Math 704 Methods of Applied Mathematics II: Spring 2017

 Lecture Room: 1333 Sterling Lecture Time: 12:05–12:55 MWF Lecturer: Jean-Luc Thiffeault Office: 503 Van Vleck Email: Office Hours: Wed 13:00–14:00, Thu 13:15–14:15

Final exam

Here are the solutions to the take-home final, which contain other approaches to solving the problem.

Syllabus

See the official syllabus.

Textbook

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 conformal mappings is Complex Variables: Introduction and Applications by M. J. Ablowitz & A. S. Fokas. [erratum]

If you're interested in learning more about the rigorous theory of homogenization of PDEs, there are a few textbooks available, such as Homogenization of partial differential equations by V. A. Marchenko & E. Y. Khruslov.

Prerequisites

An undergraduate course in ODEs (on the level of Math 319); an undergraduate course in Linear Algebra (on the level of Math 340); an undergraduate course on PDEs (on the level of Math 322).

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 02/06 2.1: 8,10; 2.2: 1,6,11,13,21,26,29; 2.3:3,7,13,20,21 [partial solutions] 2 02/20 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 04/12 8.1: 1,17,18 8.2: 7,8,10,16 8.4: 2,8,11 8.5: 3,4,14,18

There will be a midterm exam and a cumulative final exam. The final grade will be computed according to:

 Homework 40% Midterm exam 30% Final exam 30%

Exam Dates

The midterm exam will be given in the evening on the date below.

 Midterm exam Monday March 6, 2017 at 17:30–19:00, room Van Hise 114 [solutions] (average 87.6%, standard dev 10.6%) Final exam TBD (possibly take-home)

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

Note: there is not necessarily a one-to-one correspondence between lectures numbers and dates.

 lecture date(s) sections topic 1 01/18 1 Introduction 2 01/20 2.1–2.2 Transport equation 3 01/23 2.2 Transport equation (examples) 4 01/25 2.3 Nonlinear transport 5 01/27 2.3 Shocks 6 01/30 2.4 Wave equation 7 02/01 3.1 Eigensolutions 8 02/03 3.2 Fourier series 9 02/06 3.2–3.5 Convergence of Fourier series 10 02/08 4.1 Heat equation 11 02/10, 02/13 4.1 More on heat equation 12 02/13 4.2 Separation of wave equation 13 02/15, 02/17 4.3 Laplace equation 14 02/17 4.3 More on Laplace 15 02/20 6.1 Weak convergence 16 02/22 6.2 Green('s) functions ["The Green of Green functions"] 17 02/24 6.3 2D Green's functions 18 02/27 – Complex variable methods 19 03/01 – Conformal mappings 20 03/03 – Conformal mappings (cont'd) – 03/06 – Discussion of homework 21 03/08 8.1 Fundamental solutions 22 03/10, 03/13 – Janus particles [Legendre's equation (supplement)] 23 03/15 8.2 Symmetry and similarity 24 03/17 8.4 Burgers' equation 25 03/27 8.5 Dispersion 26 03/29 8.5 Solitons 27 03/31 9.1 Operator theory 28 04/03–07 – Multiscale analysis 29 04/10–14 – The exit time equation [extra notes] 30 04/17 – Optimization of exit times 31 04/19 – Eikonal equations 32 04/21 – Jet impact 33 04/24–26 – Winding around a point 34 04/28 – Laplace transforms 35 05/01 – Applications of Laplace transforms 36 05/03 – Singular perturbation theory