Jean-Luc Thiffeault's Homepage

Math 704 Methods of Applied Mathematics II: Spring 2020


Lecture Room: 1333 Sterling
Lecture Time: 9:55–10:44 MWF
Lecturer: Jean-Luc Thiffeault
Office: 503 Van Vleck
Email: jeanluc@[domainname], where [domainname] is math point wisc point edu
Office Hours: M 11:00–11:45; W 2:20–3:10

Syllabus

See the official syllabus.

Textbooks and resources

The required textbook for the class is Introduction to Partial Differential Equations by Peter Olver. Homework problems will mostly be assigned from this book, so it's important to have access to it.

A good optional textbook for complex variables and conformal mappings is Complex Variables: Introduction and Applications by M. J. Ablowitz & A. S. Fokas. [erratum]

If you'd like to learn more about fluid dynamics a good introductory book is D. J. Acheson's Elementary Fluid Mechanics (Oxford University Press). ISBN: 0198596790.

There is a large archive of papers that use the Uniform Transform Method.

Homework

Every two weeks or so I will assign homework from the textbook (or other sources) and post it here. The homework will be due in class about two weeks later.

homework due date problems
1 02/07 2.1: 8,10; 2.2: 1,6,11,13,21,26,29; 2.3:3,7,13,20,21
2 02/21 2.4: 5,12–15; 3.2: 17,22,24,30,40,42,43,59,60,61; 3.3: 2,10; 3.5: 5,6,20,22
3 03/06 4.1: 1,4,8,13,14,16,17 4.2: 8,11,22,28 4.3: 1,18,30,31,43,49,50
4 03/27 6.1: 9,20,41 6.2: 2,4,12 6.3: 4,5,6,17,18,31 Additional questions on conformal mappings
5 04/15 8.1: 1,17,18 8.2: 7,8,10,16 8.4: 2,8,11 8.5: 3,4,14,18

Piazza

We'll use Piazza Q&A for discussions about the class and related topics. Feel free to post questions and answers there about homeworks and exams, logistics, or relevant interesting things you found on the web. Note that we will only use Piazza for the Q&A feature, not for posting the actual homeworks.

Schedule of Topics

lecture date(s) sections topic
1 01/22 1 Introduction
2 01/24 2.1–2.2 Transport equation
3 01/27 2.2 Transport equation (examples)
4 01/29 2.3 Nonlinear transport
5 01/31 2.3 Shocks
6 02/03 2.4 Wave equation
7 02/05 3.1 Eigensolutions
8 02/07 3.2 Fourier series
9 02/10 3.2–3.5 Convergence of Fourier series
10 02/10 4.1 Heat equation
11 02/12, 02/14 4.1 More on heat equation
12 02/17 4.2 Separation of wave equation (read notes)
13 02/17 4.3 Laplace equation
14 02/19 4.3 More on Laplace
15 02/20 6.1 Weak convergence
16 02/24 6.2 Green('s) functions ["The Green of Green functions"]
17 02/28 6.3 2D Green's functions
18 03/02 – Complex variable methods
19 03/04 – Conformal mappings
20 03/06 – Conformal mappings (cont'd)
21 03/09 8.1 Fundamental solutions
23 03/11 8.2 Symmetry and similarity
24 03/13 8.4 Burgers' equation
25 03/23 8.5 Dispersion [video]
26 03/25 8.5 Solitons [video]
27* 03/25, 03/27 – Laplace transforms [video part 1] [video part 2] [video part 3]
28* 03/30 – Applications of Laplace transforms [video]
29* 04/01 9.1 Operator theory [video]
30* 04/03 – Multiscale analysis [video]
31* 04/06 – Multiscale analysis (cellular flow example) [video] [Matlab code]
32* 04/08 – Homogenization for a perforated domain [video]
33* 04/10 – The Fokker–Planck equation (Kolmogorov forward equation) [video]
34* 04/13 – Laplace transform solution of FP [video]
35* 04/15 – The adjoint equation (Kolmogorov backward equation) [video]
36* 04/17 – The mean exit time equation [video]
37* 04/20 – Mean exit time examples [video]
38* 04/22 – The narrow exit problem [video]
39* 04/24 – The Unified Transform Method [video]
40* 04/27 – The Unified Transform Method (cont'd) [video]

* There is not necessarily a one-to-one correspondence between actual lectures numbers and those written on the notes.

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