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Math Help - Do you recognize this equation?

  1. #1
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    Smile Do you recognize this equation?

    Hi,

    have you seen an equation like this?:


    d^2(w)/dt^2 + A (dw/dt)^s + B w = 0


    where w = dependent variable, t=time, A and B = constants, s < 1

    Any hint about this nonlinear ordinary diff. equation?
    Thanks for your help!

    Victor
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  2. #2
    MHF Contributor chisigma's Avatar
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    Although hazardous may be you can write the DE in term of Laplace Transform as...

    p^{2}\cdot W(p) + a\cdot p^{s}\cdot W(p) + b\cdot W(p)=0, 0<s<1 (1)

    In that case the w(t) can be [virtually] computed using the 'Inversion Complex Formula' as...

    w(t) = \frac{1}{2 \pi i} \int _{c-i \infty}^{c + i \infty} \frac{e ^{ p t}} {p^{2} + a\cdot p^{s} + b} \cdot dp (2)

    ... where c is a constant that lies on the right of all singularities of the complex function...

    H(p) = \frac{1}{p^{2} + a\cdot p^{s} + b} (3)

    In any case the question has to be further examined because I can have made some erroneous procedures...

    Kind regards

    \chi \sigma
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  3. #3
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    thanks, but...

    ... where such kind of equation can be found? when s=1, you have an linear ODE, but what is its name? I've seen it a couple of times and I was intrigued about it. It can be actually solved numerically, but I'm more interested on the equation as such. What more can you, community, say about it?

    Thanks again and kind regards!

    Victor
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  4. #4
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    Quote Originally Posted by quezadav View Post
    Hi,

    have you seen an equation like this?:


    d^2(w)/dt^2 + A (dw/dt)^s + B w = 0


    where w = dependent variable, t=time, A and B = constants, s < 1

    Any hint about this nonlinear ordinary diff. equation?
    Thanks for your help!

    Victor
    As the dependent variable is missing, this equation can be reduced to first order

    if p = \frac{dw}{dt} then p \frac{dp}{dw} = \frac{d^2 w}{dt^2}

    and your equation becomes

     <br />
p \frac{dp}{dw} + A p^s + B w = 0<br />
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  5. #5
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    Quote Originally Posted by quezadav View Post
    ... where such kind of equation can be found? when s=1, you have an linear ODE, but what is its name? I've seen it a couple of times and I was intrigued about it. It can be actually solved numerically, but I'm more interested on the equation as such. What more can you, community, say about it?

    Thanks again and kind regards!

    Victor
    The second order term represents diffusion in general (of w w.r.t. time)
    The first order term to the power s represents convection, and the fact that s<1 implies that it is a sort of damped out convection. Such an equation is more generally seen in fluid dynamics when s=2, but I have worked with cases when 0<s<2. For example while modeling heat flow in electronics cooling applications, due to severe space restraints, the convection is modeled as a damped phenomenon owing to the fact that hot air passing over a processor does not lose all its heat content before it is made to pass over the processor again. This can be modeled by using an s which is slightly less than 1 (~0.8-0.9)

    Hope that helps!
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  6. #6
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    great!

    Quote Originally Posted by tanujkush View Post
    The second order term represents diffusion in general (of w w.r.t. time)
    The first order term to the power s represents convection, and the fact that s<1 implies that it is a sort of damped out convection. Such an equation is more generally seen in fluid dynamics when s=2, but I have worked with cases when 0<s<2. For example while modeling heat flow in electronics cooling applications, due to severe space restraints, the convection is modeled as a damped phenomenon owing to the fact that hot air passing over a processor does not lose all its heat content before it is made to pass over the processor again. This can be modeled by using an s which is slightly less than 1 (~0.8-0.9)

    Hope that helps!
    many thanks! Do you have any reference that you can recommend where such modelling is made?
    Thanks again.
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