Matthew L. Wright
Assistant Professor, St. Olaf College

# Partial Differential Equations

## Math 330 ⋅ Fall 2017

Prof. Wright's office hours: Mon. 2–3, Tues. 9:45–10:45, Wed. 9–10, Thurs 1–2, Fri. 10:30–11:30, or by appointment in RMS 405

Help sessions: Tues. 7–9pm RNS 204 (Oct. 24 through Nov. 14)

Thursday
Sep. 7
Introduction
Heat equation
Do the following before next class:
• Complete the Syllabus Quiz.
• Read §1.1 through §1.4 in the textbook. Be sure you understand the derivations of Equations (1.2.4), (1.2.5), (1.2.9), and (1.2.10).
• Begin Homework 1.
Tuesday
Sep. 12
Heat equation: equilibrium solutions
Do the following before next class:
• Read §1.5. Note similarities between the derivation of the heat equation in one dimension and in multiple dimensions.
• Finish Homework 1 (due 4pm Thursday).
Thursday
Sep. 14
Multidimensional heat equation
Homework 1
due today
Do the following before next class:
• Finish deriving Laplace's equation in polar coordinates (solution in the notes from Thursday).
• Read §2.1 and §2.2. Come to class knowing the definition of a linear operator and the principle of superposition.
• Begin Homework 2.
Tuesday
Sep. 19
Separation of variables
Do the following before next class:
• Read §2.3. This is a long section, but the the first half (or so) should be somewhat familiar from class. Note what it means for functions to be orthogonal. Also note how infinite series are used in the solutions toward the end of this section.
• Finish Homework 2 (due 4pm Thursday).
Thursday
Sep. 21
Orthogonality and initial conditions
Homework 2
due today
Do the following before next class:
• Read the §2.3 Appendix (pages 54–55). Also read §2.4, and make sure you understand the two examples in this section.
• Begin Homework 3.
Tuesday
Sep. 26
Time-dependent solutions to the heat equation
Do the following before next class:
• Read §2.5.1 and §2.5.2. Observe how separation of variables can be used to find equilibrium temperature distributions on a 2-dimensional region.
• Finish Homework 3 (due 4pm Thursday).
Thursday
Sep. 28
Laplace's equation and separation of variables
Homework 3
due today
Do the following before next class:
• Read §3.1, §3.2, and §3.3 up to page 100.
• Begin Homework 4.
Tuesday
Oct. 3
Fourier series
Do the following before next class:
• Read the rest of §3.3. Pay careful attention to the convergence of Fourier (sine/cosine) series.
• Finish Homework 4 (due 4pm Thursday).
Thursday
Oct. 5
Differentiation of Fourier series
Take-home exam assigned
Homework 4
due today
Do the following before next class:
Tuesday
Oct. 10
Fourier series and Eigenfunction expansion
Take-home exam
due today
Do the following before next class:
• Read §3.4. Make sure you understand the conditions under which Fourier series may be differentiated term by term.
Thursday
Oct. 12
Eigenfunction expansion
Fall break! No class Tuesday, October 17.
Do the following before next class:
• Read §3.5 (it's short!). Note what happens when you integrate Fourier series.
• Do the problems on Homework 5. Typing solutions and turning them in for a grade is optional.
Thursday
Oct. 19
Wave equation
Homework 5
due today (optional)
Do the following before next class:
• Read §4.1–4.4. Make sure you understand the derivation of the wave equation and the solution of the wave equation by separation of variables.
• Begin Homework 6.
Tuesday
Oct. 24
Wave equation
Intro to Sturm-Liouville problems
Do the following before next class:
• Read §5.1–§5.3. Observe that the Sturm-Liouville equation generalizes most of the differential equations that we have considered in this course. Also note the six theorems for the regular Sturm-Liouville problem (in the big box on page 157).
• Finish Homework 6 (due 4pm Thursday).
Thursday
Oct. 26
Sturm-Liouville problems
Homework 6
due today
Do the following before next class:
• Read §5.4. Note how the Sturm-Liouville theorems are applied and how the author shows that all eigenvalues are positive without knowing the eigenfunctions.
• Read §5.5. Take special note of the linear operator notation that is used to simplify the Sturm-Liouville equation. Also note Lagrange's identity and observe how the proofs in this section follow from it.
• Read the Final Project Information sheet and start thinking about what topic you might want to study.
• Begin Homework 7.
Tuesday
Oct. 31
Sturm-Liouville problems
Operators, orthogonality, and self-adjointness
Do the following before next class:
• Re-read §5.5 to understand the proofs within. If you want to better understand connections between differential equations and linear algebra, read the Appendix to 5.5.
• Read §5.6. Observe how the Rayleigh quotient can provide a bound on the lowest eigenvalue.
• Finish Homework 7 (due 4pm Thursday).
Thursday
Nov. 2
Sturm-Liouville problems
Rayleigh quotient and eigenvalue bounds
Homework 7
due today
Do the following before next class:
• Read §6.1 and §6.2. Observe how Taylor series can be used to approximate the value of a derivative of a function using values of the function at nearby points.
• Continue thinking about what you might want to work on for the Final Project.
• Begin Homework 8.
Tuesday
Nov. 7
Rayleigh quotient
Finite difference methods
Do the following before next class:
• Read §5.7. This example should look familiar now!
• Read §6.3.1–§6.3.3. Observe how finite difference approximations for derivatives can be used to approximate solutions to the heat equation. Then take a look at the stability analysis in §6.3.4.
• Finish Homework 8 (due 4pm Thursday).
Thursday
Nov. 9
Finite difference methods
Approximate solution to the heat equation: Mathematica notebook
Homework 8
due today
Do the following before next class:
• Read §6.3.4, which expands on what we said in class about stability analysis. Read §6.3.6, about matrix notation, noting connections to linear algebra. Also take a look at the short subsections §6.3.7 and §6.3.8.
• Complete the Project Planning Survey on Moodle.
• Begin Homework 9.
Tuesday
Nov. 14
Finite difference methods
Richardson's scheme and Crank-Nicolson scheme
Do the following before next class:
• Read §6.5. Observe how finite differences can be used to approximate the wave equation.
• Finish Homework 9 (due 4pm Thursday).
Thursday
Nov. 16
Finite difference methods for the wave equation
Take-home exam assigned
Homework 9
due today
Do the following before next class:
Tuesday
Nov. 21
Higher-dimensional PDEs
Take-home exam
due today
Thanksgiving break! No class Thursday, Nov. 23
Begin working on your project.
Tuesday
Nov. 28
Work on projects
Work on your project. Identify sources, gather information, and make an outline for what you will include in your paper.
Thursday
Nov. 30
Work on projects
Work on your project.
Tuesday
Dec. 5
Work on projects
Work on your project.
Thursday
Dec. 7
Work on projects
Work on your project.
Tuesday
Dec. 12
Work on projects