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# Heat Conduction Equation--Disk

To solve the heat conduction equation on a two-dimensional disk of radius , try to separate the equation using

 (1)

Writing the and terms of the Laplacian in cylindrical coordinates gives

 (2)

so the heat conduction equation becomes

 (3)

Multiplying through by gives

 (4)

The term can be separated.

 (5)

which has a solution

 (6)

The remaining portion becomes

 (7)

Dividing by gives

 (8)

where a negative separation constant has been chosen so that the portion remains finite

 (9)

The radial portion then becomes

 (10)
 (11)

which is the spherical Bessel differential equation.

Consider disk or radius with initial temperature and the boundary condition . Then the solution is

 (12)

where is the th positive zero of the Bessel function of the first kind (Bowman 1958, pp. 37-39).

Heat Conduction Equation

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## References

Bowman, F. Introduction to Bessel Functions. New York: Dover, 1958.Carslaw, H. S. and Jaeger, J. C. "Some Two-Dimensional Problems in Conduction of Heat with Circular Symmetry." Proc. London Math. Soc. 46, 361-388, 1940.

## Cite this as:

Weisstein, Eric W. "Heat Conduction Equation--Disk." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HeatConductionEquationDisk.html