# Fourier Series, Diffusion

Cuthbert Nyack
Fourier Series can also be used to help find solutions of the Diffusion equation shown below.

This equation can be solved by separation of variables along with boundary conditions. Assuming a bar of length and initial boundary conditions given by
y(x,0) = 1 for 0 < x £ L/4 y(x,0) = 0 for L/4 < x £ L.
y(0,t) = y(L,t) = 0 for t ³ 0.
Then the solution is shown below.
The part of the solution in red is the Fourier Series of the initial distribution of y.
In the applet below, the initial distribution is shown in magenta and can be changed by changing y(0) and y(L/4). y can be assumed to be temperature. red curve shows the variation of the distribution in time and yellow shows the evolution of the spectrum. There is a very rapid decay of the higher frequencies because of the high temperature gradients they contain so that at large t only the fundamental remains regardless of the starting distribution. N is the number of terms in the Fourier Series which have been summed to obtain the distribution.

When activated the following gif image show how the applet should appear.