Partial Differential Equations for Scientists and Engineers by Farlow

I vividly remember the first time partial differential equations were introduced in university when speaking about chemical reactions. It was not followed through or taught on that course, but revisited later on in other another separate class that combines ODE and PDEs. The issue I faced when thinking about PDEs was trying to remember what types of equations (hyperbolic, elliptic,…) coincided with what solution existed. This issue stayed throughout my masters and into my Ph.D even after taking advanced PDE with the great Dr. Partha Guha. I realized the detailed proofs that we started deriving in the Ph.D class was the simpler approach compared to what was taught in the B.S. level. Nonetheless, not much from the class stuck, other than the beautiful geometry of certain classes of PDEs. (check surface tension)

Walking out with shame from spending time in classes with only 10% understanding, I complain to one of my mentors, Mokhtar Kirane on this issue. With open arms he proudly referred me to the PDE book by Farlow. By then my knowledge in probability, ML techniques, stochastic differential equations, and linear algebra were much higher than what I knew in PDE because of avoidance. One of the things I urged my younger fellow students was to hone in the basics of a mathematical concept rather than being excited on newer fields that sound invigorating, but wouldn’t be a useful endeavor. I put the time where my mouth is and sat in a bachelor level course for PDE after short of a decade and relearned the topic.

On top of Kirane’s excellent teaching, the book was a treat to read. I would catch myself reading a chapter more often like a novel rather than a textbook and forget to do questions or redo proofs. One of the helpful videos that I came across from a top-down level was the lectures by Dr. Grinfeld. https://youtu.be/-j6Em60JbyU?si=8gx0XMPDuCsVTQ1c

I would recommend a front to back read of this book to any inspiring or revising applied mathematician.

If you’re in a rush, I would say chapters 1-7 for basics, 24-26 for background on the black-scholes eqn. 36-38 for non linear equations and most importantly Green’s function