A Partial Differential Equations Qualifying Exam Planning Guide

Well, here it is, folks. Tomorrow is my long-anticipated PhD qualifying exam in partial differential equations (affectionately known as PDE for short). Alongside the examination in analysis later this week, this exam will tell the University of Tennessee’s math department whether I have mastered the fundamentals of the area in which I wish to perform research. I have spent the better part of the past year preparing for these, including a combination of reading texts, reviewing old homework and test solutions, and of course, working lots of problems. For both subjects I was provided with past exam problems, which I used as the core of my development as a problem-solver. I also sought problems from other textbooks, and even created a few of my own along the way.

Anyway, I realized that knowing the material is only half the battle with such an exam. The other half, and the half this article focuses on, is organizing the information that is important. I always like to say, “you might have memorized 60 formulas, but if you can’t figure out which one is useful when, you won’t pass the exam!” Throughout my studying I’ve tried to figure this out, and have made a rough breakdown of some of the topics I’ve seen recur in many PDE problems.

Even though the creation of this article is part of my last-minute studying for my qualifying exam, I expect this guide will be helpful for anyone studying PDE at an advanced undergraduate or beginning graduate level. My goal is to list as few formulas…