# MATH 551 (Spring 551)

## Instructor's contact information

Michael Robinson
226 Gray Hall
michaelr at american {dot} edu
My research website http://www.drmichaelrobinson.net/
Feel free to contact me with any questions (course-related or not)

Course description
Homework assignments
Course schedule
Concerning programming in this course
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

# Homework

Homeworks are posted on BlackBoard.

Homework 1 due January 28
Homework 2 due February 11
Homework 3 due March 7
Homework 4 due March 25
Homework 5 due April 11
Homework 6 due April 25
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 14: 12.2.2: The transport equation
January 17: 1.2: Derivation of the heat equation
January 24: 1.3: Boundary conditions for the heat equation
January 28: 1.4: Equilibrium temperature distributions
January 31: 1.5: Heat conduction in higher dimensions
February 4: 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 11: 2.4: Other boundary conditions
February 14: Review for exam 1
February 18: 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 25: 2.5.2: Solving Laplace's equation on a disk
February 28: 3.1-3.2: Introduction to Fourier series
March 4: Dr. Robinson on travel
March 7: 3.3, 3.6: Fourier series coefficients
March 18: 3.4-3.5: Calculus with Fourier series
March 21: 4.1-4.3: The wave equation
March 25: 4.4: Solving the wave equation
March 28: Review for Exam 2
April 1: 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

April 4: 4.5: Wave equation in higher dimensions
April 8: 4.6: Snell's law
April 11: 12.1-12.2: Method of characteristics
April 15: 12.3: Solving the wave equation, infinite string
April 18: 12.4: Reflections
April 22: 12.5: Solving the wave equation on a finite string
April 25: 12.6-12.7: Characteristics and quasi-linear PDE
April 29: Review for final

Final exam: May 6, 2:35pm-5:05pm

# Course policies

## Exams

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.

## Absences

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