File:Heatequation exampleB.gif

Here the horizontal axis represents the location along a bar of metal and the graph records the temperature at that location. It begins with an initial temperature which is hot at one side and cool at the other, and then shows how the temperature of the bar approaches an equilibrium. It is assumed that no heat is lost from the bar and that there are no heat sources. This demonstrates two key properties of the heat equation: approaching an equilibrium, and the maximum principle. The maximum principle says that the temperature will always have a maximum either earlier in time or at the ends of the bar.

Description:
Graphical representation of the solution to the heat equation for an "infinite slab" of width 1 given by:

${\displaystyle \ u_{t}=ku_{xx}}$

where k = .061644 subject to the boundary conditions:

${\displaystyle u_{x}(0,t)=0,\ \ u_{x}(1,t)=0}$

and with the initial heat distribution given by:

${\displaystyle \ u(x,0)=\cos(2x)}$

In this case, the left face (x=0) and the right face (x=1) are perfectly insulated. This image shows how the heat redistributes, flowing from the warmer left edge to the cooler right edge, then equalizing to a constant temperature throughout. This temperature happens to be the average value of cos(2x) over [0,1], as one might expect.

The solution:

${\displaystyle u(x,t)={\frac {\sin(2)}{2}}+\sum _{n=1}^{\infty }A_{n}\cos \left(n\pi x\right)\exp \left(-kn^{2}\pi ^{2}t\right)}$

where:

${\displaystyle {\begin{aligned}A_{n}&=2\int _{0}^{1}\cos(2x)\cos(n\pi x)dx\\&=(-1)^{n}{\frac {-4\sin(2)}{n^{2}\pi ^{2}-4}}\\\end{aligned}}}$

Solution details: