ABSTRACT
Since is a complete set, we may consider for the solution an ex pansion of the form
OO u(x , t ) = y ^ c n( t )X n (x). (7.1)
oo oo ^ 2 c ' n ( t )Xn (x) = k ^ 2 c n( t ) X ”(x) + q(x,t)
= - k ^ 2 c n (t) X„Xn (x) +q(x , t ) 71= 1
or OO
K W + k \ ncn( t )]Xn(x) = q(x,t). n = l
Multiplying this equality by X m (x), integrating from 0 to L, and taking the orthogonality of the X n on [0, L\ (see Theorem 3.8(iii)) into account, we find tha t
kXmcm (t)] j x ^ ( x ) d x = J q ( x , t ) X m (x) d x , 0 0
which (with m replaced by n) yields the equations
L f q ( x , t )X n(x) dx
c'n (t) + kXncn (t) = -— ------------------ , t > 0, n = l , 2 , . . . . (7.2) / X%(x) dx o
The BCs are automatically satisfied since each of the X n in (7.1) satisfies them.
