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    Boundary Value Problems for Circular Symmetry

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    Heat equation with circular symmetry.

    Please see attached.

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    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.