Modern Applied Mathematics I

Math 413-001

Fall 2023

The following table contains the schedule for the course. This page will be updated regularly throughout the semester.

Week Date(s) Notes Pages Book Sections
I. Dimensional Analysis and Scaling
1 08/21 - 08/25     1. Basic Simplification 1-6 [H] 1.1, [LS] 6.1, [L] 1.1.1
    2. Conditioning and Sensitivity 7-11 [LS] 6.1
2 08/28 - 09/01     3. Dimensional Methods 12-17 [H] 1.2, [LS] 6.2, [L] 1.1.2
    4. The Buckingham Pi Theorem 18-24 [H] 1.3, [L] 1.1.3, 1.1.4
3 09/04 - 09/08     5. Scaling 25-32 [LS] 6.3, [L] 1.2.1, 1.2.3
II. Perturbation Methods
    1. Formal Approximations for Root Finding 33-37 [H] 2.2.1, 2.2.2, [M] 1.2, 1.3
    2. Expansions via Computer Algebra 38-40 [H] 2.2.2
4 09/11 - 09/15     3. The Implicit Function Theorem 41-43
    4. Justification and Error Estimates 44-48 [M] 1.4
    5. The Newton Polygon 49-56 [M] 1.5
5 09/18 - 09/22     6. Rescaled Coordinates 57-61 [H] 2.3, 2.4, [M] 1.6
    7. Bifurcations 62-68 [L] 1.3.2
III. Perturbations of Differential Equations
6 09/25 - 09/29     1. Perturbations of Second-Order Linear Equations 69-71 [M] 2.1
    2. Regular Perturbations of Initial Value Problems 72-76 [H] 2.2.3, [L] 3.1.1, [M] 2.4
    Review for the Midterm Exam
7 10/02 - 10/04     3. Regular Perturbations of Boundary Value Problems 77-79 [M] 2.5
10/05     Midterm Exam, 12:00pm-1:15pm
8 10/10     No class!
10/11 - 10/13     4. Oscillatory Problems and Secular Terms 80-83 [L] 3.1.2, [M] 4.1
9 10/16 - 10/20     5. Poincare-Lindstedt Expansions 84-90 [L] 3.1.3, [M] 4.2
    6. Boundary Layer Analysis 91-94 [H] 2.5, [L] 3.2.3, 3.3.1
10 10/23 - 10/27     7. Matched Asymptotic Expansions 95-103 [H] 2.6, [L] 3.3.2, 3.3.3, 3.3.4
IV. Stability and Bifurcations
    1. Qualitative Study of Dynamical Systems 104-112 [L] 1.3.1, 2.1
11 10/30 - 11/03     2. Dynamics of Scalar Flows 113-116 [L] 1.3.1, 1.3.2
    3. Effects of Parameter Variation 117-123 [L] 1.3.2, 2.4
12 11/06 - 11/10     4. Equilibrium Stability in Higher Dimensions 124-130 [H] 3.2.3, 3.5, [L] 2.2, 2.3
    5. Case Study: The Tacoma Narrows Bridge 131-139
V. Modeling with Differential Equations
13 11/13 - 11/17     1. The Law of Mass Action 140-144 [H] 3.2.1, [L] 2.5.1
    2. Conservation Laws for the Kinetic Equations 145-148 [H] 3.2.2, [L] 2.5.1
14 11/20 - 11/21     3. Michaelis-Menten Enzyme Kinetics 149-156 [H] 3.3.1, 3.6, [L] 2.5.2
11/23     No class!
15 11/27 - 12/01     4. The SIR Model for Epidemics 157-161 [H] 3.1.3, [L] 2.6.3
    5. Epidemics with Reinfection and Vaccination 162-167 [H] 3.3.2, 3.5.3, [L] 2.6.1
16 12/07     Final Exam, 10:30am-1:15pm

For the course, I will draw material from the following books: