Hausdorff mQt chi~u ma ta d3: gioi thi~u d phftn

2.2 Bai toon "hiet "glide thai gian:

Xet phuong trlnh nhi~t ~u - Ut=0 V (x, y, t) E R2 X R+.d day ta chQn dQ d~n nhi~t la 1 va u=u (x,y, t) la nhi~t dQ t~i di~m (x, y) vao thai di~m t

Ta xet bai loan xac dinh phan bo nhit$t dQ tC).ithdi di~m t = 0, u(x,y,O)= v(x,y) khi bier phan bo nhi~t dQ tC).ithai di~m t =1, u(x,y,l)

Truoc he't, ta xet bai loan thu;%nxac dinh phan bo nhi~t dQt~i thai di~m t >0, u(x,y,t) khi bier phan b6 nhi~t dQ t~i thai di~m t= 0, u(x,y,O).Ta co, u la nghi~m

cua h~ phuong trlnh:

{

~u-Ut=0 V (x,y,t) E R2 X R+

Voi ~u = UXX+ Uyy

5

Chuang II: MQt 56 bfli loan quy v€ bfli loan moment

{

-(X-~)2+(Y-1])2

} 4JZ"(t - -r) 4(t - -r)

{

}

-8JZ"(t- -r)- 4(t - r)

[

(X-~)2+(Y-'7)2

]

exp -1

1

4(t - r)

8Jl"(t-r)

(x-~)-[

(X-~)-+(Y-'7)-]

-2(t - r) 4(t - r)

~

- (x - ~)2 + (y -11)2

}

(x - ~)2

{

- (x - ~)2 + (y -11)2

}

Tu'dng tlj:

~

(X-~2 +(Y-11)2

}

(y_~2

{

(X-~)2 +(Y-11f

}

~

G -~ 1 (x-4)- +(y-'7t

4;r (t - r)2 4(t -

[

- (X-~)2+(y-~2

]

exJ- (X-~)2+(y_~2

}}

.

(t - -r) 4(t - -r)2 L 4(t- -r)2

} 4n(t - -r)2 L 4(t- -r)2

(x _~)2 +(Y-11)2

[

(x _~)2 +(y - ~2

]

Tli cae ke't qua lIen ta du'<;1C:

Xet u=u (~, 7J, T )

. (2.2)

Ne'u u If!nghit%mcua ht%(1) thl:

U;;;;(1;,17, r) + U1/1l(~, '7, r) -ur (~, 77,r) = 0 ' (2.3)

(Do ~u - ut= 0)

Chuang II: MQts6 bfli roan guy v~ bfli roan moment

Xet tru'ong vec to:

F(~,11,1) = (u G; - u~G,uGIl - ullG, uG) (2.4) Taco:

divF = u;G; + uG;; - u;G; - UqqG+ u'!G'!+uG,!,! - u'!G'! - u'!'!G + u,G + ltG,

= uG;; - u;;G + uG,!'l- u'!'!G+ u,G + uG,

= u(Gq; + G'!17+ G,) - G(Uq; + U'l'!- u,)

(2.5) V~y divF =0

f)~t 0 C R3 la miSn dinh boi I ~I ~ R , I 11 I ~ R 0 <1"< t -E ; 0 < E < t

Ta co bien cua 0 la ao =ao~ u aoll u aD,

Trang do: ao;; aoll; aD, l~n lu'Qtla cac m<';itvuong goc vdi cac tr\Ic

0~,01l, 01:

->

Ky hi<$u: n la phap vec td ngoai cua a 0

Khi do tu (2.4) va (2.5) va do dinh Iy divergence ta co:

l'

l -£

~

-R

11

7

Chuang II: MQts6 b~liroan quy v~ bai roan moment

ldiv F = f(F , n) + f(F , n) + f(F , n)

ao" ao" ao,

= f f(uGq - UqG) 1~=Rd7]dT

~

(vi n = (1,0,0))

~

(vi n = (-1,0,0))

>

(vi n = (0,1,0))

- f f(uG" - u/]G) I,,=-Rd~dT + f f(uG )Ir=t-e d7]d~

~

(vi n=(O,-l,O))

>

(vi n = (0,,0,1))

-R -R

(2.6)

* Nhan xet:

