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Math Help - PDE using method of characteristic strips

  1. #1
    Senior Member
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    Feb 2008
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    PDE using method of characteristic strips

    Hi guys. I'd like to solve the following:

    u=xu_x+yu_y+\frac{1}{2}((u_x)^2+(u_y)^2),

    u(x,0)=\frac{1}{2}(1-x^2).
    This corresponds to the problem F=xp+yq+\frac{1}{2}(p^2+q^2)-z=0.

    Then the characteristic equations are given by

    \frac{dx}{dt}=x+p, \frac{dy}{dt}=y+q, \frac{dz}{dt}=p(x+p)+q(y+q), \frac{dp}{dt}=0, \frac{dq}{dt}=0.

    And it's easy to find the curves (there are two) induced by the initial condition:

    \Gamma=(x(s,0),y(s,0),z(s,0),p(s,0),q(s,0))=(s,0, \frac{1}{2}(1-s^2),-s,\pm 1)

    Solving for z we have

    z=-xs\pm y+\frac{1}{2}(s^2+1)

    (where the signs \pm correspond to the signs in \Gamma).

    But I don't know how to put this in closed form (i.e. in the form z=z(x,y)). It may just be a simple elementary calculus/algebra block on my part. Any help would be much appreciated!
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  2. #2
    Senior Member
    Joined
    Feb 2008
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    Re: PDE using method of characteristic strips

    NEVERMIND...

    I'm an idiot.... : (
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