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Math Help - PDE Separation of variables

  1. #1
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    PDE Separation of variables

    I'm a little lost and need a little help figuring out where to start.

    ut = uxx, 0 < x < 1, t > 0
    u(x,0) = f(x)
    ux(1,t) = u(1,t)
    ux(0,t) = 0

    The fact that u(x,0) is equal to f(x) is throwing me off and I'm not sure How I go about solving for lambda = 0, lambda > 0 and lambda < 0.

    Thanks!
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  2. #2
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    ut = uxx = K implies a solution of the form

    u = K(x^2/2 + t) + Bx + C

    First condition : Kx^2/2 + Bx + C = f(x)
    Second condition: K + B = K/2 + Kt + B + C
    or Kt + C = K/2
    or C = K/2 - Kt
    Third condition: B = 0

    Thus u(x,t) = K/2(1+x^2)

    K is found from condition one.
    K = 2f(x)/(x^2 + 1 - 2t)

    Thus u(x,t) = (1+x^2)f(x)/(1-2t+x^2)
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