
Heat Diffusion Equation
The heat diffusion equation is $\displaystyle u_t = Du_{xx}$. A rectangular block of dimensions $\displaystyle 0 \leq x \leq L, 0 \leq y \leq W \ \text{and} \ 0 \leq z \leq H$ has an initial temperature profile of $\displaystyle f(x,y,z)$. If all sides are held at a fixed temperature $\displaystyle u = 0$, find a formal expression for the temperature at later times.
For the case $\displaystyle f(x,y,z) = C$ obtain an explicit expression for the coefficients in the Fourier series.