上游通大学 SHANGHAI JIAO TONG UNIVERSITY 4,=a2ua, 0<x<l,t>0 类齐次边界条件 u(0,t)=0=u(l,t),t>0 u(x,0)=(x), 0≤x≤1 解：令 九=0→X=B。 九>0时，X"+九X=0→ u(x,t)=X(x)T(t) X=Asin几x+Bcos几x, X"+X=00<x<1 X'(0)=0,X'(I)=0 X'(0)=AV元=0，X'(U=-BV元sin√l=0 T'+a2T=0 因有他人司0=123礼 2<0,无非零解. →固有函数X,(x)=B.cos":u(x,t) = X (x)T(t) 2 T aT ′ + = λ 0    ′ = ′ = ′′ + = < < (0) 0, ( ) 0 0 0 X X l X λX x l II类齐次边界条件 解：令 λ < 0, 无非零解. 0 λ =⇒ = 0 X B 0, 0 sin cos , X X X A xB x λ λ λ λ > + =⇒ ′′ = + 时 2 , 0 ,0 (0, ) 0 ( , ), 0 ( ,0) ( ), 0 t xx x x u au x lt u t u lt t ux x φ x l  = << >   = = >  = ≤ ≤  X A Xl B l ′ ′ (0) 0, ( ) sin 0 == = λ λλ − = 2 , 1,2,3, n n n l π λ   ⇒ ==     固有值 L ( ) cos n n n Xx B x l π ⇒ = 固有函数