{

( j:: 2 , 8,,( t - T)' x - 4').exp - x-, 4(1- r) ) + (y - '7)2}

2 Ta co th€ giii sa : li, li~, liT]bi ch:;in boi C > 0 vi nhic$t GQcung nhu GQbie'n thien cua nhic$tGQkhong th€ tang ra vo h?n

Voi cae giii thuye't tren, ta se chung minh:

t-E: R

1~ f fcuGq) Iq=:tRdryd,=0

0 0

lim r-&r (u.G ) <- dlld1: = 0

R~"'! ",,-:tR

ChuangII: MQt so bfli toan guy v~ bai toan moment

(-Ii R

lim f feuG,,) I - dc;dr= 0

0 -R

(-Ii R

1~ f Jeu"G)1,,=:tR dc;dr =0

0 -R

Xet If SeLiG;) j;=:tRd17dr~ f f(uGq)lq=:tRd17~r

{

}

f fu(:tR, 17,r) x+ 2 exp - x+ , - Y-17 2 d17ldr

< t-fE clx:+ RI _(X+R)2

{

00 _(Y-Il)2

}

0 TCt-"C -00 11 "C (do gia thuyet lul::; c )

~

I-IE: clx:+~I.e- ~(~~;~

f Ie-;~~:~' d 'I}

d r

8.1Z"E -00

0

(do gia thuyet 1:< t -8 ~ (t-1:)2> 82 => , < :;-)

(do 1:>0 Den e 4(I-r) ~ e 41 , e 4(I-r) ~ e~

< c~lx +:R I

_(x+R)2

00 (.1'-'7)2

(d A"?'

h A- a Y - 17)

0 uOl leD tU-00 " ang cae u<;it = 2~

V~y ta co:

t-E R

(t - )

9

Chuang II: MQtso bfii toan guy v~ bfii roan moment

Cho R ~ 00thl v€ phai cua bat a~ng thuc tren ti€n v~ 0, \;f t >0

V~y 1~ f feu G;;) 1;;=:tR d7]dr = 0

Xet IIJ f (u;;G) 1;;=:tRdTJdr s clJ IIGI~=+RdTJdr

(do gia thuy€t IUqI::; C)

t-E R

- c f f 1 _(X+R)l -(Y-'l)C

.e 4(t-1:) e 4(t-,) d d

-(.dR)l t-f: 00 -(Y-IOl

(do't < t - E => t - 't > E va 't > 0 nen

_(X:tR)2 _(X:tR)2

e 4(t-,) < e 4t )

C _(X:tR)l t-E 00 {y_~)2

S 4m::.e 4t 0f f-00e -;;-.d11d't

_(Y-'l)l _(y_q)2 (do 't > 0 nen e 4(t-t) < e :it

< ~.e 4t Ji f fe-al dadr

2JZ"£ 0 -00

T' _(Y-'l)2

(do d6i bi€n a = 11l2 t trong -T'.fe 4t d11)

-2m:: .e 4t .(t - E) Jn = c-ft~ I .e_(X+R)2 4t (

(2.7)

Chuang]I: !v19ts6 bai roan ~ v~ bai roan moment

r: _(X+R)2

I

-V~y ta c6: 1 lR (u~G) I~=:':RdlldT I::; 28-J; e 4t (t - 8).

Cho R d~n den vo cling thi ve phch cua bat d£ng thlic tren d~n den 0 ,

V t >0

(-f: R

0 -R

(2.8)

Do vai tro cua uG~ va uGll nhu' nhau, u~G va UllG nhu' nhau nen ta cling

c1uQC:

{-f: R

lim f f eu GJ 1 - d~dT = 0

R-,>oo 'I 1]-t.R

va 1~ f f(uI]G) 11]=t.R d~dT = 0

Til' (2.7), (2.8), (2.9), (2.10) suy fa:

t-E R

Ji~oo f f(u Gi; - ui;G) 1~=:tRdYjd1 = 0

0 -R

1~ f feuGl] -UI]G) 1'7=t.Rd~dT=O

Bay giG ta xet hai rich phan cu6i cib (2.6)

Theo dinh 1;' hQitv bi ch~n ta c6:

Ji~ f f(u G) 1,=odYjd~ = f fG(x,Y,~,Yj,t,O).U(~,Yj,O)dlld~

(dung ham d~c tru'ng)

00 00

= f fG(x, y,~, TJ,t,O).v(x, y )dTJd~ (do u(x,y,O) :=v(x,y»

-<XJ -00

] !

