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Math Help - PDE trouble - I'm a little rusty

  1. #1
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    PDE trouble - I'm a little rusty

    Hello,
    Its been a long time since I've done PDE's or even ODE's for that matter, and I'm quite rusty. Could anyone please help me with the following?

    Solve the following PDE and explain why inverting the equations for  x(s, t) and y(s,t) may cause problems:

    u_x + u_y = u subject to u(x, ax) = F(x)

    Find the Jacobian of the coordinate transform to determine when the transformation is invertible.
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  2. #2
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    If we let

    u_s = u_x x_s + u_y y_s

    then choosing

    x_s = 1, y_s = 1\;\;(1) gives  u_s = u_x + u_y = u \;\;(2) (your PDE)

    The boundary conditions. If we let y = ax be s = 0 and x = r then we have

    On s = 0, x = r, y = ar and u = F(r)\;\;(3)

    Integrating (1) and (2) gives

    x = s + A(r),\;\;\;y = s + B(r),\;\;\;u = C(r)e^s

    where A, B and C are arbitrary. Imposing the BCs (3) gives

    x = s + r,\;\;\; y = s + ar,\;\;\;u=F(r)e^s,\;\;(4)

    noting Jacobian of the transformation for x and y vanishes if a = 1.. Now solving the first two of (4) for r and s and substituting into the third of (4) gives your solution.
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