MATH 451/651 (Spring 2017)

Instructor's contact information

Michael Robinson
226 Gray Hall
michaelr at american {dot} edu
Office hours: My research website
Feel free to contact me with any questions (course-related or not)

Quick Links

Course description
Homework assignments
Course schedule
Concerning programming in this course
Information about exams
Some useful links
Course policies

Course description

Fourier series, orthonormal systems, wave equation, vibrating strings and membranes, heat equation, Laplace's equation, harmonic and Green's functions.

Prerequisite: MATH-321

The course textbook is Elementary Applied Partial Differential Equations, by Richard Haberman, Prentice Hall. I'm teaching out of the Third Edition, but you can get any version you like.

In this course, students will

  1. Learn to solve the transport equation, heat equation, wave equation, and Laplace's equation
  2. Explain elementary physical derivations of each of these equations
  3. Interpret the solutions to these equations in terms of measureable, physical implications
  4. Describe the kinds of boundary conditions that typically arise, and their meanings


Homeworks are posted on BlackBoard.

Homework 1 due January 31
Homework 2 due February 14
Homework 3 due March 7
Homework 4 due March 28
Homework 5 due April 11
Homework 6 due April 28
Homework 7 due at beginning of the final

Course schedule

Unit 1: The heat equation

After this unit, you should be able to
  1. Explain the physical meaning of the heat equation, its boundary conditions, and its solutions
  2. Solve the heat equation using the method of separation of variables

January 17: 1.2: Derivation of the heat equation
January 24: (Dr. Robinson on travel; no class)
January 27: 1.3: Boundary conditions for the heat equation
January 31: 1.4: Equilibrium temperature distributions
February 3: 2.1-2.3.2: Separation of variables on the heat equation
February 7: 2.3.3-2.3.8: Solving the heat equation
February 10: 2.4: Other boundary conditions
February 14: Review for exam 1
February 17: Exam 1 (in class)

Unit 2: The Laplace's equation and the wave equation

After this unit, you should be able to
  1. Explain how to derive Laplace's equation, and where it can be used
  2. Explain the physical meaning of the wave equation, its boundary conditions, and its solutions
  3. Solve Laplace's equation and the wave equation on rectangular domains
  4. Explain the meaning of the Fourier series representation of a function

February 21: 2.5.1: Solving Laplace's equation on a rectangle
February 24: 2.5.2: Solving Laplace's equation on a disk
February 28: 3.1-3.2: Introduction to Fourier series
March 3: 3.3, 3.6: Fourier series coefficients
March 7: 3.4-3.5: Calculus with Fourier series
March 10: 4.1-4.3: The wave equation
March 21: 4.4: Solving the wave equation
March 24: Review for Exam 2
March 28: Exam 2 (in class)

Unit 3: Characteristics

After this unit, you should be able to
  1. Compute scattering angles for waves incident on a planar boundary
  2. Model the solution to the one-dimensional wave equation as a superposition of traveling pulses
  3. Explain how traveling wave expansions can be used to model wave propagation in complicated, higher-dimensional environments

March 31: 4.5: Wave equation in higher dimensions
April 4: 4.6: Snell's law
April 7: 12.1: Method of characteristics
April 11: 12.2: The transport equation
April 14: 12.3: Solving the wave equation, infinite string
April 18: 12.4: Reflections
April 21: 12.5: Solving the wave equation on a finite string
April 25: 12.6-12.7: Characteristics and quasi-linear PDE
April 28: Review for final

Final exam: May 9, 5:30-8:00pm

Course policies


For each exam, you will be permitted to bring one 3"x5" index card with handwritten notes on it. Electronic calculators will not be permitted. However, mechanical calculators (such as this or these) are fine.


You are generally expected to attend the lectures. Please contact me in advance if you cannot attend, especially in the case of an exam.


The course grade will be determined as follows:
25% Homework
25% Exam 1
25% Exam 2
25% Final exam

Late homeworks will not be accepted, unless there is an official (University-approved) reason for doing so.

The difference between MATH 451 and MATH 651

Giving cogent technical talks is an important skill for all scholars. The sooner and more often you present and defend technical ideas in front of an audience, the better. As part of your training, if you're taking the graduate version of the class, you must give one of the class lectures. If you're taking the undergraduate version instead, you have the option to present one of the lectures but are not required. Here are the parameters for your presentation:
  1. Your presentation will be graded solely on whether you have done it or not.
  2. You must work with me to select which lecture you'll give (see the schedule above), since I have several lectures that I would like to give myself.
  3. You may present the material from your own notes or the book. It's your choice. (A word of advice: the proofs and derivations are important, but enlightening examples are much more important! Spend time gathering or creating examples. Fully understand them in advance.)
  4. Since the class is highly interactive -- even energetic -- you'll have to be ready defend whatever you present. I'll help you if you get stuck -- so you should not feel apprehensive -- but expect to get questions from the audience.
  5. You should expect to present for about 20 minutes; I will stop you if you run past 30 minutes. After you've introduced the material, I'll follow up with my own examples and additional detail.

Academic dishonesty

Academic dishonesty is a serious offense. Read our University policies. You may work together on homeworks and projects, but the work you submit must be your own.

There is to be no collaboration with other people using any means during an exam. Doing so risks starting academic dishonesty proceedings. However, you may ask me questions during an exam, which I may answer at my discretion.

You may collaborate with other students in this class on homeworks and projects, but the work you submit must be your own. You may not collaborate with others outside of the class on homeworks or projects without my express permission.

At a minimum, honesty consists of presenting your ideas clearly and in your own words, possibly orally. On the other hand, the creation and writing of software, proofs, examples, or counterexamples is a creative process. Here are a few typical cases that are relevant for homework:

  1. If you happen to create an example, counterexample, or proof for your work on your own, you need not notate it as such. (It is expected that you will rediscover standard examples, and these will not cause any concern.)
  2. If a colleague shows you an example, counterexample, or proof that you like, please credit that person by name in your write-up. You should expect that I may challenge you to explain your writing in your own words, possibly orally.
  3. If you write an example, counterexample, or proof as part of a group, please credit all members of your group by name in your write-up. You should expect that I may challenge you to explain your writing in your own words, possibly orally.

The mathematics community has a unique perspective on academic honesty and priority. The mathematics community has strict social guidelines for assigning credit, which I expect you'll adhere to. Due to the canonicity of certain mathematical results and constructions, sometimes well-intentioned people end up in priority disputes when they independently discover something. If you have any questions on this matter, you are expected to consult me directly for advice.

In so far as computer programs are concerned, you may not simply copy another student's program and put your name on it! I may use various technical countermeasures (for instance, MOSS to detect instances of copied computer software. I am obligated to investigate suspected violations of academic integrity, and may ask you to explain or clarify sections of your program if I am unable to follow it.