[aows61]

the problem is
solve the following heat problem using FFCT:
A metal bar of length L, is at constant temperature of [tex]U_0[/tex], at t=0 the end x=L is suddenly given the constant temperature of [tex]U_1[/tex] and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any point x of the bar at any time t>0, assume k=1



Equations used:
heat eq.

[tex]\dfrac{\partial^2 u}{\partial x^2} = \dfrac{1}{k} \dfrac{\partial u}{\partial t}[/tex]

with the additional equations shown on the attached image.



my attempt:
my attempt goes like this:

$$ \frac {\partial^2 u} {\partial x^2} = \frac 1 k \frac {\partial u} {\partial t} $$

[tex]\mathcal{F}_{fc} \left[ \dfrac {\partial u} {\partial t} \right] = \mathcal{F}_{fc} \dfrac {\partial^2 u} {\partial x^2}[/tex]

[tex]\dfrac {dU} {dt} = {-\left( \dfrac {{n} {\pi}} L \right)}^{2} * F(x,t) + \left( {-1} \right)^n \dfrac {\partial{f(L,t)}} {\partial x} - \dfrac {\partial{f(0,t)}} {\partial x}[/tex]

[tex]\dfrac {dU} {dt} = - \left( \dfrac {{n} {\pi}} L \right)^2 * F(x,t) + \left( {-1} \right)^n \dfrac {\partial{f(L,t)}} {\partial x}[/tex]


and i dont know how to continue...

the attached Image:
FFCT.jpg