Luận văn Chỉnh hóa một số phương trình tích chặp

Chu'ang4: Plu(o'11gplu1pchlnhhoaphU'(fngtrlnhtichch(ip CHU'dNG 4: ltIIU"ONG PIIAp ClfiNII IIOA P IIU ONG....TItiNII -TI CIIC II~ P Trongchu'ongnay,chungtaapd~lI1gphllOngphapTikhonovdechInhhoamOt sO'phl(ongtrInhd,';lI1gtichch?pvadanhgiasaisO'gi[Yanghi~mchInhhoavanghi~m chinhxac. 4.1Phu'dngphaRthinhhocTikhonovtho bciitoan~~n tinh Gicl sa A : X -+ Y la mOtloan tt( tuye'ntlnh lien tt,ICgiiTahai kh6nggian Hilbert. Xet phuongtrlnh Av=u (v lafin). (4.1) Chungta thU'anh~nphuongtrlnh(4.1)co nghi~mduynhit VoEX. Ne'utmln ttI nglIQcA-I clIa loan ttI A kh6nglien tt,lCtIeDRangeA till phl((jngtrlnh(4.1) la kh6ngchInh.M?t khclc,ne'ukh6ngbie'tdl(QCgia tr~chinhxaccua u machI bie'tUn v6i ~nQtsai sO'0, tucla IluB- ul!s;0, till phlIongtrlnh(4.1)co thekh6nggiili dlIC}CVI Unco thekh6ngthuQCRangeA. Do v~y,chungta phai chlnhhoa phuongtdnh (4.I), t((cIa tlm mQtx£J'pXI vI)E.X chonghi~mchinhxacVosaGchosaisO'giuachungtu'ongdO'inhava nghi~m duQCchQnlamxip Xlphaiph~lthuQclien t~lCVaGduki~nno' D;nhnghia4.1.1 Phie'mhamTikhonovla(v) v6ia> 0 lingv6i phuongtrlnh(4.1)dlIQCxac c1tnhnhlIsau: la(v)=IIAv--uI12 +aI1vll2,VEX ! (4:.2) Hai loan tlm C~IClieu phie'mham(4.2)la co nghi~m,c16ngthai nghi~mnay clingla nghi~mduynhit clIamOtphlIongtrlnhchInh.Cu theb~ngc1tnhIy sauday: D;nhIt 4.1.2 Phie'mham Tikhonov (4.2)co duy nhit mQtqlc lieu VaE X va vacling ]a nghi~mduynhit clla phuongtrlnhbie'nphanchInh: * " av+AAv=Au (4.3) --- 28 Clnro-ng4: PhuO'ngphapchlnhhoaphuongtdnhtichch(Ip Chu'ngminh: 1.Tinhchlnhcllaphuongtrlnh(4.3). Xet anhx~songtuye'nHnha :X x X --+ R vdi: a(v,y)=a +. Anh x~a lien t\lCVl voimQiv,y thuQcX, taco: la(v,y)1=la +1::;al1+11<; <;allvll.llyll +IIAvll.IIAyil <;allvll.llyll +IIAI12 .11vII. Ilyll =(a+IIAI12 )1vII.llyll Anh x~ala khangtUvi voi mQiv EX taco: a(v,v)=a += al1vll2+IIAvll2~al1vll2 (4.4) . M~Hkhac,phie'mham h: X --+ R voi h(y)=la tuye'ntinhlient\lC. , Ap eI\lflgc1~nh1:9Lax - Milgram, tan t~ieluynh5'tVrxEXthoa: a(vu,Y)=hey),Vy E X =>a +=,vy E X => =,VYEX =>ava+A*Ava =A\, Ron nila,nghi~mVexplw thuQclien t\ICvaoell(ki~n.Th~tv~y: Gia sO'voi cI[(ki~nill, taco vulthoa: a(vexl'Y)= va voi elilki~nU2,tacoVex2thoa: a(va2,y)=,VYEX. SHYra: * a(vul-vu2,y)=:, Vy EX L5'y y =Val - Va2EX, ta duQc: allval - Va2112:::;a(Val - Va2' Val - Va2) =(A*(UI- U2)' Vnl - Vn2):::; :::;llval-vu211.IIAlllul-uJ IIA*II ~IIVnl--Va211<;-.IIUl-U211a -- 29 Chli(!ng4: Phli(Jngphdpchlnhh6aphli(Jngtrlnhtfchchr;ip 2. Chungminhcvc tiSucuaphiSmh~lln(4.2)clingla nghi~mcl'taphlicJngtrlnh (4.3)vangl(Qcl~i. Ju(V)-Ja(vu)=IIAv-lf -IIAvu -If +ex~1vII2-IIYuI12) -Sl(d\1l1gcong th((c: IIal12-1!bI12=Iia - bl12+2(b,a- b), taduQc: Taco: Ja(y)-Ja(Va)::::IIA(v-vaf +2(Ava -u,A(v-vJ)+-cxllv-vaI12 +- (4.5) +2ex(vu,v-vu)=IIA{v--vuf +exllv-vuI12+2(a.vu +-A*Avu -A"u,v-vu) * NSu Vala nghi~mcllaphudngtrlnh(4.3)tIll tU'(4.5)taco: Ju(v)-Ju(vu)=:IIA(v-VcJI12 +exllv-vuI12;::0, \Iv E:X =:>Vala C\(ctiSuClLaphiSmham(4.2) * NguQcl~i,nSu Vala q(c tiSuclla phiSmham(4.2)tIll khi thayv =Vn + ty ~t>0,Y E:X) vaG(4.5), ta du'(!C: t211Ayl12+ t(exva+A*Avu ---A\I,y)+ex.t21IyI12;::0 Ddn gianchot,saudo Iffygidi IwnhaivS khit ~ 0 tadu\jc: (A*Avu +exva-A*u,y);::O, \lYE:X (4.6) VI (-y) E:XDentaclingco: (A*Avu +exva-A*u,(-y»);::O, \lYE:X (4.7) TU'(4.6)va(4.7)suyra: (A*Avn+exva-A*u,y)=O, \lYE:X '" " =:>AAva +exvu-A u=O V~y:Vala nghi~mcllaphl(dngtrlnh(4.3) K9 hi~u: V~tla nghi~mCllaphlicJngtrlnh: exv+-A"AV =A"un PhepcblnhhoaTikbonovseIffyv~langbi~mchlnhhoachonghi~mchfnhxac VocuaphudngtrlnhtuySnHnhAv=u,Lrngvoisaiso'trendllki~nlao. exduQcgQila thamso'chlnhhoa. ]0 Chuang4: PhuO'ngphdpchlnhh6aplncO'ngtrlnhtichch(1P * Tn(dnghQpto<1ntli A Ia tuyC'nHnhcompact,nghi~mchlnhh6a tlmc1t((Jc6 th~khongph~lthuQclien tl,lCvaodCi'kj~n. M~111Jc1~4.1.3 Cho A : X -+ Y Ia loantli tuyC'nHnhcompactgiiJ'ahai khonggiandjnhchu5n, trongd6 dim ()kerA)=OO. Khi d6 t6nt~iday {vJc X maAvn-+ 0 nhtl'ngvn+ 0 khi n -+00.Hon nCi'a,tac6thcichQn{vn}eX SilOcho Ilvnll-~00. C/llJ'l1gminh: Vi KerA Ia mQtkhonggiancond6ngcllaX Hen)kerA la mQtkhonggian dinhchu5n,vdi chu5n: II[ v]11 = inf Ilv+zllzEKerA Xac d1nhloantli A: %erA -+Y nht(sail: Vdi mQi[V]E )kerA' A[v]=Av Ta thayngayloantti A laxacd1nh,tuyC'ntinhva1- 1. Bay gid taclll'fngminhA Ia loantt?(compact.Th~tv~y: D~tB= {vEx:llvll<l}va13={lv]E)kerA:IIIV]II<t} (B,13lacachinhcaudonvi m0IanIuQtrongX vatrong)kerA) Ta phaichungminhA(13)la t~pcompactt(cingd6i trongY. Do A la loantt(compactnenA(B) la t~pcompacttfdngc16itrongY. Vi A(13)={Av: III vlll<I}c {Av:Ilvil<I}=A(B) nen A(13)la t~p compact!tt(ong d6itrongY, nghlaIa Acompact. Ta chungminhA-I: RangeA =RangeA -+ )kerA khong lien t\lC. NC'u A-I lien tl,lC thi to<lntll c16ngnhat A-IA =I :)kerA -+ )kerA Ia compact.(dodinhIy 1.2.2).Tasuyradim)kerA <00 (dodjnhly 1.2.3),trclivdi giel thiC't.V~y A-I khonglien tl,lC. 31 ChuCYng4: PhuO'rlgphdpchlnhh6aphuC!ngtdnhtfchchc;ip Di~u do co nghia Ia t6n t<;liday {vn}c RangeA,un~ 0 nhlingA-I Un --f~ 0 khi n ~ 00. GQi U" ==Av"==A[v"J,the'thlt6ntaiday{[v"J}C%erA thoa: A[v,,]~O vaII[v"JII~8>o, n =1,2,3,... Voi mQi n, IflY ZnEKerA saGcho llYn+Znll~~ vac1~tv'11= Vn+Zn SuyraAv'n=Avn~Okhin~oova Ilvlnll~~ , n= 1,2,3,...2 , ;!"" ~.- v'n Co theglasl1Av n T- 0 \in. Tac1~tVn= rFI--- Ilv' ,,11'\j IIA v' "II IIA v~ s2 ,~ IIAV"II==]Vnll)IIAV'nll Ilv'nll_s va Ilv"II= Ilv'"11.5v'n II IIAv'nll . -- ~ 0 khl n --~00 8 ~ 00 khi n ~ 00 IIAV'nll Tuy nhien chungta co th~h9-nch@Gt(Qcsai so'gdTanghi~l11tinh loan va nghi~mchinhxac khongvliQtquaso'hC(ul19.nnao(10,n@utab6 sungthongtin thu hypv~nghi~mchinhXclC. 