Heat equation with circular symmetry.
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(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,
Since is a constant (independent of r), it can be pulled ...
The boundary value problems for circular symmetry are determined.