Math 3993 Directed Study: Introduction to Partial Differential Equations
The (Unfinished) PDE Coffee Table Book
Course Structure
You will ``work ahead'' of me, in the sense that I will not be lecturing on the topics being studied. I will provide some notes and Mathematica computer files to help you learn the material, but you will for the most part be working independently, or in collaboration with other students taking the directed study. There will be a weekly meeting time scheduled to discuss with me and other students any questions that have arisen in the previous week, but you are of course encouraged to work with other students in the directed study outside of this meeting time.
The topics have been chosen to give a chance to encounter the most important aspects of partial differential equations.
You might be interested in seeing what goes into a typical partial differential equation preliminary exam in graduate school (note these are very advanced and being able to answer
these preliminary exam questions is not the goal of our introduction, our goal is to lay the foundation for which these more advanced problems arise from):
Registration
This page was created for a 2cr course (Math 3993 Directed Study) for Spring 2019. If you would like to take this course see me before Dec 10, 2018. There is a form to fill out (which I have) which must be signed by Peh Ng (the Division Chair) before you can register and I want to give Peh all the forms at once.
Course Prerequisites
Prerequisites: Multivariable Calculus, Ordinary Differential Equations, Mathematica experience, a willingness to work hard, and most importantly an ability to work on your own.
Recommended: Linear Algebra. A smattering of Analysis would not hurt in parts either but these aren't as important.
Time Commitment
University policy says ``one credit is defined as equivalent to an average of three hours of learning effort per week (over a full semester) necessary for an average student to achieve an average grade in the course''. This course is a two-credit course, meeting one hour per week: 2 credits times 3 hours/week/credit - 1 hours/week meeting with instructor = 5 hours/week outside class. Thus, you are expected to spend 5 hours per week working outside of class, reading the textbook and working problems. You will probably need to spend more than that do do well.
Textbook
Partial Differential Equations and Boundary Value Problems, Nakhle Asmar (first, second, or third edition; what's below is based on the second edition). Fabulous book, in my opinion and the third edition is inexpensive (Dover).
Nakhle Asmar homepage with information related to his textbooks (Mathematica notebooks which are probably out of date, and Students' Solution Manual).
Course Components
Meetings. We will meet once a week to talk about what you have been working on. There will be no formal lectures. The meeting time will be determined based on the schedules of whoever is taking the course. This course should be thought of as an independent study, where you will work on material at your own pace throughout the semester. I will keep you on track and assist as needed (and, of course, provide the general outline of what you should focus on).
Assignments. The questions listed below will be turned in for grading, and you can also turn in other questions that you have been working on that you have found interesting. Each week turn in what you have completed so far. We can adjust as we go. I am looking for eight assignments of approximately 3 or 4 questions each. Assignments should be well explained and well written (they should not be so brief that a reader cannot follow the steps of your solution--think of writing it to explain it to a differential equations student who is not in the class). I foresee more hand computations on assignments than Mathematica computations, but your assignments will typically involve both.
Final Paper. You will write a final paper containing topics from a section that hasn't been covered:
- Chapter 5 (Partial Differential Equations in Spherical Coordinates), or
- Chapter 6 (Sturm-Liouville Theory with Engineering Applications), or
- Chapter 8 (The Laplace and Hankel Transforms with Applications), or
- Chapter 9 (Finite Difference Numerical Methods), or
- Chapter 10 (Sampling and Discrete Fourier Analysis with Application to PDEs), or
- Chapter 11 (An Introduction to Quantum Mechanics), or
- a topic of your choosing related to partial differential equations.
The paper should contain a summary of the concepts studied, and some example problems. It may contain some Mathematica, and if it does the Mathematica should be seamlessly integrated into the flow of the paper. The paper should be typed up using Word, LaTeX, or Mathematica. Think of the paper as a 50 minute lecture on a topic that you are going to give to your peers who are interested in the topic. We might even consider some presentations at the end of the course!
Grading
Here is the University-wide uniform grading policy.