4.2 Sai 56tnJonghdp xdu nhat(worst- caseerror) Dinhnghia4.2.1 Cho A : X ~ Y Ia loan tU'tuy@ntinhlien t\)CgiUahai khonggianBanachva Xl c X Ia l11Qtkhonggianconvoi chu§'n11.llxll11<,lnhho chu§'n1I11trenX. Voi 8,E litcachhngso'duongchotrudc,tagQi: ..sY\8,E,II.IIx,)=sup{llvII:v E XI ,IIAvllS 8,IlvIIxI s E } Ia saiso'trlionghQpxa'unha'tlingvoi so'8 trenda ki~nva thongtin them IlvllxlsE. Y nghiaciia khai ni~mnayIa : N@uco thel11thongtin v E Xl, Ilvllx]sEthI sai so'giUanghi~mchlnhhoava nghi~mchinhXclCciiapht(ongtrlnh(4.1)khongIOn h(5n..9-(0,E, 11.llxl). 32 Chu:ong4: Phuongphdpchlnhhoaphuongtrinh tlch chr;ip Nhuv~ymQiday{vn}eX thoaAvn-+0 t1~uhQitl,1v~o.Mauthl..anvai ke't quaa m~nhd~4.1.3.M~nheM4.2.2duQckh£ngdjnh. Ne'uclwn XI =A* Y ho~cXl =A* AX la ca.ckhonggianconclIaX vaicac chuffn11.1Ix1m~nh dnchuffn11.11vachQnthallis6chlnhhoaa trongphu'dngtrlnh (4.3)thichhQP,taco caedanhgiasais6giG'anghi~mchlnhhoava nghi~mchinh xacnhusail: DinhIt 4.2.3 ChoVoEX,uoEYthoa: Avo=Uo (4.9) a)Gia su VoE A"Y tUGla Vo=A"ul'u, E Y. Vai 0 tachQna«5)=<5va ne'u HEY la dli ki~ndod~cthoaIIu - uoll:::;<5thltaco : IIv8-vall:::;~(l-l-IIulll)8~ ,b) Gia su voEA"AX, tUGla Vo=A*Avl,VI EX. Vai 0 ta chQn a«5)=<5X va ne'uUEY la dli ki~ndo d~cthoa Ilu- lIo!! s <5thl ta co : V,r, -VoIIS(IIAII+~hll)oX Chungminh: TU'phl(dngtrlnh(4.3)vad£ngthuc(4.9)tasoyra: { a:a -I-A*A:a =A"u AAv =Au0 0 TrU'haid&ngthuctheotU'ngve'tac1l(QC: * " ava-l-A A(va-vo)=A (ll-Uo) " " => av(x -avo +A A(va -vo)= A (u-uo)-avo , Tich vo huanghaive'vOi( Va- Yo)tanh~nc1uQc: a -I- " =-a+<A (U-lIo),(Va -Yo»~ => allva -vo112-I-IIA(va -vo)112=~a-I-<lI-uo,A(va -Yo»~ (4.to) 34 Chuang4: Phuangphdpchlnhhoaphu'angtrll1hrichch(lP a)Biend6ivephcliclh (4.10),tac6: =<u-uo,A(vu -vo»-a<u[,A(va -Yo»~ ~llu-uoll.IIA(va -v(JII+allulll,IIA(va -vo)II=~lu-uoll+allu,II).IIA(va -vo)1I ~ ~~lu-uoll+alluIIIY +IIA(va -vo)II2 (nhobfttd~ngthl(c:4a2+b2~4ab) Thaya =a(o) =0va liB- Boll~O. Tli' (4.10)ta c6: olivo-vo112~~(o+81IuIIIY4 ~llvo-voll~ 8fC.(1+llu,II)=~(1+llulll~~ 2-v8 2 b)Tli c1jnhIy 1.1.3tac6: IIA(va:-vo)II2 == =112 Thayd5ngthti"cnayvao(4.10),vdi ChU)T Vo= A*Av) =Cv), ta UlCQC: allva -voll2 +IIC(v((-vo)112= ~IIAII.llu-uoll.llva -voll+al<v"C(va -Yo»~! ~ a Ilva - V0112+~IIAI12.llu - Uo112+5£IIV I 112+IIC( Va - V0)1122 2a 4 ~ ~IIVa - V 0 112~ 2~ IIAI12,liB - Uo112+:2 Ilv1112 ( 1 2 2 a 2 J ~ ~lIva -voll~ a211AII.llu-uoll +211vlll Thay liB- Uoll~0 va a =a(o) =oX,ta C1l(QC: v,x-Voll~(IIAI12bX+~1Iv,112b%t hay 11v,%- v"II S; (IIAll'+~hll' Joy, . Nho bfttc15ngthLrca2+b2~(a +b)2, (a ;:::0, b;:::0) tac6: 3S ChuO'ng4: PhuO'ngphdpehlnhh6aphuongtrlnh({eheh(ip v,%-Voll~(IIAII+~hll}oy, 4.3 Ung dung phlidng phap Tikhonovchinh hoc phlidng trinh tich chap GiasliA: L2(RI1)~ L2(RI1)gWahaikhonggianHilbertvdi