Luận án Chỉnh hóa một số bài toán ngược trong khoa học ứng dụng

~ PHAN I / , / CAC BAI TOAN CAUCHY CHO PHUdNG TRINM POISSON Trongph§nnaychungtaxet babai toanCauchy chophu'dngtrmh PoissontrongcacmienkhacnhaucuaR2va R3.Cacmiennayl§n lu'<;:it1f hlnhtronddnvi ma,111l'am~tphgngtrenma,lllIakhonggiantrenma.Bieu ki~nCauchyg6mgiahi cuahamu va d<;\ohamrheahu'dngphaptuy€n o9 "u en (d6ivdi bien cuamien)chotrenmQtph<1nbienma khactr6ngcuamien dangkhaosat. CacbairoannaythuQclo<;\i"lienthongtin"naysinhtrongV~tly Bia duoCacmienk€ trenlacac"mahmh"khacnhaucuatraid5t. MQtsO'k€t quacuaph<1nnay du'<;:iccangb6 trong[1],[2], [3],[4]. 1.BAI ToAN CAUCHY CRO PHUONG TRl~h POISSONTRONG HINHTRON BON V+. 1.Bai tmin : GiD = {<x,y):X2+/ <1} D={<x,y): x2+/S;1} Tlln hamu thoa: ~= f trongD u E C2(D)nC1(I» u (ei6)=uo(0), au (e'B)=U (0)" 1 .,on' I 0<0<a 0<0<a (1.1)' (1.2) (1.3) (1.4) ydi f chotru'dc!rangD ~ UO,Ul chotru'octren (0,a) , (0<a<211:) - 11- ~: - d~ohamtheehudngphaptuytnngoaitrenS=aD={(x,y):~+l=l} Ky hi~u r ={eiB:0<B<a} r= S \ Y Ntu ta bitt ~:(eiB) =v(e) ~ a ~e~ 21t thl ta bitt di~uki~n NeumanntrenS va bai toantlmu thoa(1.1)voi di~uki~nNeumanneho tru'dcla mQtbaitoanchllh,dadU<;1cduavaocaegiaotrmhphuongtrmhd~o hamrieng(xem[7],[15]). Dov~ytach9n v(e)=~:(eiB) , a.~e~21t lam.in ham. 2.Thie'thipphu'dmztrInhdehphaneuav. Trongph~nnaychungta dungphuongphapGreend€ duavi~etlm nghi~mu cuabaitoanCauchyk€ trenv~vi~cgiaiphuongtrmhtichphan Fredhohnlo~imQtddivdiv. Xet hamGreeneuabai toanNeumann ehophuongtrmhLaplace tronghmhtrondonvj G(X,Y;;,11)=~ In[eX-q)2+(Y-17)2]41r' [ 7t:'\7 7 7 ] +~In ex-P-?;-+(Y- P-17t 41r' p4 1 2 7 --Ine; +17-) 41r' (1.5) B~t M =M(x,y)ED. Gqi M' =M'(~, .~) (voi p=~X2 +y' ),13 p- p- di€m d6iXU'ngvdiM quaphepnghjchdaocuavongtrondonvj S . B~t P =p (;,11) la diemch<;1Y I 2 ( )2 1r =IMP! =~(x- ~) +Y- '1 I I r'=1M'?!=j(~_~)2 +(.L,- '1)2 ~~ p~ ~ P- I I I I 1 I p= OM'=p J (1.6) D~thay pr' =r lieU P(~,ll) E S (1.7) Khid6 1 rr' G(X,Y;~,ll)=G(M,P)=2;rIn lOp! =J..-( G1+Gz +G3)2n: (1.8) dday G1=G1(M,p)=Inr =InJ(x- ~t+(y- 7]? I( 12 ( '\2 Gz=Gz(M',P)=Inr'=In,!1 ~-~ j +1 yz - 7]) ' ~\p ',p G3 =G3(0, p) =InIOP!=In ~~2+7]z (1.9) TrudehettachU'ngminh oG(.M,p)1 =0 0n I PES (1.10) Th~tv~y, gi}101a veetd ddn v~phap tuyen ngoai d6i vdi S ~i PES, me no =OP .Ta e6 ......--.... oG11 =cos(lvlP,OP)=cosOMP=1+r2- p2 0n Is r r 2r2 I ~ ~ oGz =cosUvl'P,OP)=cosOPM'=l+r'~-p'- on IS r' r' 2r'z (1.11) - 13-