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Thread: Half Line Wave Equation

  1. #1
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    Half Line Wave Equation

    Hi,

    Solve $\displaystyle \ \ \ \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 $\displaystyle 0<x<\infty$.

    I know you are supposed to put this into the equation $\displaystyle \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 $\displaystyle \ \ \ \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 $\displaystyle 0<x<\infty$.

    I know you are supposed to put this into the equation $\displaystyle \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 $\displaystyle \Phi (y)$?
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  3. #3
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    Basically $\displaystyle \Phi(y) = e^{-y}$
    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:
    $\displaystyle \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 $\displaystyle \Phi(y) = e^{-y}$
    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:
    $\displaystyle \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 $\displaystyle (x-y)^2$ and $\displaystyle (x+y)^2$ then distribute over by $\displaystyle e^{-y}$. That should do it.
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