A Variation of parameters, Green's functions, Wronskian

joshmccraney

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.

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PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

"Variation of parameters, Green's functions, Wronskian"

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