Av(x)= fK(x-t).v(t)dt, VvEe(RI1) R" Khi d6A la loantli tuye'ntinhlient\ICvaplnidngtrlnhAv(x)=lI(X) c6 cI<;lng tkhth~p: (K*vXx)=fK(x-t)v(t)dt=u(x) R" (4.11) Bai loantlmvex)vdi da ki<%nu(x)clla pln(dngtrlnh(4.11)(nhll'oa trlnh bayCJ 2,3,Chuang2) n6ichungla khongchlnh.Ap d\lllgphepchlnhh6aTikhonovchob~ti toantuye'nHnh,tatlmnghi<%mchinhh6acuaphlidngtrlnh(4.11). * Tn(oche't,taxacdjnhtoantt(lienhQpA" cuaA. VfEL\RI1), taco: == f(K * vXt).f(t)c\t= f [ fK(t - 1:).v(r )d1: ] f(t~lt RII RI1 RI1 =JUK(H)f(l)dt]V(.jd. Xet <p(1:)= fK(t-1:)f(t)dtEL2(R"). (djnh 19Fubini) RII SHY ra = M~Hkhac " = (djnh191.1.2) :::; =, Vv EL2(HI1) :::; =O, VvEe(Rn) Lffy v=(A"f-(p)Ee(R"),tac6: IIA*f-<pll:=0 36 Chuang4: Phuangphripchlnhh6aphuCJngldnh tichch4p ~A'f(T)=(p(T)= JK(t-T)f(t)dt h.k.ntrenRo. R" £)~tp(x)=K(-x), \/x E Ro, tac6: A '[(T) =(P * OCT) h.k.ntren RI1. /\ /\ /\ /\ /\ /\ =>A* f =P* f =p.f =K. f (4.12) Khi d6 phlidngtdnhchlnhh6a (Xv+A*Av =A\lo tlidngc1twngvoi phlic1ngtr}nh: <xv+ p* (K * v) = p* Uo (4.13) Bi€n L16iFourierhaiv€ cua(4.1.3)va sll'd\lflg(4.12)taC1liQc: /\ /\ /\ /\ 1\ /\ a. v+K K. v =K. uo (4.14) VI phlidngtrlnh(4.3)lachlnhBen(4.14)IamOtphlidngtrlnhchlnh. Ki hi~u: -- /\ /\ ~J1~(t)= K(t).uo(t) a +IK(tf 1\ VI ~ISEe(R")val K(t) 2 1 ~ I, Hen~p~Ee(RI!) 1 /\ 1 2.ya a + K(t) V /\ £)~tv~(x)=~J1~(x)=>v~(t)=\F~(t)~v~Ee(R") V~y nghi~mchlnhh6a v~E e(RI!) clJa phli(jngtrlnh tich ch~p(4.11) GliQctinhtheoc6ngthuc: S 1_- J ~(tLt~~(t;-.eXP(ixt)dtv (x)=- ( )"/2 1 /\ (~ a 2n R" a +K t1 Tli L1!nhly 4.2.3,tac6h~quasanclay: He gull4.3.1 Cho VoE e(RI1), UoE 1}(RI1)thoaphlidngtrlnh (4.11). 37 Chu:c'Jng4: Phu'(Jngphdp chlnh h6aphu'(/ngtrlnh tfchchr;ip a)GiclSl(VoE A*(e(Rn )), luc la vJx)= fK(t x).f(t)dt, f E C(R"). Vdi R" 0>0, tachQna(o)=0, Khi do,ne'uuE L2(R") la di1ki~ndoc1~cthoaman: Ilu - Uo112::;;0 thl ta co live-voI12::;;~(1+llfI12~'/2 (4.15) b) Oi:! sli v0 E A' A(L' (R' )), tlie lit v0 (x)~ J(JK(t- x).K(t- T)f(T)d+t , fEL2(R"). Voi 0>0, tachQna(0)=02/3.Khi do ne'uuEI}(Rn) la dC(ki~n doc1~cthc)aman(4.15)thltaco: hili -voll, S(IIAII+~llfll,)01/1 Chungtabie'trhngne'uthongtinthuh~pvenghi~mch1nhxactrenmQtkhong gianconcangnhothlsaiso'giUanghi~111ch1nhoavanghi~l11chinhxclccangnhc). TuynhiencackhonggianA"(L2(R"))va A"A(C(R")) rit nIle),thl(dngla g6mcac hamthuQclOpCO'Jnenvi~cx~icdinhchungfit khokhan. Phftndl(oiday,chungtase danhgia sai so gii1anghi~rncblnbboava nghi~rn chlnhxac CUdphl(ongtrlnhHchch~p(4,11)voi gia thie'tnghi~mch1nhxac thuQc khonggianconSobolevHI (Rn)(rQnghoDcackhonggiannoitren)vdichutln11.11111: IHIII =(IH~+IIDvll~)~=(!HI: +IIDVII:)~.=(IHI: +lllll.