8H.V\H.Tl!N~:!E;.,j\

- ~ I l"' Ii)

- n~~~-1

ChuangII: MQtso b?ti roan guy v~ b?tiroan moment

Tu'dngtV :

R R

Ji-Too fJ(uG) 1'=1-8dl1d~

CD oc

= f fCCx,y,C;,TJ,t,t-E).UCC;,TJ,t-E)dTJdC;

-00 -,x

Ta chung minh ke't qua sau:

00 00

lim f fG(x, y,~, 11,t, t - E).U(~, 11,t - E)dl1d~ =u(x, y, t)

8~O

-00 -00

*B6 d~:

1 CD

{

2

a j;;, fexp - (x - ~) }

d>:

'7'8> 0

{

- (x - ~)2

}

2& i~-xl>o 48 d~ ~ 0 khi 8~ 0

TiI d6 suy fa:

{

- (x - ~)2

}

(b.l)

(b.2)

c lim f fG

I - - d~dl1=1

d Bi;it Vo={(~,T]) :I ~-xl ::;; 0, I T]-Y I ::;; o} la hlnh vuong trong mi;itph~ng

p (~, T])

Khi d6 :

~~ f G 1.=1-8d(~, 11)= 0

P\ vo

Tu d6 suy fa: lim f G

I - d(~, 11)= 1

8~O -1-8 Vo

ChuangII: MQtso bai roan guy v~ bai roan moment

* Chung minh:

a E>~t a = I => da = I => d~ = 2Fcda

i 2-v & 2-v &

{

(X-~)2

}

1 00 2

Khi d6: 2;;;;ll:& -00 fexp - 4& d~=2 ;ll:& -00r- fe-a 2Fcda

1 00

= j; fe-a'da-00 1

= j;J;

=1

b Xet x,~ co dinhthoaI X- ~I ~ 8

{

(x - ~)2 1

}

Ham so h: 8] H 2fn exp - 4 £1

{

J

~]

" 1 (x-t)- 2 (x-C;)- 2 (x-C;)- 2

1

{

( ;::2

}{

2

}

X, ~

(5

Ix-c;12 (5::::>Ix-c;12 252 ::::>_(X-C;)2 ::;_(52

a &1 E -,CO nen& ] 2-::::> c

2 (52 (52 2

::::>~ 21::::> 1-~::; 0

13

Chuang II; MQt so bai roan quy v~ bai roan moment

2

Tli (2.11) va (2.12) ta co: 1- ~I (x - ~)2 ~ 0

Voi 0 < £ ~ - => - > ? => - >

}

Ma heEl)la ham giam

=> h ( ~) ~h(J2)

=> ~ exp

{

- (x - ~)2

}

{

- (x - ~)2

}

M~t khac: 1 exp{

- (x - ~)2

}

~ 0 khi [; ~ 0

Thi;lt vi;lyd~t t = l khi [;~ 0+ thlt ~ (f)

X, 1. ( -12

et 1m t.e = lIn~ = Im-, =

I->en 1->«0el I->en

2t.el-Tli (2.13) va (2.14) ta duQc:

1 fexp{

- (x _~)2

}dl; ~ 0 khi E~ 0

Ket hQp voi (a) ta duQc: ~ fexp

{

- (x - ~)2

}d~ ~ 1

2'\jTtE 1.;-\IS6 4£

khi E ~ 0

Vi;ly ta chung minh xong (b)

Xet foo fd) G 1,:1-&d17dl;

{

- (x- 1;)2 + (y - 17)2

}d17dl;

00 a) 4Jr& 4&

{- (X-c;)21.exp

{

- (Y-17)2

}d17dl;

Chuang II: MQt s6 bai roan quy v~ bai roan moment

(y-ry)2

}dry)

-00 -00

d Ta co:

:yo

fG

I d(~, 11)= r d~ fG

1

,=t-o ~~-xl2:O

+ txi$l1 dl;f'1-Y!2:o1'=Hdll (Tinh trang mi~nS2)

- 1

1

Y) - (x-ci+(y_q)2

1

~

- (x-~f (y-Il)2

4 41t8 ' -xi~o ~ Ie E.e 4E dl1

11-\ ~o

1 _1<-;)2 1 Ct) _(y_~)2

-~

.

