This Demonstration shows the solution to the heat equation for a one-dimensional rod. The rod is initially submerged in a bath at 100 degrees and is perfectly insulated except at the ends, which are then held at 0 degrees. This is a a Sturm–Liouville boundary value problem for the one-dimensional heat equation

,

with boundary conditions , , and , where is time, is distance along the rod, is the length of the rod, and .

The solution is of the form

,

where is the conductivity parameter (a product of the density, thermal conductivity, and specific heat of the rod) and

.

If you increase the number of terms , the solution improves as long as the time is small. As (the final state), the entire rod approaches a temperature of 0 degrees. You can see the effect of the thermal properties by varying the conductivity parameter .

[1] R. Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 4th ed., Saddle River, NJ: Prentice Hall, 2003.

[2] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.