chỉnh hóa một số bài toán Moment 7_2

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

Chuang IV : Chinh hoa mQt so bai toaD moment

CHU<JNG IV: CHlNH HOA IVH)TSO BAI TOA.N MOMENT

4.1 Chlnh hoa bai tmin moment Hausdorff mOtchi~u:

-Xet bai tmln moment Hausdortl mQt chi~u : fun ham u(x) trong L2(0,1) sao

I

cho: fu(x).xkdx=,llkVai k=0,1,2 ;

0

(4.1)

/.l=(/.lk) la day bi ch~n cho tntoc

Nguoi ta chung minh duQcbai to<ln(4.1) la khong chlnh nghia la khong 1uont6n t~i nghi~m va trong truong hQpt6n t~i nghi~m thl chung khong phl,lthuQclientl,lCvao da ki~n cho truoc

M1,lcdich cila chuang nay la chlnh h6a bai roan moment boi phuong phapmoment hliu h~n Ta ap d1,lng phuong phap ch~t Cl,ltachuang truoc trong vi~c

Chuang IV: Chlnh hoa mQtso bai toaD moment

(

d'l1

- 2- "_;

Do ,(x 1) - (_n), daodx-11

(4.3)

ham cap 2n coa da thue boc 2n J

Xet : In = L (x2 - l)n dx Ta tinhln b~ng tkh phan truy h6i

Chuang IV: Chinh h6a mot s5bai roan moment"" .

=

2" ,n! 2n + 12".n! 2

Chuang IV : Chinh hoa mQtso bfli roan moment

.: Binh nghia: 1n (x) = J2n + 1 Pil(1- 2x )

Tli chung minh tren : Lpn (t)Pm(t)dt =0 voi n=1=m

Tli Ln la da thuc b~c n va tli (4.5), (4.6) ~ (Ln) la day tn!c chu5n d~y dli

Cho ~ =(~k) If! day so tht,I'c.

Ta dinh nghla day:

i

j=1

k = 0, 1, 2,

Chuang IV : Chlnh hoa mQtso bili roan moment

D~t

n

pn = pn (}.t)= I Ak(~)Lk (Lk dong vai tra ej C5chuang III)

k=O

Khi d6 pn lel nghi<$mcua bai toan moment hU'uh~n (4.2)

Truong h<;1pdli ki<$nchinh xac ta co ket qua sail :

neu j ::::: i va Cij=0 neu j > i

Neu u lelnghi~m duy nhat cua (4.1) thl pn (!J.)~ u trong L2(0, 1) khi n' ~ 00

(pn la nghi~m cua bill toan moment hUi.:thi;ln(4.2) iu(x )Xk dx =jLk )

Hon nila neu nghi~mu la trong HI(O,1) thlll P" (jL)- ull~ (1 )Il ull HI

Cha Un E L 2 (0, 1) 1a nghi<$m cua (4.1) tu'ong ling voi !J () = (!J.~ )

D~t : f(t) =sz:Js (2t +1"'.3" vdi 0< £ <1; n(£) = [CI (£ ~IJ]

([x] 1elso nguyen Ion nha't:::::x)

Khi d6 t6n t?i mQt ham so Yj(f:);0 < f: < 1 saG eho Yj(f:)~ 0 khi f: ~ 0 vadoi voi nhling day !J.thoa: ll~- ~oL, =S~pl~k- ~~I < E

Taco: IIpn(E)(!J.)-uoll:::::Yj(f:)

Chu'ang IV : Chlnh hoa mQtso bai toan moment

Han nila ne'u Uo E HI (0, 1) thl :

.: Ynghia: Trong tru'dng hQp dil ki~n khong chinh xac dinh 1)1da chI ra Cl;l

th€ ham fer)va so n (c:)sao cho nghi~m cua bai roan moment hil'uh?n

(4.2) xffp Xlnghi~m clia bai roan (4.1) Ngoai ra ta con co danh gia duQcsai so

D€ chung minh dinh 1)14 I,4.2 ta dt1avao b6 d~ sail :

Trude he't ta xet tru'dng h< Jpda thuc Ta co:

Chuang IV : Chlnh hoa mQts6 bai toan moment

(VI [x(1 - x)]j= (x - x2y co 56 mn nho nha't la xj ~ d~o ham de'n ca'p j co M s6

tl! do Hij!, cae sf) con l(,lic-ochua Xknen the- x =0 VaG ta chi con l(,lij! ~ L/o) = (2j +1Y/2.\J..j!= (2j + 1)1/2

Chuang IV : Chinh h6a mQt so bfli toaD moment

Chu'ongJY : Crunh hoa mQtsO'bai toaD moment

Tli hai ket qua tren ta du'QC:

Chuang IV : Chinh hoa mQt so bfli roan moment

Pm la day da thuc sao cho Pm ~ Vtrong HI (0, 1)

Do (Pm) la day c1athuc nen ap dl,lng (4.22) ta c1U<;1c:

Th~t v~y : do Pm~ v trong HI (0, 1) nen :

Chuang IV: Chinh h6a mQtsf) bai roan moment

Ma c;> Ilpm- vll~I(OI)= !{(Pm - vr + [(Pm- V)']2 }dx:2: i[(Pm- v)']2dx

= i[(p'm-v'W dx

= IIp':l- vf

=> Ilp~ - v'II ~ ° khi m ~

00 hay P,:, ~ v'

M~t khac do tint lien Wc cua rich vo huang ta co :

(Pm,Lk)~(v,Lk)khi Pm ~V trong H'(O,I} Tucla Akm ~Ak khim~oo Tli'

hai ket qua tren va tIT(4.23) ta duQC:

Ilvf :2:6a; + I (2k+ lr a~

k~2

Vai

ak = iv(x)Lk (xjix =AkV~y ta dff chung mint xong b6 d~

.: Chllng minh dinh IV4.1:

'l:

Do (Lk) la h~ tn,tc chuin nen u =LAkLk vai u E e (0,1) (Lk dong vai tra

k~O

nhu ek; (1, X, X2, ) dong vai tra nhu <g" gz, >a ChuangIII).

- Chung mint tuang tl;1'nhua ChuangIII ta co u la nghi~mcua (4.1).

V~y ta duQc pn~ u trong L2 (0,1) khi n ~ 00,

.:. D€ chung mint Ilpn (11)- till ::; 1 Ilull, n = 1,2, ta dl,l'a vao b6 d~ 4.1.

2(n+1) H

Ta co: 6a~ + I(2k+lYk~2 a~::; !Ivfdx

=> I4k2A2kk~O ::; !juf dx

Chuang IV: Chlnh hoa mQt so bai roan moment

M~t khac: !Iur dx ::;llull~I().I)

I 4k2A2k ~ Ilull~I(O,I)' k"O

Chuang IV: Chinh h6a mQt so bai roan moment

.: Chz?ng minh dinh IV 4.2:

Theo tinh chat cua chu:1nta co :

Ilpn~)-uoll ~ Ilpn~)-pn~o)I+llpn~o )-uoll

Chuang IV: Chinh h6a mQt 56 bili lOan moment

Chuang IV: Chlnh hoa mQt so bai toaD moment

Ilpn(E)(Il)- uoll ~ 11(8)

bang phan chung)

.: Cu6i cling ta cho Uo E HI (0, 1).

Chuang IV: Chinh hoa mQts~bai toan moment

D1fa vao bang bien thien ta c6:

2t -t-1~O V'tER

~2t~t+1

Tli 2tE + 1::; 2tE + 2

Chuang IV: Chinh h6a mQts6 bai toan moment

The' vaoike'tqua lipn(E) (/l 0) - u0II

,; 2 n 8 + 1[( \ ].llua IIH' (01) ta dli<!c:

=> n(o)= t, + 1<0(21n3 +~ In 2 .In 27.fi r. +]

Tli bat d~ng thuc tren cho phep ta u6c 1uQngduQc s6 neE)

V~y ta da chung minh xong dinh 19 4.2

4.2 Chinh hoa bid toon moment cua bh~n d6i Laplace nglidc:

Chuang IV: Chinh hoa ffiQtso b:ti loan moment

Do do ne'u uEL2(O,co)=> flu(xldX <00

Ma x~ 0 => e-3x ::;1

-, 2

=> [Ju(x)1 e-3xdx::;r iu(xf dx <-co

Tlidotaco:ne'u UEe(O,X) thl \FEL2(OJ)

,

Ta vie't l<;1i:llw(tf ell= r:1I(x)12e-;'dx

Tli (4.36) va ap dlfng dinh 19(4.:2)ta du'QC:

Khi do t6n t<;1imQt ham sC; ll(E) (0 < £ < 1) sao cho 11(£) ~ 0 khi £ ~ 0

vadoivoimQiday ~L=(~L,,~l2, )th()a II,U-,Li"!/,,:::;E;taco:

Ilq"(l:)(,u)-u"t :;//(1;) (4.37)

Voi q"(C)(,u) = eXp"(C) (Ji).e-r (4.38)

jI=(/12,/13"")

IIIII:= [II(x)12 e-:' c£'\ (4.39)

va pn(E)(/1) nhu' trung l!inh 19 4.:2

Han nua ne'u Uo E HI (0,:1' ) thl:

Il qn(&)(J.!)- u II ::; f)< -+- 311Uo11H' (0, co)

Chuang IV: Chlnh h6a mQt so bai roan moment

Chung minh

Tli dinh 1;' 4.2 ta du'<;jc:

lip n(E)(II) - w 0 II ::; T\(e) voi II= (1l2' 1l3, ) (4.40)

Tac6: wo(t) = t.uo(-Int)

=> Ilpn(L)(]7) - woll= Ilq/(C)(,u) - uot

Tli (4.40) va (4.41) => Ilqn(E)(Il)- uot ::;T\(e)

(do (4.39))(4.41 )

V~y ta da chung rninh xang (4.37)

Bay giCita xet Uo E H1(O,oo). Khi d6:

w'(t) = u(-Int) - u'(-Int)}.tt

=u(-In t) - u'.(-In t)

=> W'D = uo(-Int) -u'O (-Int)

Thea dinh nghla chuffn trong HI (0,1) ta c6:

IlwoIIHI(O.l)= \!(w~ + w ~~tr

Chuang IV: Chinh h6a mQt sf) bfli toan moment

(do ~a + b ~.Ja + -Jb voi a;:: 0, b;:: 0)

,,;(iII.U" (-In nl' dtr +( i[u"(-In t) - u'" (-1n t)]' dtt

M~tkhac tli' llw(tfdt= [lu(x)j".e-1xdx va w(t)=t.u(-lnt)

=> flLuo (-In nl2dt=[Iuo (x)!2.e -3xdx

flu'o(-Int)ldt= [u'o(x).e-xdx

Thay vao (4.42) ta du'<;5c:

IlwoIIHl(O.I)~(!luo(x)12e-3XdXr+ ([IUo(x)\2.e-Xdxf +([lu'(x)12e-Xdxf

~31IuoIIHI(O,OO)

(do IluI12HJ(O.OO)= [(U2+ U'2~X ;::llul12va IIUI12Hl(O.oo) ;::lluf, x;:: 0 => e-3x ~ 1) V~y : IlwoIIH"(O,I) ::;31IUoIIH'(O.OO) (4.43)M~t kbic tli'dinh Iy 4.2 ta co:

IlpO(E)(iI)- w011 ~ EYz +IlwoIIH'(O,I)C(E) (4.44)(do thay Uo bai wJ

Chuang IV: Chlnh h6a mQtso bfli roan moment

4.3 Cblnb boa bai toaD nbH~tngu'dc tbOi gian

Tli bai roan nguQc tim v(x,y) (nhi~t GQt~i t=0) ne'u biet

(X-;)2 +(Y-'1) 2

[ [) v(~,l1)e- 4 d~dll = 4nu(x, y,l)

Ta xet truong hQpv(x,y)=O v6i x< 0, y< O.Khi do d&ngthuc tren co th~viet thanh :

C;-+'1-e-(m2+n2) r r v(;, ll)e-~ e-(I11~_n'1)d;dll= fmn (4.45)

voi lOIn= 4;ru(-2m,-2n,l)

V6i co(s,t) =v(-lns,-lnt).e 4

f-lij = 47Z'.e(i+l)2+U+I)2 u(-2i - 2,-2j - 2,1) (vi m=i + 1, n=j + 1)

EHiyla bai roan moment Hausdorff hai chi~u, ta co th~ dung phuong phap dChuang III d~ chlnh hoa no

Ta nh<1cl~i vai ky hi~u :

voi I =(0,1) x (0,1)

B6i v6i m6i day s6 thvc /-l=(Ili) 1, J = 0,1 ,2, ta xac dinh day

A= A(/-l)= (Ai) nhusau:

Chuang IV: Chlnh h6a mQtso bai roan moment

( +

')1

voi (theo s1,I'xac dinh cua khong gian L~)

?v~E L~

ChU'ngminh

Tli VEL"'(R+,RJsuy fa ham sO'