~II:)~ { 2 2 } 1/2 { 2 } 1/2 = J~ dt+Jt2~dt = J(1+t2)~(t~1t chotnidnghc;1Pd~cbi~tvoi li«ti =M.exp(-PltlY), (4.16) ( ) 1/2 voit=(tl,t2,...,tJER",ltl= tt~ vaM,P,rIacaehangsodu'eing. Nh?n/\xet: IK(nl cod~ng(4.16)lahich~nvaItl.IK(t)1cungbich~n (doIK(t)1~ 0theocip exp(-PIW).Tacoke'tCjmlsaisotn(dnghQIJxiu nhfll 38 Chuang4: Phuongphdpchlnhh6ap/1l((Jngtrll1hrichch~lp vac1<1l1hgiaSRiso'gilia nghi~mchinhhoava nghiemchillhxac clla p11lJ'ongtrlnh (4.11)trangL\RIl) nhtisan: Mt?nh c1~4.3.2 Giil stI nghi~mchinhxac VoCttaphu'ongtrlnh(4.11)tlltlQcHI(RIl) va II«(l1co :l?ng(4.16).Khi do SRiso'tnionghc;1pxilunbiltthoamanbiltc1~llgtlltI'c: "r(8,E,/it,I)< ~ 1/2khi6~ 0' .' (1U8) trongdo C Ii]hhngso'chIphl;!tbuQcvilaE. Chu'ngminh: Xet v E HI (RIl) thoa: IIAvll2::;;8 va IlvIInl::;E 1\2 f(l+t2)v (t)dt=llvII:rl::;E2 R" =:> J.M,.eXP(2~ltl');'(I)dt=IIK;I[= IK~Iv[ =IIK*vll:=IIAvll:<;6' Vdi a> 0 tily Y (ta se cllQUl?i a tbichhc;1p)till: 1\2 l+t2 1\2 1 ) 1\2 E2 fv (l)dl::;f ~.v (t)dt::;~ f(t+t2v (t)dl::;~ 1'1>" Itl>ol+<1 1+a R" l+a (4.17) 1\2 M2exP ~ 2~ l t I Y)1\2fv (l)::; f . 2' - -:y v (t}lt::; 1'1:;" Itl<:oM .exp -2~<1Y) 1 2 ( . Y)1\2 82<. M exp- 2~t v (l)dt<-:-- - M2.exP(-2~ay)J' II -, M2.cxp(-2f3aY) (4.18) Xet p1n(ongtrlnhin a: F2 82 F2M2 ~~ = ( )(1+ a2)exp(2~aY)=-~l+a2 M2.exp-2~aY 82 E2M2 (chQnsRiso'8 tren dli ki~nc1t1be c1€ 7~ > 1) Hamso'y(x)=(1 +x2 )exp~~xY) tren(O,+«J)la hamso'tangvaco mi~ngiatrj a (1,+00)Henp1niongtrlnh(4.19)co nghi~mduynhflta=al) thoa linl al) =+00 . 1)-70 (4.19) 39 Chuang4: Phu'ongphcipchlnhh6aphuongtrlnh{{chCh(ljJ ~(1 2y )/-' +a , B~t A= 2 >0, taco:l+a { A(I+a2)=~(I+a2Y)22~aY 1+a2 >In(1 +a2) => ) E2M2 (I+A)(I+a2 >2~aY+In(l+a2)=ln z8 ,=> I 1+A -<-- 1+a2 E2M2 In-- 82 (4.20) TiY (4.17), (4.18)va (4.20),voi a =al) tac1ttQc: 1111 2 11 /\ 11 2 /\2 /\2 /\2 2 V 2 = V = fv (1)01= fv (1)dt+ fv (1)otS 2E <2E2 (1+A)= (1+A)E2 2 R" l'l>as Itl-;;as l+a~ E2M2 ( ~ ) 1 In-- In bM +In- 02 0 )IP C +(1+A 13 khi 0 --+0 =>11vII2< 1 1/2~ ( 1 ) 1/2 (In(EM)+In8) In8 V~y:.!r(o,E, 11.1111')< ( ~ ) 1/2 In-- 0 khi 8 -t 0+ fJ Neu stt d\lllg phttdng trlnh(4.14)voi a=8 d€ chlnhh6a phttdngtrlnh(4.11) trong0611«(11c6d~ng(4.16)tacoketquac1:1nhgii saisagifi'anghi~mchll1h6a vanghi~mchinhxactrongL2(Rn)nlntsau: Menhd~4.3.3 Gia Sttnghi~mchlnhxacVocllaphtt'dl1gtrlnh(4.11)thllQCHI (R"),trong(16 11«(1)1,c6 d~ng(4.16) va lll) la dfi' ki~n do d~c voi !IllI)-Un II~8 . Khi d6 t6n tai nghi~mchlnhh6a vI)(li'ngvoi a =8)saocho: c Ilvo -vo112 <1 In- 0 khi 8 -t 0+ ,trongd6C la h~ngsachiphl,ltl1llQCvao!lvolllI' 40 Chuang4: Phuangph6pchlnhh6aphuangtrinhtichdt(lp Chu'ngminh: Ghl Sl(a =0, tUcacphuongtrlnh(4.11)va (4.14)suyra: ! I«(t).~o(t)=~I()(t) &~,(t)+IK(t1'.~,(t)~k(t)~,(t) (4.21) (4.22) (4.21) => I) ~'" (f)+<.({~o(t)=~(l)~o(t)+I) ~o(t) (4.23) Tt( (4.22)va (4.23),tadt(Qc: ( ) 11 2 ( ) - ( ) A A A A A AA A A 0 vs-vo +K . VS-Vo =K. US-Uo -o.Yo (4.24) NMn hai vi! cua (4.24)v<Ji(~8(t)-~"(t)J f6i 15y Uehphonhai vi! tfen R" ta duQc 2 2 ( J -- ( -~- J A A A A A A A A A A A A A °llv.