- A: Represents achievement that is outstanding relative to the level necessary to meet course requirements.
- B: Represents achievement that is significantly above the level necessary to meet course requirements.
- C: Represents achievement that meets the course requirements in every respect.
- D: Represents achievement that is worthy of credit even though it fails to fully meet the course requirements.
- F: Represents failure and indicates that the coursework was completed but at a level unworthy of credit, or was not completed and there was no agreement between the instructor and student that the student would be temporarily given an incomplete.
- I: See the catalog.
The grade for the course will be calculated by the following formula:
| Assignments | 80% |
| Paper | 20% |
Your numerical grades will be converted to letter grades and finally Grade Points via the following cutoffs (see the UMM Catalog for more on Grades and Grading Policy):
| Numerical | 95% | 90% | 87% | 83% | 80% | 77% | 73% | 70% | 65% | 60% | Below 60% |
| Letter | A | A- | B+ | B | B- | C+ | C | C- | D+ | D | F |
| Grade Point | 4.00 | 3.67 | 3.33 | 3.00 | 2.67 | 2.33 | 2.00 | 1.67 | 1.33 | 1.00 | 0.00 |
Course Calendar
| Date | Lecture Topic | Homework | Resources |
|---|---|---|---|
| Week 1 | Course Introduction | ||
| 2.1 Periodic Functions | Periodic Functions (MMA) | ||
| Week 2 | 2.2 Fourier Series | 7, 24, 26 | Fourier Series (MMA) Gibbs Phenomenon Applet |
| 2.3 Fourier Series and Functions with Arbitrary Periods | 31, 32 | Fourier Series for Functions with arbitrary period (MMA) | |
| Week 3 | 2.4 Half-Range Expansions: The Cosine and Sine Series | 17 | |
| 2.5 Mean Square Approximation and Parseval's Identity | 18 | Short note on orthogonality of sines and cosines (MMA) | |
| Week 4 | 2.6 Complex Form of Fourier Series | Complex Form of Fourier Series (MMA) | |
| 2.7 Forced Oscillations 2.8 Proof of Fourier Series Representation Theorem |
Read these sections. | ||
| Week 5 | 2.9 Uniform Convergence of Sequences and Series of Functions 2.10 Dirichlet Test and Convergence of Fourier Series |
Read these sections. If you are into Analysis, you might try a problem or two from 2.9 and/or 2.10. | |
| 3.1 Partial Differential Equations in Physics and Engineering Read Classification of PDEs (elliptic, parabolic, hyperbolic) |
14 | Method of Characteristics: Notes | Mathematica Clairaut's Theorem |
|
| Week 6 (updated) | 3.2 Modeling: Vibrating Strings and the Wave Equation | 6 | If you want to practice building models, this is a good section. I have a solution to the Hanging Chain later. |
| 3.3 Solution of the 1D Wave Equation: Separation of Variables | 12 (long but intriguing!) | ||
| Week 7 | 3.4 D'Alembert's Method | 17 | |
| 3.5 The One Dimensional Heat Equation | |||
| Week 8 | 3.6 Heat Conduction in Bars: Varying the Boundary Condition | 19, 20 | Orthogonality |
| 3.7 The Two Dimensional Wave and Heat Equations | 17 | ||
| Week 9 | 3.8 Laplace's Equation in Rectangular Coordinates | ||
| 4.2 Vibrations of a Circular Membrane: Symmetric Case | 10 | Symmetric Vibrating Drumhead (MMA file) General Vibrating Drumhead (MMA file) (based on 4.3) |
|
| Week 10 | 5.2 Dirichlet Problems with Symmetry | 12 | |
| 6.1 Orthogonal Functions | |||
| Week 11 | 6.2 Sturm-Liouville Theory | 35 | Hanging Chain with Kick (MMA file) (based on 6.3) |
| 7.1 The Fourier Integral Representation | 23, 24 | ||
| Week 12 | 7.2 The Fourier Transform | 1 | |
| 7.3 The Fourier Transform Method | 23, 24 | ||
| Week 13 | Finish Paper |