Matthew L. Wright
Assistant Professor, St. Olaf College

# Partial Differential Equations

## Math 330 ⋅ Fall 2018

Prof. Wright's office hours: Mon. 1–2, Tues. 10–11, Wed. 2–3, Thurs 10–11, Fri. 1–2, whenever the door is open, or by appointment in RMS 405

Help sessions: Tuesdays 7–8pm in Tomson 186

Thursday
September 6
Introduction
ODE review
Do the following before next class:
Tuesday
September 11
Heat equation
Do the following before next class:
• Read §1.3 and §1.4. Note three possible boundary conditions discussed in §1.3. Then note how the heat equation, with certain boundary conditions, can be solved for equilibrium solutions in §1.4.
• Finish Homework 1 (due 4pm Thursday). You may want to use the LaTeX template on Overleaf.
Thursday
September 13
Heat equation
Homework 1
due today
Do the following before next class:
Tuesday
September 18
Multidimensional heat equation
Do the following before next class:
• Read §2.1 and §2.2. Note the definition of a linear operator and the principle of superposition.
• Finish Homework 2 (due 4pm Thursday).
Thursday
September 20
Separation of variables
Homework 2
due today
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. Answer the reading questions (TeX source), and bring your answer to class on Tuesday.
• Begin Homework 3.
Tuesday
September 25
Separation of variables, continued
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.
• Finish Homework 3 (due 4pm Thursday).
Thursday
September 27
Orthogonality and initial conditions
Time-dependent solutions to the heat equation
Homework 3
due today
Do the following before next class:
• Re-read §2.4. Note how orthogonality of sine and cosine functions is used to find the coefficients of the series solutions in this section.
• Begin Homework 4.
Tuesday
October 2
Laplace's equation and separation of variables
Do the following before next class:
• Read §3.1 and §3.2. Note the convergence theorem for Fourier series.
• Finish Homework 4 (due 4pm Thursday).
Thursday
October 4
Fourier series
Take-home exam assigned
Homework 4
due today
Do the following before next class:
Tuesday
October 9
Fourier series
Take-home exam
due today
Do the following before next class:
• Read §3.3. Pay close attention to the definitions, examples, and convergence properties of Fourier sine and cosine series.
• Read §3.4. Note the conditions under which a Fourier (cosine/sine) series can be differentiated term by term.
• Take a look at Homework 5.
Thursday
October 11
Differentiation of Fourier series
Fall break! No class Tuesday, October 16.
Do the following before next class:
• Re-read §3.4. Make sure you understand the conditions under which a Fourier (cosine/sine) series can be differentiated term by term. Also note the method of eigenfunction expansion.
• Read §3.5 (it's short!). Note what happens when you integrate Fourier series.
• Finish Homework 5.
Thursday
October 18
Eigenfunction expansion
Homework 5
due today
Do the following before next class:
Tuesday
October 23
Eigenfunction expansion
Wave equation
Do the following before next class:
Thursday
October 25
Wave equation
Homework 6
due today
Do the following before next class:

For two extra-credit points, attend one of these two talks by Minah Oh on Monday or Tuesday, and complete these two questions on Moodle.

Tuesday
October 30
Finish D'Alembert's solution to the wave equation
Intro to Sturm-Liouville problems
Do the following before next class:
Thursday
November 1
Sturm-Liouville problems
Homework 7
due today
Do the following before next class:
• Read §5.4 and §5.5. To better understand connections between differential equations and linear algebra, read the Appendix to 5.5.
• Continue thinking about what you might want to work on for the Final Project.
• Begin Homework 8.
Tuesday
November 6
Sturm-Liouville problems
Do the following before next class:
• Re-read §5.5. Note the role of Lagrange's identity and Green's formula in the proofs presented in this section.
• Read §5.6. Observe how the Rayleigh quotient can provide a bound on the lowest eigenvalue.
• Finish Homework 8 (due 4pm Thursday).
• Continue thinking about what you might want to work on for the Final Project.
Thursday
November 8
Sturm-Liouville problems
Rayleigh quotient and eigenvalue bounds
Homework 8
due today
Do the following before next class:
• Read §5.7. This example should look familiar now!
• 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.
• Complete the Final Project Planning Survey on Moodle. See also the Final Project Information.
• Begin Homework 9.
Tuesday
November 13
Finite difference methods
Do the following before next class:
• Re-read §6.2. Note how the finite difference approximations can be applied to second derivatives.
• Read §6.3.1–§6.3.3. Observe how finite difference approximations for derivatives can be used to approximate solutions to the heat equation.
• Finish Homework 9 (due 4pm Thursday).
Thursday
November 15
Finite difference methods: Mathematica notebook
Take-home exam assigned
Homework 9
due today
Complete the take-home exam before next class: PDF file, TeX source, Moodle link for file upload.

For two extra-credit points, attend the Research Seminar by Jasper Weinburd (Nov. 16, 3:40pm, RNS 204), and complete these two questions on Moodle.

Tuesday
November 20
Finite difference methods
Take-home exam
due today
Thanksgiving break! No class Thursday, Nov. 22
Do the following before next class:
• Re-read §6.3. Focus on §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.
• Begin Homework 10.
Tuesday
November 27
Guest presentation
Finite difference computational investigation: Richardson Scheme and Crank-Nicolson Scheme
Do the following before next class:
• Read §6.5. (It's short!) Observe how finite differences can be used to approximate the wave equation.
• Finish Homework 10 (due 4pm Thursday).
Thursday
November 29
Finite difference approximation of the wave equation
Homework 10
due today
Tuesday
December 4
Work on your final project. Prepare an outline of what you intend to write in your paper. One person per group should upload the outline here.
Thursday
December 6
outline of paper
due today
Work on your final project. Prepare a rough draft of your paper to hand in on Tuesday. One person per group should upload the rough draft here.
Tuesday
December 11
rough draft of paper
due today
Finish your final project. One person per group should upload the paper here. Each person must complete the Final Project Evaluation (on Moodle).
Wednesday
December 19
Project presentations
2:00 – 4:00pm