Heat Transfer along a Rod

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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 .


Contributed by: Stephen Wilkerson (April 2011)
(Department of Mathematical Sciences at the United States Military Academy, West Point, NY)
Open content licensed under CC BY-NC-SA




[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.

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