# Boundary Conditions for 3 Coupled First order ODE

1. Jul 19, 2012

### tau1777

I am trying to solve four coupled equations. Three of them are first order differential equations and the fourth is a algebraic one. The equations look something like this:

V$_{l}$(r) = f$_{1}$(r)W'$_{l}$(r) (1)

h''$_{l}$ + f$_{2}$(r)h'$_{l}$ + f$_{3}$(r)h$_{l}$(r) = U$_{l}$(r) (2)

f$_{4}$(r)U'$_{l}$ + f$_{5}$(r)h'$_{l}$ + f$_{6}$W'$_{l}$ + f$_{7}$V'$_{l}$= \$ (3)

U$_{l}$+ h$_{l}$ + f$_{8}$(V$_{l}$ + W$_{l}$) = 0 (4)

I didn't explicitly write this out for every term but everything is a function of 'r', and they are defined on the domain {0, R}. Also I did lie about having only four equations, the index l runs from l=2, to as large as my computer can handle (lmax). So I have (lmax-2)*4 equations.

I am trying to use a finite difference scheme and the following boundary conditions:

W$_{l}$(R) = 0 , V$_{l}$(0) = U$_{l}$(0) = 0 and
h$_{l}$(R) = houtside$_{l}$(R). What I mean by this is that we know what the function h$_{l}$(r) is outside the domain so the solution inside should match the know solution outside.

The function houtside$_{l}$(R) =$\Sigma \frac{(l+s)!(l-s)!}{(s+2)!}... (2M/R) h_{l,0}$

The '...' in the equations above means that there are more terms such as (l+s)! before (2M/R) but they are all multiplicative only.

Thanks for reading thus far, so here is the question. How can I determine $h_{l,0}$?

I know how to implement the other boundary conditions in my code but unless I know $h_{l,0}$, I do not believe I can continue.

I have tried taking the know outside solution and plugging it into equation (2) above, but then the trouble is what is U$_{l}$(R)? I know that U$_{2}$(R) =1 but I'm not sure I can extend this to all values of l.

Thanks for the help.

Last edited: Jul 19, 2012