Solving the heat equation

  1. 1. The problem statement, all variables and given/known data

    Let [itex]s = x/\sqrt{t}[/itex] and look for a solution to the heat equation [itex]u_{t} = u_{xx}[/itex] which is of the form u(x,t) = f(s) and satisfies the IC u(x, 0) = 0 and the BC u(0, t) = 1 and u(∞, t) = 0.

    2. Relevant equations

    [itex]∫e^{x^{2}} = \sqrt{\pi}[/itex]

    3. The attempt at a solution
    Let f(s) = u(x, t)
    Substituting f(s) into the heat equation:
    [itex]\frac{-xf'(s)}{2t^{3/2}} = \frac{f''(s)}{t}[/itex]
    [itex]s = x/\sqrt{t}[/itex]
    [itex]-0.5sf'(s) = f''(s)[/itex]
    It's now just an ODE.
    [itex]f'(s) = g(s)[/itex]
    [itex]-0.5sg(s) = g'(s)[/itex]
    [itex]ln(g(s)) = -0.25s^2 + constant[/itex]
    [itex]g(s) = Ke^{-0.25s^{2}} = f'(s)[/itex]
    [itex]f(s) = ∫f'(s)ds = Ke^{0.25}∫e^{-s^{2}}ds = Ke^{0.25}\pi^{1/2}[/itex]

    So now I'm left with a trivial solution of f(s) = u(x,t) = constant. I don't think this is what u(x,t) is supposed to be. Which part did I do incorrectly?
     
  2. jcsd
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