《数学物理方程》课程PPT教学课件（讲稿）第八章（8-1）习题3

19 q=qsnφ 在边界上,热流等于自由冷却都热流 加太阳光投射产生的热流。故 (ku,+ Huloso=f(o f(o) (0mp"-(Am cos mo+Bm sin mo) (ku,+Huea=H(Co+2a"(An, cos mo+Bn sin mo ))+k2ma-(An cos mo +Bn sin mo)=f(p) m=l HCo+>(Ha"+kma"-)(Am coSmo+Bm sin mp)I,=f(o)

19 a q    q' = qsin  在边界上，热流等于自由冷却都热流 加太阳光投射产生的热流。故 ( ) ()   k u Hu f + =a =        = 0 ( 2 ) sin (0 ) ( )        q f ( , ) ( cos sin ) 1 u   C0  Am m Bm m m m = + +  = ( , ) ( cos sin ) 1 1 u   m Am m Bm m m m =  +  = − ( 边界条件 ) { ( cos sin )]} ( cos sin ) ( ) 1 1 1 k u Hu  H C0 a A m B m k m a Am m Bm m f  m m m m m m + a = + + +  + =  = −  = = ( )( cos sin )]} ( ) 1 1 HC0 Ha k ma Am m Bm m f  m m m = + + + = −  = 

HCo+>(Ha"+kma"-)Am cosmo+ Bm sin mp )1)=f(p) (Ha +kmd")Am=f()cos modo=sin p cos mado q qp 丌2(1+m)2(m-1)2(m-1)2(m+ 丌2k+12k 丌2k+12k-1) 丌4k q (h"+km-)Bnm=兀 f(o)sin modo p sin madp=2 (m=1) B q 1) 2(Ha+k) )/(Ha2+kmd l(,q) H2加+行Sm0 cos 2np za2m1(4k2-1)(Ha+2mk)

( )( cos sin )]} ( ) 1 1 HC0 Ha k ma Am m Bm m f  m m m = + + + = −  =      d q  = 0 sin 2     HC f d  = 2 0 0 ( ) 2 1    0 2 con q = −  q = H q  C0 =      Ha kma Am f m d m m ( )cos 1 ( ) 0 1  + = −      m d q sin cos 0  = ] 2( 1) ( 1) 2( 1) ( 1) 2( 1) 1 2(1 ) 1 [ 1 1 + − − − − + − − + = − + m m m m q m m  ] 2 1) 1 2 1 1 [ − − + = k k q  4 1 2 1 ] 2 1) 1 2 1 1 [ 2 − = − − − + = k q k k q        Ha kma Bm f m d m m ( )sin 1 ( ) 0 1  + = −      m d q sin sin 0  =      = = 0 ( 1) ( 1) 2 m m  )/( ) 4 1 2 1 ( 2 2 1 2 − + − = − k k m Ha kma k q A         n a k Ha nk q Ha k q H q u n n n cos2 (4 1)( 2 ) 2 sin 2( ) ( , ) 2 1 2 2 1 − + − + = + −  =  2( ) 1 Ha k q B + =

20 uep. =f() ulpeps =f i(o) u(p,)=Co+Do In p+>(CmP"+DmPm)(Am cos mp+Bm, sin mp) p2 f()→f(q)=C+Dhn+∑(CnP+Dn)(A1 cos mpp+ b sin m 2m=2=f2(9) f2()=Co+DoIn p2+2(CmP2"+DmP2m)(Am cos mo+Bm sin mp) Co+Do In p,=o f()dp=fie (CmP+Dmp)a f()cos mondo=f +dIn f2(lo=f2 nPi)B do=fim (CmP2+DmP2")A、2 f,()cos mado=f, (CmP"+DmP2m)Bm ==f2(p)sin modo=f2

20. 1 2 ( ) u  =1 = f 1  ( ) u = 2 = f 2  ( , ) ln ( )( cos sin ) 1 u   C0 D0  C  D  Am m Bm m m m m m = + + m + +  = − ( ) u  =1 = f 1   ( ) ln ( )( cos sin ) 1 f 1  C0 D0 1 C 1 D 1 Am m Bm m m m m m = + + m + +  = − ( ) 2 2 u = = f   ( ) ln ( )( cos sin ) 1 f 2  C0 D0 2 C 2 D 2 Am m Bm m m m m m = + + m + +  = − 1 10 2 0 0 0 1 ( ) 1 C + D ln = f d = f       20 2 0 0 0 2 2 ( ) 1 C + D ln = f d = f       1 1 2 0 1 1 1 ( )cos 2 ( ) m m m m m m C + D A = f m d = f  −        2 2 2 0 2 2 2 ( )sin 2 ( ) m m m m m m C + D B = f m d = f  −        2 1 2 0 2 2 2 ( )cos 2 ( ) m m m m m m C + D A = f m d = f  −        1 2 2 0 1 1 1 ( )sin 2 ( ) m m m m m m C + D B = f m d = f  −       

D In PI Co=fro-Do In p=fio-hno fro f2 In plp2 In Plp2 In p2-fio Cm AmPIm+Dm Am=fimIp2" Cm Am P2m+D Am=f2mlPp2 CA CmBmPi +DmBm=fim2P' CmBmP2+DmBm=f2m2P2 Am, +Anprm=fimf +DaMp fmlp, fr

10 20 2 1 1 0 D ln = f − f   1 0 1 2 2 2 0 1 2 1 1 2 1 0 2 0 0 1 0 0 1 1 0 1 ln / ln ln / ln ln / ln ln f f f f C f D f           = − − = − = − m m m m m m m m f f C A 2 2 2 1 1 1 1 2 1 2     − − = m m m m m m m C B D B f 2 2 2 2 2 + =  m m m m m m m C A D A f 2 1 2 2 2 + =  m m m m m m m C B D B f 1 2 1 2 1 + =  m m m m m m m C A D A f 1 1 2 2 1 + =  m m m m m m m C A D A f − − + = 1 1 1 2 1  m m m m m m m C A D A f − − + = 2 1 2 2 2  m m m m m m m m f f C B 2 2 2 1 1 2 1 2 2 2     − − = m m m m m m m m f f D A 2 2 2 1 1 1 1 2 1 2 − − − − − − =     m m m m m m m m f f D B 2 2 2 1 1 2 1 2 2 2 − − − − − − =    