# The Heat Equation

This demonstration illustrates the behaviour of solutions of the heat equation

u_{t} = a^2 u_{xx}
u(0,t) = 0
u(l,t) = 0
u(x,0) = f(x)

In this example a=1, l=10 and the intial amplitude consists of one bump centered on x=5, just as in the wave equation demonstration. The demonstration plots the solution given by separation of variables that you have found in class. When the demonstration starts, the initial amplitude is plotted. By clicking the "Advance time" button, you instruct the computer to increase the time by an amount specified in the "time interval window". Separation of variables expresses the solution as a sum

b_1(t) sin(pi x/l) + b_2(t) sin(2 pi x/l) + ...

of modes. The demonstration also plots the values of the first six coefficients b_k(t). Contrast the time dependence of these coefficients with that of the corresponding coefficients for the wave equation.

The demonstration below is an applet. Google Chrome, Firefox, Safari and Microsoft Edge no longer execute applets natively because of security issues with NPAPI plugins. However applets can be played in the Google Chrome browser by using the CheerpJ Applet Runner extension.