-voll, + K{v-V.), ~-ojvo V5-V" eltIJK{U'-uo) v.v.ell Sa d~lngbatd~ngthlicHoldervathayII~IS-- ~IOt =Ilus- Uo112so, chuy I( (1)hi chi;ln.Sail d6rutgQntaduQc: °ll~'- ~'II:+IIK(~'-~"{s OCIHI,+III<lljl~' ~"II, (4.25) Bo botsOh:;1ngthlihai(j vfi traiclla (4.25)tac1tiQc: °ll~,-~oll,~IHI,+11.='1', (4.26) Bo botsOlwngthlinhatd vfi traiCUd(4.25)vadung(4.26)suyra: IIK(~5-~oJII:~8f,' 41 Chuang4: Phuongphdpehlnhh6aphu'o'ngldnh tiehdl(ljJ Ti€p tl,lenhanhai v€ eua(4,24)voi ItI2(~o(t)-~o(t))r6i lily tiehphanhai v~ t A RII dA -'*A'fen , an lien: () ll ltl,(~o-~o) 11 2 + ll ltl,I«~o-~o) 11 2 ==-()fltI2~O(V8-Vo)dt+ fltI21«2t8-2to)(~o-~o)dt 2 2 R" R" Ap dt,l\lgbiltcHingth((eHolderehocaetiehphana vi ph{tivoiehuy 1+«t)1 bieMn,sand6thay II~'-~,tso vaIlltl;°11,~ IIDq,00 IIDv,ll,. ta at(de: 2 2 ollltl(~8-~0)11 + ItII«(~8-~0) soIIDVoI12 11 It1C~8-~0) 2 2 112 +0111111[(1Jltl(~"-~"t, .. (4.27) Eo botso'hangth((haia v€ tnlietlabilta£ngth((e(4,27)taco: 111(;'-;0)11,S; IIOY0 II, + It I Kt='1', Lily a> 0 thyY ( taseehQnl?i athichh9P),taco: 2 2 2 I 1 2 f I vo-vo l dt= f ;0-;0 dt~ fItI2.a-2;0-;0 dt~a-2flf ;0-;0 dt Itl>:I Itl>:I Itl>:I R" , II /\ /\ 11 2 T2 =a-2Itl(vo-vo) ~-+ 2 a II 1 2 /\ /\ 2 2 "',Vo-vo dl= f V'-Vo dl'; f M exp(-2fJlll')IA _A 1 2 11- Itl~a II M2ex (- 2R. Y) ,Vo Vo dt~ t~a IJa 1 ~ f 2 Y 1 /\ /\ 1 2 M2exP(-213aY)R"M exp(-2~ltl )vo-vo dt /\ /\ /\ 2 K(vo- Yo)II = .. 8T2 22 <---L M exp(-2~aY)- M2 exp(2~aY) 42 Chuang4: Pln((jngphdpchlnhh6aphuangtrinhtfchch~lp GQi TJ = max rrj2,T;} (T3ChIph~1thuQcvao JJv011111), ta SHYra: IIVI)-voll~ = flVI)-v,Jclt+ flVI)-Vol\lt:::;T3 [-;-+--;eXP(2f3aY) ] 1'1>:1 1'1$" a M Xetphltclngtrlnh5na: 18M2 -- =-exp(213a Y) a2exp(2pa Y) =-- a2 M2 8 (4.28) THullsOy(x)=x2 exp (2pxY)tren(0, +co)lahamsOtangvami0n giatri la (0,+co) Henphl(clngtrlnh(4.28)c6 nghi~mduynhata=aothoa lim al)=+00 1)->0' ChQna =ao,khid6 Ilvl)- v()ll~s ~~3 al) (4.29) ~ pa1 - Y J-)~t Ie=- >0, ta SHYra leal)=pal) al) M2 ~2(1+Ie)al)=2al)+2leao>21nao+2Pa1=ln8 2 --2 ~ ~ < 2(1+Ie) '1 2 <0 M2In---- 0 2(1+Ie)=1 1 In- +21nM 0 (4.30) Tli (4.29)va(4.30)chofa: IIVI)-Voll~<2T3 [ ~(I+A) ] 2 In -- +2111M 0 II II )2 rI~- 2(1+11,) C -kl 'S:: ( - ) " V -v < ~- 11u-t I) °2 3 1 1 In- +21nM In-- 0 8 C la hhngsO'chIph~thuQcvao Ilv011111 . Bay giGxetcacphl(clngtrlnhtichch~ptL(clnglingtuhaib~tiloannhi~tdffneu trongchuclng2. Phuclngtrlnhctiabai loannhi~tnguQcthaigian: (K,*VXx) = II=-+1exp[=(x- ~2)].V(~)d~= u(x), X E R2-v1[ -co 4 (4.31) 43 Clu[(Jng4: Phli(jngphcipchlnhh6aphu'(Jnglrinh tfchch~lp Pl11idngtdnhcuabai loannhi~ttrang16khoanthamclo: (K *vXt)=~=ft .v('"C) exp [ --1 lh=u(t),t>O2 2-Jno(t-'"CY/2 4(t-'"C)J (4.32) Menh d~4.3.4 Ghi sli nghi~mchinhx~.cVocuacacphttdngtduh(4.31)va (4.32)IhuGCH1(R) U'ngvdiciaki~n U va lIo la ciaki~ndo d~cvdi Iluo-lIII2 ::;;O. Khi do 1ll6iphu'dllg Irlnhd€lI 16nt~tiIlghi~mchlnhhoa v0(U'ngvdi a =0)saoclIo: live-vet<~ khi0 -+0+ 111- 8 trongdoC la h5ngso'chithuQcIlvolllli. Chung l11inh: Ta chic~nch((ngto 11(1(1.1va11(2(~ico cl~lng(4.16),saudo apcI~lJlgk€t qua Cllam~nhd€ 4.3.3. ( 2 ) 1 -x Ta co : K, (x)=2FnexP 4- B€ y f(x)~ eXJ1( -:2 }os(tx)U1hams6 chanva g(x)~ ex{-:~}iII (IX)la hams6le. f\ 1 +00. 1 +w ( - 2 JKI (t)= r;;- fKI (x).exp(--itx)dx= r- fexp.f- .exp(-- itx)dx...;2n._" 2n...;2__" 1 [ +00 ( 2 ) +00 ( 2 J ]=2n-.Ji ,£exp -;- cos(tX)clx-i,£exp-;- sill(lx)clx ~ n~2'[exr(-:}OS(tX)dX Su'clt.lJlgCGlIgIh(tc tich phfll1: "," 1 - t2 fexp(-ax2 )cos(tx)dx=- r- Fn exp(--)0 2...;a 4a ChU'O'f1g 4: Phu'angphdpchlnhh6aphwJ'ngtrinhtfchch~lp A 1 ( 2\Ta CO:K,(t)= ~exp --t };2n ~11(I(ti=J~n exp(-t2} v?y IK,(ti eod~ng(4.16)yoi: M =k,p =l,y=2 TaeoK2(t)=J 2t'!~j;exp(- ~J,t>0 10 ,t~O 1\ 1'00 1 +00I ( -1 )~ K2 (~)=-~ fK2(t)exp(-it~)dt=~ f-'/2exp- .exp(-it~)dt;2n 0 2n ;2 0t 4t 1 [ +00 I ( - I ) +00 ( ]~ 2nfi J, V, exp4r cos(,~)dt- i J [~,exp ~~}in(t~)d[ . St(d~lI1geaee6ngthueHehphan: +1 ;/2 exp( - a ) cas(tx)dx=Eexp(--J2at)cas-J2a! oX X ~~ 'J-i2exr ( -a_ ) sin(tx)dx=Inexp(--J2a!)sin-J2at oX X ~a ta Ol(QC: 1\ 1 [ ~'. fl~l jK2(~)=J2;exp -~2'-1~2 ~ 1 1<2(~~=~ exp [ - rl~l-i {jij ] =~exp ( - rl~l ]1 .12n ~.2 ~2 -J2n ~.2 1 1\ ~ . 1 1 1 V?y K2(t1cod~ng(4.16)yoi: M =-J2n-,B=Ii' y=2- /1 Chuang4:Phuangphdpchlnhhoaphu(Jl1gtrlnhrichch(ip 4.4.It-nitngb.iem. rheincu6icungcualu~nvannay,chungtachIc6g~ngtimhiOOdangcuanghic$m chinhxaca6ivdihaibailoanngu<;1ccuaphuongtrlnhnhic$ta5neuC1ChlfC1ng3 000chuangtrlnhMathematicavdigiiithie'tdingsai56trendl!kic$nla sais6sinh radocaebudcxa'pxitrungianvatutiOOloancuamaytinh. I. Bditoannhietngu(1thaigian. Giii sau(x,O)=vO(x)la hamdu<;1cxacdinhnhtfsan: ~[~ :=»p[-(2/3)*~x] 2x2 e---Y BiSudiSnd6thicuahamvO(x): .-/_.J.- --',. / " / 0.' " / "- / n, -'" /' . / "- // n 4 "". =~~=:~~/./ 0 2 "',,-. "', - , - 1 ~--- ~._.:::~-=-~.::::_.=- Theocongthlic(3.10)tac6u(x,l)=uO(x): t'D[x-.J:=(1/ (2*91tt[Pl]»* n,l~rale[~[-«x-z) * (x- z» /4]*vO[z], {z,-:rn5ni.qr,+Infiniq.} ] )3 -~-ell11 Nhuv~yvO(x)langhi~mchinhxaclingvdidll ki~nchinhxacuO(x). Bie'nd6iFouriercuau(x,I)hi: U2[z.J := ~1r'i~~[t'D[XJ , x, ~ 1 . 11z2 - -J3 e-8- 2 Bie'nd6iFouriercuanhanK1(x): 46 C/lltang4:Phuangphdpchlnhh6aphuangtrlnhrichch(ip. k2[z..J:=~ -z"2] /9:;Ir:t[211'Pi] e-7l- ..;2; BlnhphttdngcuamodulicuanhanK1(x): kZ2[z]:=E}p[-2*z"2]/ (2*Pi) -2:02e 271" Choa =0,001,tac6bie'"nd6iFouriercuanghi~mchinhh6a: ~[z] :=k2[~*tQ[~/ (O.<Dl+k22[~) _197l-[ie s- ~2.