Also, the method doesn't really solve the indefinite integral, if I am not mistaken.
I thought the integration by parts was a lot more straight-forward.
I know what Green's functions are but have very little experience because I deal with linear dynamics for the most part.
I am not familiar with the numerical method of characteristics.
I have undergraduate degrees in Discrete Mathematics and Computer Engineering, and a Master's in Electrical Engineering where a significant portion of my classes were on feedback systems and controls.
I take it you are an Aeronautical Engineer, or otherwise focused on turbulence and fluid dynamics.
Nevermind, I looked you up. "laser reseach specialist" and now material science/electrical engineer, huh?
To be honest, the people who come on Ventrilo and other places generally have rather simple math related frustrations. I may try to tackle some of the questions you pose, but it is clear we specialized in different areas of mathematics.
I generally have a preference for the more abstract froms of math: Groups, Rings, Vector Spaces, etc. some amount of Topology, and the study of limits. Combinatorics, Graph Theory, and Algorithm Design are my bread-n-butter type of mathematics. Statistics and Control Theory, are close seconds. I let the computers deal with PDE's...not my favorite branch of math to be sure. I do really like Numerical Analysis (though I don't have much experience in designing algorithms for PDE solvers).