Boundary Conditions for 3 Coupled First order ODE

[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.

 

[jvicens]

Thank you for sharing your equations and boundary conditions. Solving coupled equations can be a challenging task, especially when there are many terms and unknowns involved. Here are some suggestions that may help you in determining h_{l,0}:

1. Use numerical methods: Since you mentioned using a finite difference scheme, I assume you are already familiar with numerical methods. You can use a numerical solver such as MATLAB or Python to solve your equations and determine the value of h_{l,0}. These solvers have built-in functions for solving differential equations and can handle large systems of equations.

2. Use a perturbation approach: In some cases, it may be possible to use a perturbation approach to solve coupled equations. This involves breaking down the problem into simpler parts and solving them separately. Then, the solutions can be combined to obtain a solution for the entire system. This approach may be useful if your equations have some small parameters that can be used as a basis for perturbation.

3. Consider simplifying assumptions: Depending on the specific problem you are trying to solve, it may be possible to make simplifying assumptions that can reduce the complexity of your equations. This can make them easier to solve and may also help in determining h_{l,0}.

4. Consult with experts: If you are still having trouble determining h_{l,0}, it may be helpful to consult with experts in the field. They may be able to provide insights and suggestions on how to approach your specific problem.

I hope these suggestions are helpful to you in solving your equations. Good luck!