Welcome to Partial Differential Equations! For course info and policies, please see the syllabus. For grades, log into Moodle. If you need help or have questions, please contact Prof. Wright

**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

- Complete the Syllabus Quiz.
- Read §1.1 through §1.2 in the textbook. Answer the reading questions, and bring your answers to class on Tuesday.
- Begin Homework 1.

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

- 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.

- Read §1.5, answer the reading questions, and bring your answers to class on Tuesday.
- Begin Homework 2.

- 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.

- 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.

- 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.

September 26

Time-dependent solutions to the heat equation

due today

- 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.
- Read §2.5.1 and §2.5.2. Answer the reading questions, and bring your answer to class on Tuesday.
- Begin Homework 4.

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

- Read §3.1 and §3.2. Note the convergence theorem for Fourier series.
- Finish Homework 4 (due 4pm Thursday; LaTeX solution template).

- Complete the take-home exam: LaTeX template, Moodle submission link.

**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.

- 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).

- 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).

- Read §4.1–4.4. Answer the reading questions and bring your answers to class on Tuesday.
- Begin Homework 6.

- Finish Homework 6 (due 4pm Thursday; LaTeX solution template).

- Begin Homework 7.
- Read §5.1–§5.3. Answer the reading questions, and bring your answers to class on Tuesday.

- Finish Homework 7 (due 4pm Thursday; LaTeX solution template).
- Read the Final Project Information and start thinking about what topic you might want to study.

**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.

- 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.

- 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.

- 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.

- 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!

- 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!

November 19

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.

November 21

*Take-home exam assigned*

due today

December 3

- Review the guidelines for the paper in the final project information.
- Gather sources on your project topic.
- Prepare an outline of what you intend to write in your paper. By Thursday, one person per group should upload the outline here.
- You may wish to look at two sample papers written by students in Math 330 in previous years: Navier-Stokes Equations and Tumor Growth.

- Review the guidelines for the paper in the final project information.
- Prepare a rough draft of your paper to hand in on Tuesday. Your paper doesn't need to be finished, but your draft should show the progress that your group has made. One person per group should upload the rough draft here. The professor will provide comments on the draft for your group.

- Review the guidelines for the paper in the final project information.
- One person per group should upload the paper here.
- Prepare to give a 10–15 minute presentation on your project to the class on December 18.
- Each person must complete the Final Project Evaluation.