Solving Coupled ODE eigenvalue problem

Question

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 too 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 $\kappa$?

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.

Answer 1

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 conditions probably require some more work. For second-order differential equations 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.