>1 2 (0~~~-;:-:~::~~-) -- 2n VI thongthSIffyFourierngu9ctru9ctie'"ph~llnv2(z),tadungxffpxi dathuctrtf{jc khilflyFourierngu9cnhusau: Ruoc1:T~obanggi:itris6cuahamv2(z): T= 'J1:Ib1e[ {z,~(3/ (2*Pi)] * E:Ip[-19*z"2 / 8] / (2* (0.001+E:Ip(-2* z"2] / (2*Pi) » }, {z, -10, 10,O.25}] Duoc2:Xffpxl b5ngnQisuydailiac: V12=Itd~'~m['l1 Xell1 dathucv221a'xa'pxi cuav2[z]. Duoc3:Bie'"nd6i Fourierng\J9chamv22,tadH<;1cnghi~m 47 ChU(!11g4:Phu(!11gphdp ehlnhhoaphu(!11gtrinh tfeheh(ip chInhh6a: vOO[x-.-J:'"~ar:m[v22[z] , Z, x] Bi~udi~ndBthicua nghi~mchinhxacvanghi~mchInhh6a((ngvdithams6 chInhh6aa =0,0.01trenclIngh~tn:lct9ade>: P.lct[ (VOO[)(],~[Jq}, {X,-2.5, 2.5}] / /' /' / 1. 5 // / // // ,IT-'-''''~'--,, /,/ // ,/' /' ,/ -,~" ,/ ,-,--"~/ --- "'-. 0.5 ""' ,,---,.. '-"-"'- " ..::.::::::=---'::::-"-- 21-2 '-I Choa =0,0001,tadIng HmduQcnghi~mchinhh6atlidng((ng. SosanhdBthinghi~mchinhh6avdidBthicuacaenghi~mchlnhh6a ((ngcae tham55chinhh6a khacnhau: a=1/l000,a=1/l0000 trenclingme>t h~tr~lct9adO. ./' " / " / ,.-2 '" " "// ',,' . \ // "\ {I I . .1:_" '\\;/ " ~ '/ ,\// ~'- \ / / / 0.5 .~ \\ //; // ". \'-. /' // /",/ ' , "", .~<' ...:::::?~- - -2 -1 1 2" 48 Chuang4:P/u(rfngphdpchlnhhoaphllrfngtrinhrichch(ip 11.801toClnnhl,t trong16khoanthdmdo. XIStu(1,t)=vO(t)Ia ham dtf<;1Cxac dinh nhlf sail: vO[t:J :=ItSJ:t[l+t"2] Theocongth((c(3.22)tau(1,t)=uO(t): UJ[t.J :=-l/Sqrt[l+ t"2] 1 1+t2 (Sais6dotinhgftndungb~ngchlfongtrinhMapple- Mathematicakhongtinh tr1,fcliSp dlf<;1c). Nhuv~y,vO(t)du<;1Cxemla nghi~mchinhxac((ngvdi dil ki~nchinhxacuO(t). BiSn d6i FouriercuauO(t): u2[z_J :.. fu.u::ieJ:'l':ransfonn[uO[tJ, t, zJ -J-~ BesselK[O, zSign[z]] Lien h<;1pcuaFouriercuanhflllK2(t): k22[Z_J :.. (1/ Sqrt[2",PiJ) '" E1!p[-Sqrt[1'Ibs[zJ/2J +I", Sqrt[1'Ibs[zJ/2]] e (l-i)~JlbslZJ /2 ~-- ./2 7T BlnhphuongmoduncuaFouriercuanhanK2(t): k23[z_J :=(1/ (2",Pi» "'E1!p[-Sqrt[2*1'Ibs[z]]J e-/2 ~ JIbs!z) 27T 49 ChltrJng4:PhltrJngphdpehlnhh6aphltrJngtrinhtiehehi)p Choa =0,001.Tacobie"nd6iFouriercuanghi~mchlnhhoavOO(t): v2[z.J :. k22[Z]*u2[z]/ (0.001 +k23[z]) e (I-i) ..}JIbs[Z] ..}2 BesselK[O, z Sign [z]] ( -..}2..}Jlbsrzl1 0.001 + e 2 IT ) 7T VI khongth~IffyFourierngtf'1ctrtf'1ctie"ph~llnv2(z),tadungxffpXldathli'c trttdckhi IffyFourierngtf<;1c: Booe1:T~obangghitris5cuahamv2(z): T:= Table[v2[z], {Z,0, 5, O.25}] Btio~2:XffpXlblingnQisuydathli'c: v22[z.J :=Inteqcl.aticn[T] Xemdath((cv22[z]IaxffpXlcuav2[z]. Booe3:Bie"nd6i Fourierngtf'1ccuahamv22[z]: vOO[t_]:=Inver:seE'rorier:TJ:ansfonn[v2[z]I z, t] tadu"1Cnghi~mchinhhoa: -~InverseFourierTranSfonn[7T (1+1) ..}~ 1000.e ..}2 BesselK[ 0, z Sign [z] ] ], z, t 159.155+1. e..}2..}lbsl~; Sosanhd6thjnghi~mchinhxacvanghi~mchinhhoali'ngvdithallis5chinhhoa a =0,001trencungmQth~tr\lCto~dQ: 50 ChllrJng4:PhllrJrlgphdp chinh hoaphllrJngtrinh rich ch(1p 1 ~~ ",,-, 0 . A I- -" " --"'------- -.. - -------........---- 1 . 2 0 . 6 0 . 4 0 . 2 --' nu -- -.. ~-~ -----..-- - - - 2 3 .s