Thread: How to make this integration?

1. 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 $u$?

2. Re: How to make this integration?

Originally Posted by brianx
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