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Math Help - PDE help

  1. #1
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    PDE help

    1. Consider the heat equation in a two-dimensional region with an insulated boundary. An isotherm is a line of constant temperature, i.e., a level curve of the function u(x,t). Show that the isotherms are always perpendicular to the boundary.

    I know that the level curve is going to be perpendicular to the gradient of the function, but I don't know where to go with that.

    2. Consider the heat equation with internal heat source on a rod of length L:

    u_t = u_xx + x - B
    u(x,0) = f(x)
    u_x(0,t) = u_x(L, t) = 0

    For what values of the constant B does an equilibrium temperature distribution exist, and for each such B, what is the equilibrium distribution?

    For 2, I got that the equilibrium temperature exists when B = .5L, but I don't know how to find the equilibrium distribution.

    Thanks in advance!
    Last edited by Math Major; September 2nd 2009 at 03:01 PM.
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  2. #2
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    Quote Originally Posted by Math Major View Post
    1. Consider the heat equation in a two-dimensional region with an insulated boundary. An isotherm is a line of constant temperature, i.e., a level curve of the function u(x,t). Show that the isotherms are always perpendicular to the boundary.

    I know that the level curve is going to be perpendicular to the gradient of the function, but I don't know where to go with that.

    2. Consider the heat equation with internal heat source on a rod of length L:

    u_t = u_xx + x - B
    u(x,0) = f(x)
    u_x(0,t) = u_x(L, t) = 0

    For what values of the constant B does an equilibrium temperature distribution exist, and for each such B, what is the equilibrium distribution?

    For 2, I got that the equilibrium temperature exists when B = .5L, but I don't know how to find the equilibrium distribution.

    Thanks in advance!
    For #2, at equilibrium u_t = 0 so

     <br />
u_{xx} = B - x<br />

    Now integrate twice (your condition B = \frac{1}{2} L will satisfy the insulated BC's).
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