# help in Solving PDE heat problem with FFCT

solve the following heat problem using Finite Fourier Coseine Transform(FFCT): A metal bar of length $L$, is at constant temperature of $U0$, at $t=0$ the end $x=L$ is suddenly given the constant temperature of $U_1$ 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: 1. Heat equation: $$\frac {\partial^2 u} {\partial x^2} = \frac 1 k \frac {\partial u} {\partial t}$$ 2. the following FFCT Equations ( as in the attached pic): FFCT Equations

My attempt at solutions goes like this: $$\frac {\partial^2 u} {\partial x^2} = \frac 1 k \frac {\partial u} {\partial t}$$ $$\mathcal{F}_{fc} \left[ \frac {\partial u} {\partial t} \right] = \mathcal{F}_{fc} \frac {\partial^2 u} {\partial x^2}$$ $$\frac {dU} {dt} = {-\left( \frac {{n} {\pi}} L \right)}ˆ{2} * F(x,t) + \left( {-1} \right)ˆn \frac {\partial{f(L,t)}} {\partial x} - \frac {\partial{f(0,t)}} {\partial x}$$ $$\frac {dU} {dt} = - \left( \frac {{n} {\pi}} L \right)ˆ(2) * F(x,t) + \left( {-1} \right)ˆn \frac {\partial{f(L,t)}} {\partial x}$$

and i don't know how to continue, can you provide the rest of the solution in details please, regards.

• Hello Mr. @Harry49 , here is my complete problem. – aows61 Sep 14 '17 at 16:24
• i just want to make sure you got the problem @Harry49 – aows61 Sep 14 '17 at 16:30
• Hello Mr Harry @Harry49 – aows61 Sep 14 '17 at 16:31
• Please stop the spam! The most straightforward way to solve this PDE problem is separation of variables (see e.g. [1]). – Harry49 Sep 14 '17 at 16:32
• dear Mr. @Harry49 , i already solve it using separation of variables, but am required solve it using the Finite Fourier Cosine Transform (FFCT), and i tried but stopped somewhere. – aows61 Sep 14 '17 at 16:37