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Math Help - Heat Equation

  1. #1
    Super Member Deadstar's Avatar
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    Heat Equation

    Use the formula u(x,t) = \frac{1}{\sqrt(4 \pi t)} \int^{\infty}_{\infty} e^{-\frac{(x-y)^2}{4t}} f(y) dy
    to find the solution to the heat equation u_t = u_{xx} on the real line with initial condition u(x,0) = f(x) := e^{-x}.

    My biggest, and maybe only, problem is trying to figure out what to 'do' with the f(y) in the integral, what does it represent?!? As i was typing this though something to do with f(y) = f(x-t) = e^{-x - t} popped into my head however im not sure if this applies to this situation or where i have seen/used it...

    Anyone able to give me nudge in the right direction?
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by Deadstar View Post
    Use the formula u(x,t) = \frac{1}{\sqrt(4 \pi t)} \int^{\infty}_{\infty} e^{-\frac{(x-y)^2}{4t}} f(y) dy
    to find the solution to the heat equation u_t = u_{xx} on the real line with initial condition u(x,0) = f(x) := e^{-x}.

    My biggest, and maybe only, problem is trying to figure out what to 'do' with the f(y) in the integral, what does it represent?!? As i was typing this though something to do with f(y) = f(x-t) = e^{-x - t} popped into my head however im not sure if this applies to this situation or where i have seen/used it...

    Anyone able to give me nudge in the right direction?
    But aren't they telling you that f(x)=e^{-x} ?

    and hence f(y) is just e^{-y} !
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  3. #3
    Super Member Deadstar's Avatar
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    HAHA! I fail... Thats ridiculous how i never noticed that... Should be plain sailing from here!
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  4. #4
    Super Member Deadstar's Avatar
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    The boat has hits rocks almost straight away...

    so...

    u(x,t) = \frac{1}{\sqrt(4 \pi t)} \int^\infty_\infty e^{-\frac{x^2 + y^2 -2xy + 4ty}{4t}} dy

    And i got a hint to complete the square with the y's so i went ahead and did that and ended up with...
    u(x,t) = \frac{1}{\sqrt(4 \pi t)} \int^\infty_\infty e^{-\frac{x^2 -2xy + (y+2t)^2}{4t}}dy...

    Im trying to find a way to seperate this mess so i can use the whole \int^\infty_\infty e^{-x^2} dx = \sqrt(\pi) thing...
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  5. #5
    Moo
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    Hi,

    So you have x^2+y^2-2xy+4ty and you want to integrate with respect to y.

    Group the terms containing y :
    =y^2+2y(-x+2t)+x^2=(y-x+2t)^2-(-x+2t)^2+x^2=(y-x+2t)^2+4xt-4t^2

    now substitute z=y-x+2t and get the factor e^{4xt-4t^2} out from the integral
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