丌 e cos bxe C 4at 丌1 丌 Vat 2avnt F e 2a√m(t-r) l(x,)=0(x)* 1 2a√(t-r) T'f(x, 2)2aT(t-T)
ú ú û ù ê ê ë é = ò ¥ - - 0 4 2 2 2 1 cos a e bxdx e a b ax p Q (11) 2 1 2 1 1 2 2 2 2 4 2 4 a t x a t x e a t a t e - - × = p p p ( ) [ ] ( ) w t ( t ) p t - - - - - - = a t x a t e a t F e 2 2 2 2 1 4 2 1 ( ) ( ) p ( t ) j - = * a t u x t x 2 1 , ( ) ( ) ( ) ò - - - + * t d a t x e a t f x 0 4 2 2 2 1 , t t p t t
∫0G)+7d 2a√tJ-a 1 f(,τ) dsdt a兀 t-7 a2△u=0 记小0= FloG] o()e dri=g(o)
( ) ( ) ( ) x t x t p t t x f e d d a t a t x t - - - ¥ ò ò- ¥ - + 2 2 4 0 , 1 2 1 ( ) ( ) ò ¥ - ¥ - - = j x x p x e d a t a t x 2 2 4 2 1 2. 0 2 u t - a D u = u (r ) t r = 0 = j [ ( )] ( ) ( ) òòò¥ -¥ - × F u r t = u r t e dr = u t i r , ~ , , w r r w r r r r 记 [ ( )] ( ) ( ) òòò¥ -¥ - × j = j = j w r r w r r r ~ r F r r e dr i r
注意到M! d r ∫ (01x+02y+02) dxdydz 2 OX az Ox- dy 0z 02v(01y,2),02v(o1,y,z) z)+ 10 3 ave dz
[ ] òòò¥ -¥ - × ÷ ÷ ø ö ç ç è æ ¶ ¶ + ¶ ¶ + ¶ ¶ = e dr z u y u x u F u iw r r r r D 2 2 2 2 2 2 注意到 ( ) òòò¥ -¥ - + + ÷ ÷ ø ö ç ç è æ ¶ ¶ + ¶ ¶ + ¶ ¶ = e dxdydz z u y u x u i x y z 1 2 3 2 2 2 2 2 2 w w w ò ò ò ¥ -¥ ¥ -¥ - - - ÷ ÷ ø ö ç ç è æ ¶ ¶ + ¶ ¶ + ¶ ¶ = e dxe dye dz z u y u x u i x i y i z 1 2 3 2 2 2 2 2 2 w w w ( ) ( ) ( ) ( ) ò ò ¥ -¥ ¥ -¥ ú û ù ê ë é ¶ ¶ + ¶ ¶ = + 2 1 2 2 1 2 1 2 1 , , ~ , , ~ , , ~ z u y z y u y z i u y z w w w w e dye dz i y i z - w 2 - w 3 ×
(0,02) , O1(01,02,2)-02(01O2,2)+ az dz =-O12i(o1,O2,O2)-O2l(0,0200)-03i(0,0203) O2n(o1.02,03)=02() du(o,t) 于是 a202i(0)=0
e dz i z - w 3 × ( ) ( ) ( ) ò ¥ -¥ ú û ù ê ë é ¶ ¶ = - - + 2 1 2 2 1 2 2 1 2 2 2 1 , , ~ , , ~ , , ~ z u z u z u z w w w w w w w w ( ) ( ) ( ) 1 2 3 2 1 2 3 3 2 1 2 3 2 2 1 , , ~ , , ~ , , ~ = -w u w w w -w u w w w -w u w w w = -w (w w w ) = w (w ) (*) r u u ~ , , ~ 2 1 2 3 2 于是 ( ) ( ) 0 , ~ ~ 2 2 - w w = w r r a u dt d u t (w ) j (w ) r ~ r , 0 ~u =
对照(9),(10)的解有 (0,)=q(0)°0 n(,)=F[(,1 =Fo(oe a ot F-F oG=Fero I
对照(9),(10)的解有 ( ) ( ) a t u t e 2 2 ~ , ~ w w j w - = r r u(r t) F [u ( ,t)] ~ , 1 w r r - \ = [ ( ) ] a t F e 2 2 1 ~ w j w - - = r [ ( ) [ ]] a t F F r F e 2 2 1 ~ 1 w j - - - = * r