# Boundary Value Problems for Circular Symmetry

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Heat equation with circular symmetry.

Please see attached.

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#### Solution Preview

See attached

(a) In Cartesian coordinates, if u(x, y, t) is the temperature field and H(t) is the amount of heat (in calories, say) contained in an area A, then

, where c is the specific heat of the material and is the density.

In circular coordinates, dx dy = r dr d , where r ranges from a to b and ranges from 0 to . If u(r, t) is the temperature distribution, then

= .

Plugging in the limits of integration,

H(t) =

=

Since is a constant (independent of r), it can be pulled ...

#### Solution Summary

The boundary value problems for circular symmetry are determined.

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