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Thread: How to make this integration?

  1. #1
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    Question How to make this integration?

    This is a general solution for the heat equation:

    $$ u = \frac{2}{\sqrt \pi} \int_{x2 / \sqrt (\kappa t)}^{\infty} \phi \Bigg (t-\frac{x^2}{4 \kappa \mu^2} \Bigg )e^{-\mu^2} d\mu $$
    where
    $$\mu = \frac{x}{2\sqrt(\kappa(t-\lambda))}$$

    for a boundary condition of

    $$\phi (t) = \frac{A} {1 +\exp\Big(B (C+ D t) \Big)} $$

    the general solution transforms to

    $$u = \frac{2}{\sqrt \pi}\int_{\mu=0}^{\infty} \frac{A} {1 +\exp\Big(B (C+ D (t - \frac{x^2}{4\kappa\mu^2})) \Big)} e^{-\mu^2} d\mu - \frac{2}{\sqrt \pi}\int_{\mu=_0}^{x/\sqrt{4\kappa t}} \frac{A} {1 +\exp\Big(B(C+ D (t - \frac{x^2}{4\kappa \mu^2})) \Big)} e^{-\mu^2} d\mu$$

    1. Is this equation correct? Derived correctly from the general solution by incorporating $\phi (t)$ ?
    2. How to make the integration to solve this equation for $u$?
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: How to make this integration?

    Quote Originally Posted by brianx View Post
    This is a general solution for the heat equation:

    $$ u = \frac{2}{\sqrt \pi} \int_{x2 / \sqrt (\kappa t)}^{\infty} \phi \Bigg (t-\frac{x^2}{4 \kappa \mu^2} \Bigg )e^{-\mu^2} d\mu $$
    where
    $$\mu = \frac{x}{2\sqrt(\kappa(t-\lambda))}$$

    for a boundary condition of

    $$\phi (t) = \frac{A} {1 +\exp\Big(B (C+ D t) \Big)} $$

    the general solution transforms to

    $$u = \frac{2}{\sqrt \pi}\int_{\mu=0}^{\infty} \frac{A} {1 +\exp\Big(B (C+ D (t - \frac{x^2}{4\kappa\mu^2})) \Big)} e^{-\mu^2} d\mu - \frac{2}{\sqrt \pi}\int_{\mu=_0}^{x/\sqrt{4\kappa t}} \frac{A} {1 +\exp\Big(B(C+ D (t - \frac{x^2}{4\kappa \mu^2})) \Big)} e^{-\mu^2} d\mu$$

    1. Is this equation correct? Derived correctly from the general solution by incorporating $\phi (t)$ ?
    2. How to make the integration to solve this equation for $u$?
    So far as I know you derived it correctly. (I never checked the limits.)

    Looking at the integral, I would say that it has no exact solutions. I couldn't recognize a specific integral for it (ie. It doesn't seem to be an elliptic integeral, it's not Ei, etc.)

    -Dan
    Thanks from brianx
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