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Partial Differential Equations

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Utt-Uxx+2Uxy-Uyy=0
    with the conditions:
    U(1,x,y)=cos(x)+ey
    Ut(1,x,y)=sin(x)-y2

    2. Relevant equations

    Not using separation of variables to solve.

    3. The attempt at a solution

    I've gotten the general equation to be of the form:

    U(t,x,y)=ψ(x+t,y-t)+ζ(x-t,y+t)
    but solving for the initial conditions is giving me a problem:

    ψ(x+1,y-1)= sin(x)+ey-ζ(x-1,y+1)

    ψ1(x+1,y-1)=-cos(x)-ζ1(x-1,y+1)

    ψ2(x+1,y-1)=-ey2(x-1,y+1)

    subbing that into the Ut and simplifying gives me this:

    2 ζ 1 +2 sin(X) = 2ζ2-ey +y2

    which I'm not sure what to do with...

    help greatly apreciated
     
  2. jcsd
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