Originally Posted by
linalg123 Question: Find the temperature u(x,t) in a rod of length L if the initial temp is f(x) throughout and the ends are x=0 and x=L are insulated.
Solve if L=2 and f(x) = {x , 0 < x <1
{0 , 1 < x <2
attempt:
k d^2(u)/dx^2 = du/dt ; 0<x<L, t>0
Insulated ends means: u(0,t)=0 u(L,t) = 0
u(x,0)= f(x) ; 0 <x<L
let u(x,t) = X(x)T(t)
d^2(u)/d(x)^2 = TX'' ; du/dt = XT'
kTX'' = XT'
X''/X = T'/(kT) = -λ^2
X'' + λ^2X = 0
T' + λ^2kt = 0
X'(0)=0 X'(L)=0
for λ=0 , X(x) = ax+b
but from the boundary condition a=0 , X(x)=b
for λ^2 > 0
X= acosλx + bsinλx
X' = λ(-asinλx +bcosλx)
this is where i am getting stuck. any help would be great, thanks