第5章 边值问题

1、2020)(12drdrdrdr0)(drdrdrd1Cdrdr1Crdrd21lnCrC)(r0201222drdrdrdr21CrC /UarUarraUar21CrCBrArln)(110)(12r0)(22rDrErln)(225.4 分离变量法求解拉氏方程 0.2222222zyxZYXzZyYxXzyx)()()(),(.22222222dxXdZYzyx0111222222dzZdZdyYdYdxXdXZYX0111222222dzZdZdyYdYdxXdX2221xkdxXdX2221ykdyYdY2221zkdzZdZ0222XkdxXdx0222YkdyYdy0222Zkd

2、zZdz0222zyxkkk02XkXx02xkkkxkxBkxAXcossinjkxjkxBeAeX02xk0 xkBxAX02xkjkkxkxBkxAXcoshsinhkxkxBeAeX)()()(),(zZyYxXxyx00),( ,)0 ,( 0),(), 0( 0Ubxvuxvyavyvvvyyxx00),( ,)0 ,( 0),(), 0( 0Ubxvuxvyavyvvvyyxx)()(),(yYxXyxv0 XX0 YY0)()0( , 0 aXXXX0)()0( , 0 aXXXX本征值为：), 2 , 1( )/(2nann本征函数为：)/sin()(axnxXn0 YYay

3、nnaynneBeAyY/)(于是由叠加原理得到v的通解为：1/)/sin()(),(naynnaynnaxneBeAyxv00),( ,)0 ,(Ubxvuxv10)/sin()()0 ,(nnnaxnBAuxv1/0)/sin()(),(nabnnabnnaxneBeAUbxvnannnudxaxnuaBA) 1(12sin2000naabnnabnnnUdxaxnUaeBeA) 1(12sin2000/1)/sin()()0 ,(nnnaxnBAxv01122222zuurrurrrDrCruln)(0 ZZ0)(122 RrnRrR)()()(rFNrEJrRnn是实数其中T)()(0

4、0TrFNTrEJ )sin()cos()(zDzCzZ)()()(rBKrAIrRnn)()(00rBKrAI100sincoslnmmmmmmmmBmArDrCrDC),(rummmmrmBmAru0 sincos),(mmmmrmBmAru sincos),(00)1()(cosnnnnnnPrBrABrArln)(0|Var0|br 0lnln)(Vrabrbbar,01drdrdrdrh hq qx xy yz zh hq*q*0|0z0|x0|z0|y),(),(zyxzyx),(),(zyxzyx002z*0*114),(qqqqRRqzyxh hq qx xy yz zh hq*q*qR*qR( (x,y,zx,y,z) )zqahPRzqahPR*ll*ll00*0022rMPlrPMlrdrrdrPMMPlrMPrPMl*0*02200.*所在直线称为电轴l常数MPPM*OMPMOP*PMMPlrMPrPMl*0*0.2.200常数MPPMl*0ln2*OPOMOMOPMPPM*OPaadMPPMdaOP2*qq*0*41MQMQrQrQ00*MQMQrQrQDadOQQDaQ2*cos2cos2222*222addaraDDarMQMQcos4