## Course information

Department of Mathematics

University of California

Davis, CA 95616, USA

e-mail: `jkhunter@ucdavis.edu`

Office Phone: (530) 601-4444 x4016

Office: MSB 3230

Office hours: W 3:15 – 4:30 p.m.; R 1:00 – 2:00 p.m.

Classroom: Bainer 1128

## Announcements

Solutions to the Final are here.

## Convolutions

A history of the convolution, and how it got its name, is here.

Therefore, the operation is in some sense the "revolving" of one of the input functions with the other one. Particularly in mathematics, physics, and related areas, the verb commonly used to designate such a revolving is "to convolve." This verb comes from the Latin words con and volvere, which mean "together" and "roll up," respectively; thus, convolve means "roll up together." Accordingly, the action of convolving is called convolution. This is why (3) is usually known today as the CCO, or simply convolution, although it has been given many other names in the past.

## Text

Basic Partial Differential Equations, David Bleeker and George Csordas.

## Syllabus

The course will cover Chapters 6 – 9 of the text: Laplace's equation; Fourier transforms; numerical methods; PDEs in higher dimensions; and other topics of interest to the class.

## Important Dates

- First class: Mon, Jan 8
- Last day to add: Wed, Jan 24
- Last day to drop: Mon, Feb 5
- Last class: Fri, Mar 16
- Academic holidays: Mon , Jan 15; Mon, Feb 19

## Exams

There will be one Midterm and a Final

- Midterm: Wed, Feb 21 (in class)
- Final: Mon, Mar 19, 1:00 – 3:00 p.m. (Exam Code C)

## Midterm

Solutions to the Midterm are here.

The midterm will be in class, Wed Feb 21. It will cover the following sections.

- 6.1 Laplace's equation.
- 6.2 Dirichlet problem for rectangle by separation of variables.
- 6.3 Dirichlet problem for discs and annuli. Poisson integral formula. Mean value theorem.
- 6.4 Maximum principles. Uniqueness for the Dirichlet problem.
- 7.1 Complex Fourier series.
- 7.2 Fourier transform and its properties. The convolution theorem.
- 7.3 Inversion theorem and Parseval's identity.
- 7.4 Solution of PDEs by Fourier transforms. Application to the heat, wave, and Laplace equations.

## Homework

Homework problems will be assigned each week and collected in class.

## Grade

The course grade will be based on (weights in parentheses):

- Homework (25%)
- Midterm (30%)
- Final (45%)

## Other resources

If you want to type good-looking mathematics, the standard tool is LaTeX, or one of its many variants. See here to get started.

MATLAB is a useful platform for the numerical solution of PDEs (and many other things).

## Homework Sets

Problem numbers refer to the exercises in the text.

**Set 1** (Friday, Jan 19)

- Sec. 6.1, p.348: 1, 2, 4, 7
- Sec. 6.2, p.363: 5, 8
- Optional: Use MATLAB, or another programming language, to make a surface plot of the sum in 5(b)

**Set 2** (Friday, Jan 26)

- Sec. 6.3, p.380: 1(e), 3(e), 8, 10, 12
- Sec. 7.1, p.427: 1(a), 1(f), 2

**Set 3** (Friday, Feb 2):

- Sec. 6.4, p.393: 1, 3, 4, 5, 6, 7, 8

**Set 4** (Monday, Feb 12):

- Sec 7.1, p.427: 3(a)
- Sec 7.2, p.440: 1, 3, 7, 10, 11

**Set 5** (Friday, Feb 23):

- Sec 7.4, p.478: 2, 5, 6, 8, 11, 14

**Set 6** (Friday, Mar 9):

- The problem set is here.