Matthew L. Wright
Assistant Professor, St. Olaf College

# Partial Differential Equations

## Math 330 ⋅ Fall 2019

Prof. Wright's office hours in RMS 405: Mon. 9:00–10:00, Tues. 9:30–10:30, Wed. 2:00–3:00, Thurs 1:00–2:00, Fri. 9:00–10:00, whenever the door is open, or by appointment

Help sessions: Mondays 7:30–8:30 in RNS 204

Thursday
September 5
Introduction
ODE review
Do the following before next class:

Optionally, watch the following video: But what is a partial differential equation? (3Blue1Brown).

Tuesday
September 10
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 12
Heat equation
Homework 1
due today
Do the following before next class:
Tuesday
September 17
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). You may want to use the LaTeX template on Overleaf.
Thursday
September 19
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, and bring your answer to class on Tuesday.
• Begin Homework 3.
Tuesday
September 24
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). You may want to use the LaTeX template on Overleaf.
Thursday
September 26
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.

Optionally, watch the following video: Solving the heat equation (3Blue1Brown).

Tuesday
October 1
Laplace's equation and separation of variables
Do the following before next class:
Thursday
October 3
Fourier series
Take-home exam assigned
Homework 4
due today
Do the following before next class:

Extra credit opportunity: Attend either of Dr. Eugenia Cheng's talks on Thursday October 3 (3:30pm in Tomson 280 or 7:00pm in Carleton Weitz Cinema) and answer these two questions on Moodle to earn two extra-credit homework points.

Tuesday
October 8
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.

Optionally, watch the following video: But what is a Fourier series? From heat flow to circle drawings (3Blue1Brown).

Thursday
October 10
Differentiation of Fourier series
Fall break! No class Tuesday, October 15.
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 (due 4pm Thursday; LaTeX solution template).
Thursday
October 17
Eigenfunction expansion
Homework 5
due today
Do the following before next class:
Tuesday
October 22
Wave equation
Do the following before next class:
Thursday
October 24
Wave equation
Homework 6
due today
Do the following before next class:
Tuesday
October 29
Intro to Sturm-Liouville problems
Do the following before next class:

Extra credit opportunity: Attend at least one of the student talks at the Northfield Undergradute Mathematics Symposium (NUMS) on Tuesday, October 29 (3:40–6:50pm in RNS 310) and answer these questions on Moodle to earn two extra-credit homework points.

Thursday
October 31
Sturm-Liouville problems
Operators and orthogonality
Homework 7
due today
Do the following before next class:
• Read §5.4 and §5.5. Note the role of Lagrange's identity and Green's formula in the proofs presented in this section. 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 5
Sturm-Liouville problems
Rayleigh quotient and eigenvalue bounds
Do the following before next class:
• Read §5.6. Pay special attention to the minimization principle: the Rayleigh quotient can provide a bound on the lowest eigenvalue.
• Read §5.7. This example should look familiar now!
• Finish Homework 8 (due 4pm Thursday; LaTeX solution template).
• Continue thinking about what you might want to work on for the Final Project.
Thursday
November 7
Sturm-Liouville problems
Large eigenvalues
Homework 8
due today
Do the following before next class:
• Read §5.8. This section goes into more detail about the first problem we worked on in 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.
• Decide what you would like to work on for the Final Project, then complete the Final Project Planning Survey on Moodle.
• Begin Homework 9.
Tuesday
November 12
Intro to 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; LaTeX solution template).
• If possible, bring a computer with Mathematica to class on Thursday!
Thursday
November 14
Finite difference methods for the heat equation
Homework 9
due today
Do the following before next class:
• Re-read §6.3.1–§6.3.3. Focus on §6.3.4, which expands on the stability analysis that we examined in class. 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— we will examine some of these other schemes next week.
• Begin Homework 10.
• If possible, bring a computer with Mathematica to class on Tuesday!
Tuesday
November 19
Finite difference methods for the heat equation
Do the following before next class:

Finish Homework 10 (due 4pm Thursday; LaTeX solution template). This homework involves modifying Mathematica code and producing plots. You may copy bits of Mathematica code and plots into your LaTeX document. Or, you may upload Mathematica notebooks along with your LaTeX document—if you do this, clearly state in your LaTeX document where the reader can find for your code and plots.

Thursday
November 21
Finite difference methods
Take-home exam assigned
Homework 10
due today
Complete the take-home exam before next class.
Tuesday
November 26
Finite element method
Thanksgiving break! No class Thursday, Nov. 22
Do the following before next class:
Tuesday
December 3
Final projects