In the style of Cosma Shalizi’s notebooks I’m including some links to things that I’ve read recently.
This little one is written about PDE, Generalized Functions and Differential Equations.
Historically generalized functions were first used in Physics by Heaviside and Dirac.
Schwartz the great analyst came along and added rigour.
Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics: This is an interesting NASA Technical Paper, with some excellent introductions to Aerodynamics applications.
Taylor PDE notes Particularly relevant are the Generalized functions parts and distributons.
Generalized Functions by Gelfand and Shilov I’ve not read this book from cover to cover. However I hear good things about it.
Introduction to the Theory of Distributions [Paperback]
F. G. Friedlander (Author), M. Joshi (Author) Cambridge originated textbook with a thorough Pure Mathematics introduction. Mark Joshi is well known in the world of Financial Mathematics and was once a well known Quant.
There are of course many books on this, which I’d love to read. Including a book by one of the late great Mechanicists and Fluid Mechanicists James Lighthill.