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.

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.

COPYRIGHT © 2005 Cuthbert Nyack.