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Math Help - Solution of Diffusion (Heat) equation

  1. #1
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    Solution of Diffusion (Heat) equation

    Hi, another diff. eq problem:

    In my notes, it suggests to use a similarity transformation to solve the heat equation in one dimension \frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}.

    So if \xi = \frac{x}{2\sqrt{Dt}} and u(x,t) = v(\xi), then:

    \frac{\partial u}{\partial t} = \frac{v}{\xi}\frac{\partial \xi}{\partial t} =. . .

    ...What?

    any help in explaining this would be much appreciated.
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  2. #2
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    You have  \frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}. Letting \xi = \frac{x}{2\sqrt{Dt}} then u(x,t)\equiv v(\xi)=v(x,t) then:

    \frac{\partial u}{\partial x}=\frac{\partial v}{\partial \xi} \frac{\partial \xi}{\partial x}=\frac{\partial v}{\partial \xi}\left(\frac{t^{-1/2}}{2\sqrt{D}}\right)

    and:

    \frac{\partial^2 u}{\partial x^2}=\left(\frac{t^{-1/2}}{2\sqrt{D}}\right)\frac{\partial^2 v}{\partial \xi^2}\left(\frac{t^{-1/2}}{2\sqrt{D}}\right)=\frac{\partial^2 v}{\partial \xi^2}\left(\frac{1}{4D t}\right)

    ok, you do \frac{\partial u}{\partial t} in terms of partial of v with respect to \xi. When I do that and simplify, I get:

    \frac{d^2 v}{d\xi^2}+2\xi \frac{dv}{d\xi}=0

    Solving that and converting back to x and t, I get an answer in terms of the exponential integral which, when I back-substitute, satisfies the PDE.
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  3. #3
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    Thanks for the reply. I'll work through it and see what I get
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  4. #4
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    if \xi = \frac{x}{2\sqrt{Dt}} and u(x,t) = v(\xi), then:

    \frac{\partial u}{\partial t} = \frac{v}{\xi}\frac{\partial \xi}{\partial t} =. . ..
    Hi, can somebody explain the logic behind this statement please? Thanks!
    Last edited by harbottle; October 19th 2009 at 11:14 AM.
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  5. #5
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    Quote Originally Posted by harbottle View Post
    Hi, can somebody explain the logic behind this statement please? Thanks!
    I believe what you want is

     <br />
\frac{\partial u}{\partial t} = \frac{\partial u}{\partial \xi} \cdot \frac{\partial \xi}{\partial t} = \frac{d v}{d \xi} \cdot \frac{\partial \xi}{\partial t}<br />
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