Sturm-Liouville Theory - Partial Differential Equations | Lecture 23

[@JesusSportsNature]

Thank you for these videos!

[@nopo_b3645]

Yeah actually, the higher the eigenvalues the similar the eigenfunctions are to sine and cosine especially for bigger x. I guess thats for many but not all depending on p and q

[@jasonbramburger]

What do you mean by similar to sine and cosine? How is this measured? Many famous eigenfunctions are polynomial, so not similar at all since sine and cosine are transcendental functions.

[@nopo_b3645]

@@jasonbramburger eigenvalues are similar for some of them. My experience. Has something to do with the number of zeros on the interval and the further to the right boundary the more eigenfunction resembles sine or Cousine. I just think why this is .... maybe only when q term is Zero? What I Studied was Diffusion covection with quadraticly in space dependence.... dy2dx2 -3x2 dydx + lambda y =0

[@jasonbramburger]

@@nopo_b3645 Here is an example of eigenvalues and eigenfunctions that are very different from sine and cosine examples: www.global-sci.org/oldweb/jms/freedownload/v50n2/pdf/502-101.pdf

[@nopo_b3645]

@@jasonbramburger thx. Yeah I just remembered when I was deriving some analytical solutions for convection diffusive systems and how rich all the SL theory and also generally orthogonal polynomials, including general fourier series is. And that eigenvalue similarity was probably only for some specific systems, anyway using those eigenvalues for lambda100 or more gave a much better reconstruction of the initial condition. Really interesting is also fitting of transients, where you have the sum of exponentials. Didnt yet quite understand why DMD seems to not like those, but works better for complex eigenvalues or periodic in time functions. That I namely also tried to do:: Fit eigenvalues out of measured data, but that failed horribly, I remember reading it somewhere that this is very difficult because lamdas can interswitch, and I dont exactly know if constraints in the form lamda1