the problem is

solve the following heat problem using FFCT:

A metal bar of length L, is at constant temperature of $\displaystyle U_0$, at t=0 the end x=L is suddenly given the constant temperature of $\displaystyle U_1$ 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.

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

with the additional equations shown on the attached image.

my attempt:

my attempt goes like this:

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

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

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

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

and i dont know how to continue...

the attached Image: