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Math Help - Please help on the multi variable problem

  1. #1
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    Please help on the multi variable problem

    This is not a school problem. I just learning PDE on my own. The problem is:

    Transform \frac{\partial^2 u}{\partial t^2}= c^2\frac{\partial^2 u}{\partial x^2} in to \frac{\partial^2 u}{\partial \alpha \partial \beta}=0

    where u(x,t)=F(x+ct) + G(x-ct)

    and \alpha = x+ct,\beta = x-ct

    \alpha = x+ct,\beta = x-ct \Rightarrow x=\frac{\alpha + \beta}{2}, t=\frac{\alpha - \beta}{2c}

    \alpha = x+ct,\beta = x-ct \Rightarrow \frac{\partial \alpha}{\partial x}= 1, \frac{\partial \alpha}{\partial t}= c, and also  \frac{\partial \beta}{\partial x}= 1, \frac{\partial \beta}{\partial t}= -c

    \frac{\partial x}{\partial \alpha} = \frac{\partial (\alpha + \beta)}{2} = \frac{1}{2} [\frac{\partial \alpha}{\partial \alpha} + \frac{\partial \beta}{\partial \alpha} = \frac{1}{2}[ 1 + 0] = \frac{1}{2}

    \Rightarrow \frac{\partial x}{\partial \alpha} = \frac{1}{2}

    \frac{\partial u}{\partial \alpha}= \frac{\partial u}{\partial x} \frac{\partial x}{\partial \alpha} + \frac{\partial u}{\partial t} \frac{\partial t}{\partial \alpha} = \frac{\partial F(\alpha)}{\partial \alpha}+ \frac{\partial G(\beta)}{\partial \alpha}

    I gave this a lot of thoughts. I still cannot accept  \frac{\partial G(\beta)}{\partial \alpha}=0

    Please bear with me. Let's take a look at this example:

    Let G(\beta)=-\beta = \beta - 2\beta = x-ct-2x+2ct=(x+ct)-2x=\alpha -2x

    \Rightarrow\frac{\partial G(\beta)}{\partial \alpha}= \frac{\partial \alpha}{\partial \alpha} + \frac{\partial 2x}{\partial \alpha}= 1+1=2

    Yes I know I am playing around and the argument is very thin, but never the less, it is valid. My whole point is just because  \alpha and \beta are independent variable, \frac{\partial G(\beta)}{\partial \alpha} not necessary equal 0.

    \frac{\partial \beta}{\partial \alpha}=0 do not imply at all \frac{\partial G(\beta)}{\partial \alpha}=0

    I don't think the question is good to say \frac{\partial^2 u}{\partial \alpha \partial \beta}=0 I have been struggling on this very point for two days!!!!

    Please tell me if I am correct.
    Thanks a million.
    Alan
    Last edited by yungman; February 9th 2010 at 12:36 PM.
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  2. #2
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    Anyone please?
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