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I've been trying to find some resources that would help me figure out how to numerically solve a coupled system of ODEs which is also an eigenvalue problem.

The system is something like:

$ \tag{1} \kappa h_2(r) +\kappa r h'_2(r)+ (1+\kappa) W_3(r) + (1+ \kappa) r W'_3(r)+ (1 + \kappa) r^2 W''_3(r) = 0 $

$ \tag{2} W_3 +r W'_3+ (1+\kappa) h_2 + (1 +\kappa) r h'(2,r)+ (1+ \kappa)r^2 h''_2 = 0$

I'm not looking for a solution to these equations per se, these are just the equations I have so I thought I'd put them up as a reference.

I need to solve for the functions $h_2$ and $W_3$, on the domain ($0,R$). I have the following boundary conditions:

$\tag {3} W_3(r \rightarrow 0) = r^3, W_3(r = R) = 0 $ $\tag{4} h_2(r \rightarrow 0) = r^3, h_2 (r = R) = h_{2_{outside}}(r=R) $

Where $h_{2_{outside}} $ is a know solution from the domain ($R,\infty$). I also have the following condition

$ \tag {5} h_2 (r = R) h_{2_{outside}}'(r=R) - h_2'(r=R)h_{2_{outside}}(r=R) =0 $

I am also confused about this system of equations because it would appear that I have one to many boundary conditions. That is unless the conditions on $W_3$ and $h_2$ going to zero are redundant. That is enforcing one will enforce the other just to keep equations (1) and (2) self consistent. However this issue may be digressing from my main question.

The main question is: given equations (1) and (2) what numerical method can I use to find $W_3$, $h_2$, and $\kappa$?

1ST EDIT. I should mention that I've already tried solving these equations with a spectral approach, but have so far been unsuccessful. So I'm mainly looking for new ways to attack the problem. My preference would be finite-difference methods since I know those, but I'm open to learning new methods as well.

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Reformulating (1) and (2), the system reads $$ \begin{pmatrix} \mathcal{L}&\\ &\mathcal{L} \end{pmatrix} \begin{pmatrix} W\\ h_2 \end{pmatrix} = \kappa \begin{pmatrix} -\mathcal{L} &\mathcal{M}\\ \mathcal{M} & -\mathcal{L} \end{pmatrix} \begin{pmatrix} W\\ h_2 \end{pmatrix} $$ where the calligraphic symbols are composed of some linear differential operators. You're right about the fact that this is a generalized eigenvalue problem (and although technically correct, one would probably not talk about an "ODE" here).

On a finite domain like $(0,R)$, differential equations without boundary conditions are virtually meaningless. You're right in your assumption that your boundary conditons probably require some more work. For second order differential equation like (1), (2), the set of conditions $$ W(0) = w_0\\ W(R) = w_R\\ h_2(0) = h_{20}\\ h_2(R) = h_{2R} $$ would constitute a well defined system. To my knowledge, statements like "$W(r)=O(r^3)$ for $r\to 0$" cannot be imposed on a differential equation. You may be able to show though that this is a property of any solution of (1), (2).

With an appropriate set of boundary conditions, after discretization, you can feed on a huge body of literature on numerical eigenvalue solvers, e.g., Saad's book.

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  • $\begingroup$ Hi Nico, Thanks very much for the response. I think I can definitely write down the boundary conditions as you describe them. I would have something like $W(0) = 0,W(R) =0, h_2(0) = 0, h_2(R) = h_{2_{outside}}(R)$. I should have mentioned that I've already tried a spectral approach to the problem and I've been stuck there for months. I was hoping there was a finite difference way of solving such problems. Thanks again. $\endgroup$ – tau1777 May 10 '13 at 21:55
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    $\begingroup$ How you discretize the differential operators is a matter of choice, and indeed in 1D it may be the easiest thing to start off using finite differences. You end up with an actual equation system (matrices, vectors) to which you could apply an eigenvalue solver. I don't see a problem there. $\endgroup$ – Nico Schlömer May 11 '13 at 6:13

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