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Math Help - PDE; change of variables

  1. #1
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    PDE; change of variables

    I have the PDE a(t)*(df/dx) + df/dt + b(t)f=0.
    I have to make a change of variables now where g=fc(t), y=xd(t) and s=e(t). I have no idea how this works with coefficients depending on t.
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  2. #2
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    Maybe I wasn't clear in my first post, f is a function of x and t and with df/dx I mean the partial derivative of f to x. The question is to rewrite the PDE into a PDE in g(y,s), y and s.
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  3. #3
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    It wasn't clear to me. In general, for the PDE:

    a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}+c(x,y)u(x,y)=f(x,y), we solve the characteristic DE:

    \frac{dy}{dx}=\frac{b(x,y)}{a(x,y)}

    Assume the characteristic solution can be put into the form h(x,y)=k. Then the change of variable w=h(x,y),\; z=y will convert the PDE to an ODE in z for fixed w. Probably not what you want. That's why I didn't say anything. Also while I'm sayin', this is right out of "Basic Partial Differential Equations" by Bleecker and Csordas. It's a nice PDE book that's easy to read. They go over first order PDEs like this pretty good.
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  4. #4
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    I just want to rewrite my original PDE in a new PDE of the form h(s)*(dg/dy) + k(s)*(dg/ds) + l(s)*g=0.
    Again with 'dg/dy' I mean the partial derivative.
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  5. #5
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    This is how I'd do it; maybe someone can suggest a better approach to both of us. I first write it in standard form:

    \frac{a(t)}{b(t)}\frac{\partial f}{\partial x}+\frac{1}{b(t)}\frac{\partial f}{\partial t}+f=0

    The characteristic equation becomes then:

    \frac{dt}{dx}=\frac{1}{a(t)}\Rightarrow x-\int a(t)dt=K

    So I make the change of variables:

    y=x-\int a(t)dt \quad\quad s=t

    Then define: g(y,s)\equiv f(x,t)

    and under these transformations it can be shown:

    \frac{a}{b}\frac{\partial f}{\partial x}+\frac{1}{b}\frac{\partial f}{\partial t}=\frac{1}{b(s)}\frac{\partial g}{\partial y}

    Thus I have:

    \frac{1}{b(s)}\frac{\partial g}{\partial y}+g=0
    Last edited by shawsend; October 5th 2008 at 07:29 AM. Reason: changes v to g
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