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Math Help - help with PDE

  1. #1
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    help with PDE

    Can someone help me with this, it's slightly different from other separable equation PDE's I've done before:

    Consider a thin rectangular plate (0 <= x <= pi, 0 <= y <= pi/2) with the following steady-state temperature distributions on the edges
    u(x,0) = 0
    u(0,y) = 0
    u(pi,y) = 0
    u(x, pi/2) = 60sinx + 20sin2x

    Find u(x,t) to determine the rate of change.

    Thank you very much for any help. It's greatly appreciated.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    Can someone help me with this, it's slightly different from other separable equation PDE's I've done before:

    Consider a thin rectangular plate (0 <= x <= pi, 0 <= y <= pi/2) with the following steady-state temperature distributions on the edges
    u(x,0) = 0
    u(0,y) = 0
    u(pi,y) = 0
    u(x, pi/2) = 60sinx + 20sin2x

    Find u(x,t) to determine the rate of change.

    Thank you very much for any help. It's greatly appreciated.
    The solution is given by u_{xx}+u_{yy}=0 on [0,\pi]\times [0,\tfrac{\pi}{2}].
    Look for a solution u(x,y) = X(x)Y(y).
    Notice that, \frac{X''}{X} = - \frac{Y''}{Y}.
    Therefore, \frac{X''}{X} = k \text{ and }\frac{Y''}{Y} = -k.

    Case k=0: Then XY = (ax+b)(cy+d). Does this satisfy the boundary conditions?

    Case k>0: Then XY = \left( a\sinh(\sqrt{k}x) + b\cosh(\sqrt{k}x) \right) \left( c\sin (\sqrt{k}y) + d\cos (\sqrt{k}y) \right). Does this satisfy the boundary conditions?

    Case k<0: Then XY = \left( a\sin(\sqrt{k}x) + b\cos(\sqrt{k}x) \right) \left( c\sinh (\sqrt{k}y) + d\cosh (\sqrt{k}y) \right). Does this satisfy the boundary conditions?

    Try to work out the problem from here.
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