2nd order PDE - help needed

Baluba · #1 $Old$ November 18th, 2009, 01:39 PM

H''(n) + (1/2)n*H'(n) = 0.

Boundary conditions: H(0) = 1, H(inf) = 0.

I really don't know where to start ... any help will be greatly appreciated.

Danny · #2 $Old$ November 18th, 2009, 03:28 PM

Quote:

Originally Posted by Baluba $View Post$

H''(n) + (1/2)n*H'(n) = 0.

Boundary conditions: H(0) = 1, H(inf) = 0.

I really don't know where to start ... any help will be greatly appreciated.

First separate and integrate

$\frac{H''}{H'} = - \frac{1}{2} n\, dn$

so

$\ln H' = \ln c_1 - \frac{1}{4} n^2\;\; \Rightarrow \;\;H' = c_1 e^{-n^2/4}$

then integrate again (on the second integration we'll keep it in integral form)

$H = c_1 \int_0^n e^{-t^2/4}dt + c_2$

$H(0) = 1\; \text{ gives}\; c_2 = 1$ while $H(\infty) = 0\; \text{ gives}\; c_1 \pi + 1 = 0$

giving the solution as

$H(n) = 1 - \frac{1}{\pi} \int_0^n e^{t^2/4}dt$

or in terms of error functions

$H(n) = 1 - \text{erf} \left(\frac{n}{2}\right)$ .

Baluba · #3 $Old$ November 18th, 2009, 04:04 PM

Thanks a lot!