Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 34
Текст из файла (страница 34)
HcrrOJih3YR rrpeo6pa30Batt11e <Dypbe, pelllHTb 3a,[(aq11:= a 2urx, u(x, 0) = cp(x), u,(x, 0) = \jl(x);= a 2uxx + f(x, t), u(x, 0) = u,(x, 0) = O;23) Liu= a u.rn u(x, 0) = cp(x);1) Uu2) Uu4) u1 = a uu +fix, t), u(x, 0)2=0.PellleHHe u(x, t) orrpe.[(eJieHo B rrnnyrrnocKOCTH x E R, t > 0.15.45. HcrroJih3YR rrpeo6pa30Batt11e <Dypbe, pelllHTb 3a.[(aqw21) u, = a u 0 , u(O, t) = µ(t), u(x, 0) = O;22) Ur= a u.u, ux(O, t) = v(t), u(x, 0) = 0;3) Ur= a2itx.r + f(x, t), u(O, t) = u(x, 0) = 0.PernettHe u(x, t) orrpe.[(eJietto B o6nacTH 0 < x < oo, t > 0.15.46.
HcrrOJib3YR rrpeo6pa3oBattHe <Dypbe, pelllHTb 3a,[(aq11:1) u, = a\uu+ u,.,.),u(x, y, 0) = <p(x, y);?..2) u, = cr(uu + u,_,.) +fix, y, t), u(x, y, 0) = 0.2PellleH11e u(x, y, t) onpenenetto B nonynpocTPattcrne (x, y) E R , t > 0.15.47. HcnoJib3YR rrpeo6pa3oBaHHe <Dypbe, pelllHTb 3a,[(aqw1) u, = a 2(uu + u_,._,.), u(x, 0, t) = 0, u(x, y, 0) = f(x, y);2) u, = a\uu + u,.y), u(x, 0, t) = f(x, t), u(x, y, 0) = O;3) u, = a\urx + u,y), u,.(x, 0, t) = 0, u(x, y, 0) = f(x, y).Pelllett11e 11(x, y, t) onpeneneHo B o6JiaCTH x E R, y > 0, t > 0.onEPAUHOHHOEHCqHCflEHHE22315.48. J1cnoJih3YH npeo6pa3osam1e JlanJiaca, peurnTh 3a.ua 1u1:u '. =u .\..,. +c/u+f(x),1){ u(O, y) = u)O, y) = 0, x > 0, y > O;Iu ,. = uu + u + B cos x,2) { u(O,y) = Ae-3 ",u,(0,y) =0, x > O,y > 0.115.49.
HaqaJihHaH TeMnepaTypa (npH t = 0) TOHKOfO 0.llHOpO.llHOro cTep)KHH pastta ttymo. Onpe.uemfTh TeMnepaTypy u(x, t) cTep)KHHnpH t > 0:1) CTep)KeHh HMeeT KOHeqttyIO ,UJIHHY l, T. e. 0 < x < l Hu( +O, t) = o(t), u(l - 0, t) = O;2) CTep)KeHb noJiy6eCKOHeqeH, T. e. 0 < x < Cf) Hu( +O, t) = o(t), u( oo, t) = O;3) CTep)KeHh noJiy6eCKOHeqeH, T.
e. 0 < x <Cf) 11u( +O, t) = µ(t), u(oo, t) = 0.15.50. ,UoKa3aTb, qTQ npeo6pa30BaH11e rHJib6epTa H1 cpyHKUl111fE L"(R) o6paruaeTCH B HYJih rnr.ua 11 TOJibKO rnr.ua, Kor.uaf= 0 n. B.15.51. ,UoKa3aTb, qrn llJIH JII06oH: cpyttKu1111 f E L2(R) 11MeeT MeCTO paBeHCTBO[.I Hf(x) 12 dx =[.I f(t) 12 dt,T.
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ITycTh .ueH.crn11TeJibHa5! cpyHKUH5! u E I!'(R ), p > 1. .IJ:oKa3aTb, qrn cpyHKU115!1 I~ u(t)F( z)=- --dt, Im z >O, z=x+iy,TC -z-tnp1rna.uJie)KHT npocipaHcrny Xap.uwHP (Im z > 0).15.56. )J;oKa3aTb, qTo .umr cpyHKUMM F(z) 113 npe.UbI.UYllleii: 3a.uaq11cyruecrnyeTlim:-->1.lm:>OF(z)=F(t)M np11 :noM F(t)= u(t) +iHu(t).I15.57 . .IJ:oKa3aTb, qTQ ,UJI5! 6eCKOHeqHo-,n11cpcpepem:~11pyeMOHcPYHKUMM f c KOMnaKTHblM HOCMTeJieM Ha R+ npeo6pa30BaHHe MeJIHHa Mf eCTb cpyHKU115!, aHaJIHTJ1qecKa5! BO BCeH KOMnJieKCHOHIlJIOCKOCTH.15.58.
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