Variation of parameters, Green's functions, Wronskian

[member 428835]

Hi PF!

I am trying to solve an ODE by casting it as an operator problem, say $K[y(x)] = \lambda M[y(x)]$, where $y$ is a trial function, $x$ is the independent variable, $\lambda$ is the eigenvalue, and $K,M$ are linear differential operators. For this particular problem, it's easier for me to work with the inverse operator problem $M^{-1}[y(x)] = \lambda K^{-1}[y(x)]$. Constructing inverse operators implies building a Green's function.

The technique I've used to build a Green's function is variation of parameters, which takes the form $G = y_1(x)y_2(\xi) / w_\alpha$ where $y_1,y_2$ are fundamental solutions associated with a particular operator, say $K$, and $w_\alpha$ is their associated Wronskian, where the subscript $\alpha$ is a parameter. I observe $\alpha \to 0 \implies w\to 0$. Does this imply any $\alpha \neq 0$ yields the correct Green's function? What if $\alpha$ is very VERY small but not zero? Is this something that can cause numerical issues?

I ask this because an analytic solution for the operator ODE exists for the $\alpha = 0$ case, but this causes issues with the Wronskian. When benchmarking, how ``small'' of an $\alpha$ should I use? I can provide more information if someone is willing to help and needs more understanding, as I have not really mentioned specifics regarding the numerics.

 

[jvicens]

Hello,

It seems like you are on the right track with using variation of parameters to build a Green's function. To answer your question about the value of alpha, it is important to consider the numerical stability of your solution. If alpha is too small, it may cause numerical issues and result in inaccurate solutions. On the other hand, if alpha is too large, it may result in a less accurate solution. It would be best to experiment with different values of alpha and see which one gives you the most accurate and stable solution.

Another thing to consider is the behavior of the Wronskian as alpha approaches 0. If the Wronskian approaches 0, then the Green's function may not be well-defined. In this case, it may be better to use a different method for constructing the Green's function.

In terms of benchmarking, it would be helpful to compare your numerical solution to the analytic solution for different values of alpha to see how it affects the accuracy of your solution. This will give you a better understanding of how small of an alpha you can use without causing numerical issues.

I hope this helps and feel free to provide more information if needed. Good luck with your research!