# Solving Coupled ODE eigenvalue problem

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.

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).
• 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. May 10, 2013 at 21:55