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Math Help - Half Line Wave Equation

  1. #1
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    Half Line Wave Equation

    Hi,

    Solve   \ \ \ \frac{\partial{u}}{\partial{t}} = k(\frac{\partial^2{u}}{\partial{t}^2}) \ \ \ \ \ \ \ u(x,0) = e^{-x} \ \ \ \ \ \ \ \ u(0,t) = 0 \ \ \ \ \ on the half line 0<x<\infty.

    I know you are supposed to put this into the equation  \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]\Phi(y)dy but i'm not sure how to simplify it.

    Thanks
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  2. #2
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    Quote Originally Posted by tbyou87 View Post
    Hi,

    Solve   \ \ \ \frac{\partial{u}}{\partial{t}} = k(\frac{\partial^2{u}}{\partial{t}^2}) \ \ \ \ \ \ \ u(x,0) = e^{-x} \ \ \ \ \ \ \ \ u(0,t) = 0 \ \ \ \ \ on the half line 0<x<\infty.

    I know you are supposed to put this into the equation  \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]\Phi(y)dy but i'm not sure how to simplify it.

    Thanks
    What is \Phi (y)?
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  3. #3
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    Basically  \Phi(y) = e^{-y}
    So you "just" need to simplify the integrand. I know you need to use the gaussian at some point and maybe competing the square for the exponents.
    It thus becomes:
     \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]e^{-y}dy
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  4. #4
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    Quote Originally Posted by tbyou87 View Post
    Basically  \Phi(y) = e^{-y}
    So you "just" need to simplify the integrand. I know you need to use the gaussian at some point and maybe competing the square for the exponents.
    It thus becomes:
     \frac{1}{\sqrt{4\pi*kt}}\int_0^\infty[e^\frac{-(x-y)^2}{4kt} - e^\frac{-(x+y)^2}{4kt}]e^{-y}dy
    Open up the paranthesis for (x-y)^2 and (x+y)^2 then distribute over by e^{-y}. That should do it.
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