Consider a thin rod of length [math]L[/math] with an initial temperature [math]f\left(x\right)[/math] throughout and whose ends are held at temperature zero for all time [math]t>0[/math]. The temperature [math]u\left(x,t\right)[/math] in the rod is determined from the boundary-value problem:[br][math]\frac{\partial u}{\partial t}=c^2\frac{\partial^2u}{\partial x^2}[/math] 0<[i]x[/i]<[i]L[/i] and [i]t[/i]>0;[br][math]u\left(0,t\right)=u\left(L,t\right)[/math] [i]t[/i]>0;[br][math]u\left(x,0\right)=f\left(x\right)[/math] 0<[i]x[/i]<[i]L[/i].[br][br]In the following simulation, the temperature [math]u\left(x,t\right)[/math] is graphed as a function of [i]x[/i] for various fixed times.[br][br][b]Things to try: [/b][br][list][*]Change the initial condition u(x,0)=f(x).[br][/*][*]Explore the solutions by clicking on the buttons or type a number to show the graph.[br][/*][/list]