Analytical Solution

Joseph Fourier published a significant text titled “Theorie Analytique de la Chaleur” in 1822. He proposed the expansion of arbitrary functions in infinite sums of trigonomic functions, sin() and cos(), to precisely solve the heat equation.

The solution of a $1-D$ heat equation for a simple Dirichlet boundary condition and no heat source is reproduced in the <a href=''http://en.wikipedia.org/Heat_equation''> Wikipedia site.</a>

This solution is very similar to that for the elliptic PDE. It starts with a separation of variables, $X(x)$ function of $x$ and $T(t)$ function of $t$, and imposes the Boundary conditions. This brings in the Fourier sums, and mixes the variables. The details are found in yhe Wikepidia web site.