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

In: s+ln2 t 4

Chuang IV: Chinh h6a mQt so bai toan moment

(do d6i bie'n s = e -1;, t = e -11va dint nghla p", q" )

= r r (q n (e-~, e-q)- v(~, TJ)Y p(~, TJ)d~dTJ

= IIq"- vl12L~ (do dint nghla chuifn trong L~)

Tli (4.47) va (4.48) ta du<;1C: qn ~ v trong L~

Ne'u v E W1""(R+,RJ ta xet:

In' s+ln' t -

Tli dint ly 4.1 ta co:

lip" (11)- wIIL'I') ,; 2(n 1+ 1)IIwIlH'I')

TU (4.48), (4.49), (4.50) ta c6: Ilq" - viiI; ,; 2(nC+ I) Ilvllw"

(4.50)

Chuang N: Chinh hoa mQt so bai roan moment

V~y ta da chung minh xong dinh 1'1

Truong hQpdu kil$nkh6ng chinh xac ta co ke't qua sail:

f)~tn(8)=[F-l(8-X)] ([x] 1a so nguyen IOn nhat ::';x)

Khi do t6n tqi mQt ham so' 11(8) (0 < 8 < 1) saG cho 11(8) -)0 0 khi 8 -)0 0 vavdi mQi day f= (fmn)thoa:

suplem2+n2(fmn -f;n)l<1l1,n E taco: Ilqn(E)-voll!"-p ~T](E)

(vo la nghil$m chinh xac, qn(E)la nghil$m ung vdi bai roan monent huu hqn)

Han nua ne'u V() E wl,a:>(R+, R+ )thi:

Ilqn(&) -vollz :::;8X + Cilvollw'x

Theo tinh cha't cua chu~n ta co:

lip n (J1)- W 0" :::; lip n (J1) - p n (J1() )11 + lipn (J1 0) - w0 II (4.51 )

huang IV: Chinh h6a mQtso bai roan moment

(n-k)!(k!)2 Lk,k!(Xl'XJ = Lk,(xt).Lk!(x:).

kl k!

Aklk! =Ak,k!(~)=I ICkIPICk!P!~PIP!

p,=o P2=O Tli (4,52) ta c6:

Chuang IV: Chlnh h6a mQtso bai tmin moment

2

815

(chung minh tu'dng tv d dinh 1y 3,4)

M~t khac tli dinh 1y 3,4 ta co:

Chuang IV: Chlnh h6a mQt s6 bai roan moment

voi wo(s,t) = vo(-Ins,-Int).e

In2s+ln2 t 4

Chung minh tudng t1,1'd dinh ly 4.2 ta co 11(8)~ 0 khi 8 ~ 0

Tu (4.48) va (4.57) ta co: Ilqn(E)(Jl)- vollL2 :::;11(8)"

Neu Vo E WI""(R~) thl tu (4.49) ta co:

Tu danh gia sai 56 d dinh ly (4.2) ta co:

'

I 11

Voi C(8) ta du'Qcxac dinh nhu' tren

V~y ta da chung minh xong dinh ly

* Tu bai roan nguQc Hmv(x,y) ta viet l~i:

(X-~)2 +(Y-l1)2

voi g(x,y)=4nu (x,y,l)

tHy la phuong trlnh tich ch~p d6i voi ham chu'a bier v(~,11)

_(Xl + yl )

Lay bien d6i Fourier hai ve ta du'Qc:

1\ , -, ~.

Chuang IV: Chinh hoa mQtso' b~liloan moment

Trang thl,l'cte ta kh6ng C(')clCi'ki~n chlnh xac ma chi co dCi'ki~n do duQc co

sai so, Ta se xay dl,l'ngnghiC'm \.hlnh boa 6n dinh d6i voi nhung thay d6i cua g,Cho go nhuCi (4,59) cl) nghi~m Vo E e (R 2)Wong ling ci ve phai sao cho

Jig - go t:(R ') < E

Ta xay dl,l'ngham Y (~n dinh l16i vdi s1/ thay d6i trong g, Neu Yola du trail ta

co th~ danh gia sai so giCi'aV(}V:lv

Chuang IV: Chinh hoa mQt so bfli toan moment

Chuang IV: Chinh hoa mQt 56 bai roan moment

M~t khac rhea gia thuye't jig - goIIL'(R') ~ G

=>-4>-(

E )