# Math 322 Applied Mathematical Analysis II: Spring 2017

 Lecture Room: B139 Van Vleck Lecture Time: 11:00–11:50 MWF Lecturer: Jean-Luc Thiffeault Office: 503 Van Vleck Email: Office Hours: Wed 13:00–14:00, Thu 13:15–14:15

## Final exam

The final is this Saturday May 6, 2017 at 10:05–12:05, room Van Vleck B239.

• No notes, books, calculators, .... Just a pen/pencil and eraser.
• The exam consists of four questions.
• The exam material is cumulative, but it will be heavily skewed towards the post-midterm material.
• Viscous flow and numerical methods are not on the exam.
• You should definitely know how to carry out separation of variables, both in Cartesian and cylindrical polar coordinates.
• You do not need to know the definition of Bessel functions or the equation they satisfy. If there is such a question on the exam, this information will be provided.
• You do not need to know the derivation of Legendre polynomials.

## Syllabus

See the official syllabus.

## Textbook

The textbook for the class is Partial Differential Equations and Boundary Value Problems With Applications by Mark A. Pinsky (Third Edition).

I will also give a few lectures from D. J. Acheson's Elementary Fluid Dynamics (Oxford University Press). ISBN: 0198596790. This is not a required book.

## Prerequisites

Math 319 and 321.

## 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 02/03 02/06 0.1.1: 2,4,7; 0.1.2: 1–4, 0.1.4: 1–3; 0.1.5: 1–4; 0.2.2: 1–3; 0.2.3: 1–8; 0.2.4: 1–4 2 02/20 0.3: 1,2,3,6,10,15; 1.1: 2,3,4,9,10,18,26,27,29,33,34; 1.2: 1,5,6,7,13,16; 1.3: 1,4,10–13; 1.4: 2,3; 1.5: 1,3 [Correction: in 0.3.3 for some printings f-bar should be f-hat.] 3 03/06 03/10 1.6: 1,6,8,10; 2.1: 1,2,4,5,10,11,12,13,14,21; 2.2: 1,3,7,8,12,14,18,21 4 03/27 2.3: 1,3,8,11; 2.4: 1,4,5,7,17,18 5 04/05 2.5: 1,6,14,15; 3.1: 4,5,8,9,10,18,26 6 04/19 3.3: 1,8,9; 5.1: 2,3,4,5,7,15,16; 5.2.6: 1,2,6,7,8

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 82%, standard dev 12%) Final exam Saturday May 6, 2017 at 10:05–12:05, room Van Vleck B239

## 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 0.1.1–3 Introduction: What is a PDE? 2 01/20 0.1.4 Heat equation 3 01/23, 01/25 0.1.5; 0.2 Classification; Separation of variables 4 01/25, 01/27 0.2 Separation of variables with BC 5 01/27 0.3 Orthogonal functions 6 01/30 0.3 Orthogonal functions (cont'd) 7 02/01, 02/06 1.1 Orthogonal functions (end); Fourier series – 02/03 – Homework discussion 8 02/06, 02/08 1.2 Convergence of Fourier series 9 02/10 1.3 The Gibb's phenomenon 10 02/13 1.4–1.5 Parseval's theorem; Complex form of Fourier series 11 02/15, 02/17 1.6 Sturm-Liouville eigenvalue problems 12 02/20, 02/24 1.6.4–1.6.5 General Sturm-Liouville problems [Matlab code for fig 1.6.2] 13 02/22, 02/24 4.2.2 Legendre's equation 14 02/27, 03/01 2.1.2–2.1.5 The heat equation in three dimensions [extra notes for general boundary condition] 15 03/03, 03/08 2.2.1–2.2.3 Initial Value Problem in a slab – 03/06 – Discussion of homework 17,18 03/10 2.3 Nonhomogeneous boundary conditions [example] 19 03/13 2.4.1–2.4.5 The vibrating string 20 03/15 2.4 The vibrating string (cont'd) 20 03/17 2.4 Resonant behavior 21 03/27 2.5 Multidimensional problems 22 03/29 2.5 Nodal lines 23 03/28 3.1.1–3.1.3 BVPs in cylindrical coordinates 24 04/03 3.1.6,3.1.7 BVPs in cylindrical coordinates (cont'd) 25 04/05 3.2,3.3.1 Wave equation and Bessel functions 26 04/08 5.1 Fourier transforms 27 04/10 5.1,5.2 Fourier transforms (cont'd) 28 04/12 5.2.4–5.2.6 Method of images 30 04/14–20 Acheson 229–232 Viscous flow and corner eddies (Start at bottom of page, at "Example".) 33 04/24 7.2 Numerical solution 34 04/26 7.2 Numerical solution examples (Matlab programs) – 04/28, 05/01 – Discussion of homework – 04/28–05/03 – Discussion of homework