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).

See also 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

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