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Math Help - Partial Differential Equations

  1. #1
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    Partial Differential Equations

    Hey guys,

    I am stuck on solving the following semi-infinite wave equations:

    Ut = Uxx 0<x<infinity, 0<t
    Ux (0,t) = 0
    U(x,0) = f(x)
    Ut(x,0) = 0

    I have no idea how to get started. Can anybody get me started in the right direction, thanks.
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  2. #2
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    Quote Originally Posted by spearfish View Post
    Hey guys,

    I am stuck on solving the following semi-infinite wave equations:

    Ut = Uxx 0<x<infinity, 0<t
    Ux (0,t) = 0
    U(x,0) = f(x)
    Ut(x,0) = 0

    I have no idea how to get started. Can anybody get me started in the right direction, thanks.
    What you give is a "diffusion equation", not a "wave equation". Wasn't the equation really U_{tt}= U_{xx}? How you would solve it depends on how much you already know.

    The most basic method would be to "separate" the variables by looking for a solution of the form U(x,t)= X(x)T(t) and getting two ordinary differential equations for X and T separately. The general solution, for a finite interval, would be a sum of such things and, for an infinite interval, an integral.

    Or you could "shortcut" that process by assuming the result- that the solution can be written as a "Fourier Transform" of the form \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty T(t)e^{-isx}ds. Then U_{tt}= \frac{1}{\sqrt{2\pi}\int_{-\infty}^\infty T' e^{-isx}ds and U_{xx}= \frac{-ix}{\sqrt{2\pi}}\int_{-\infty}^\infty Te^{-isx}ds so that your equation is \frac{1}{\sqrt{2\pi}\int_{-\infty}^\infty T' e^{-isx}ds= \frac{-is}{\sqrt{2\pi}}\int_{-\infty}^\infty Te^{-isx}ds or \int_{-\infty}^\infty (T"- T)e^{-isx}ds= 0. In order for that to be true you must have T"+ isT= 0 for all t. Solve that for T and then use the conditions to determine the constants involved.
    Last edited by HallsofIvy; December 10th 2009 at 06:36 AM.
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  3. #3
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    Oops! I meant heat equation, not wave equation. So I will apply apply "Separation of variables to see what I get" since I am more comfortable with this method and see if it turns out ok. I ll post my answer later.
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