"E

d~

f "E

2;:;;' 1;-XI2:0 2;:;;' -coe 11

«-;)2 00 - (y-'1):

~

4.

2& -xl~o 2& !1l-YI2:0

~OkhiE~O

(do (bl) va (b2))

15

do ket qua (a) )

52

E

Sj

-8/2 8/2

I -8/2 I

52 I

I

Chu'ongII~Jv19ts6 bii roan ~ v~ bii roan moment

V~y ~i-To fG 1.=1-$d(~, 77)= 0

P\V/j

Tli do suy fa: lim fG

I - d(~, 77)= 1

$~O -1-$

V/j

(do c)

V~y ta da: chung minh xong b6 de

Trd l~i di~u cftn chung minh:

Jim f fG(x, y,~, 11,t - E).U(~,11,t - 1:)dl1d~ = u (x,y, t)

E~O

-00 -00

Do tinh lien U,lCcua nghit$m u, vdi III - y I <bn

1~-xl<bnVa8du nhotaco:1 u(~,Il,t-8)-u(x,y,t)1 <~

n

Tli b6 de (c) va do tinh bi chi,lnclh uta co:

<:J) <:J)

I Jim E~O f fG(x, y,~, ll, t - 8).U(~,Il, t - E)dl1d~ - u(x, y, t) I

-<:J) -00

00 oc

= I 8~Olim f fG(x, y,~, 11,t - 8).U(~, 11,t - 8)dl1d~

-co -co

00

- lim fG(x, y,~, 11,t - 8).U(X, y, t)d11d~ I

8~O

-co

(Do b6 de (c) u (x, y, t) = u ((x, y, t).l)

00 oc

= I E~Olim f fG(x, y,~, 11,t - 8).{ u(~, 11,t - 8) - u(x, y, t) }dlld~1

-co -ex)

00 00

~ lim!8~O f fG(x, y,~, 11,t, t - 8).{ u(~, 11,t - 8) - u(x, y, t) }dlld~ I

-00 -a)

y 1

:::;-n

(VI G >0)

Chuang II: MQt so bai tmin quy v~ bai roan moment

~ Jim ~ fG(x, y,~, 11,t, t - E).d(;, 11)+ 2C lim fG(x, y,;, 11,t, t - c:).d(;, 11)

= ~n (do lu(~, 11,t, t - 8)- U(x, y, t.1 < ~,b6n d~ (d) va lul::;C)

-00 -00

Tli (2.7), (2.8), (2.9), (2.10), (2.15) ta duQc:

voi vex, y) la nhit%tde>t'.li t =0

V~y ta dii gi:ii xong bai roan thu~n

Trd l'.li voi bai roan nhit%tnguQc thai gian xac dinh phan bo nhit%tQQ Tli (2.16) the' t =1 ta duQc:

(x-~r,1. +[Y-17),2

-OCJ -OCJ

Gi:i su v (x, y) =0 voi x < 0 ,y < D Khi d6 ding thuc tren c6 th~ vie't thanh:

(x-~)2+(Y-Il)2

voi

{

X=-2m

y=-2n Kbi d6 tu d~ng thue tren ta duQc :

~2+')'

e-(m2+n2) r rv(~, Y]).e ;-.e -(Jll;-"n'))d~dll = fmn

voi fmn= 4nu (-2m, -2n, 1) m, n = 1, 2,

B6i bie'n voi:

{

s= e-;

t= e-'7 ta duQc

i iw(s ,t).Si tjdsdt = f.1ij ydi i, j = 0,1,2,

17

ChuangII: MQr so bai roan guy v~ bai toan moment

In2s+ln2,

w(s , t) = v(-In s ,-In t).e 4

(i+I)2 (j+I)2

Jlu =4;r:e u(-2i - 2,-2j - 2,1)

(VI ill = i + 1, n = j + 1)

Bai to<:lntren 1a bai to<:lnmoment Hausdorff hai chi~u va ta se tlm each 06 chinh hoa no (j Chuang III