Modern Applied Mathematics I

Math 413-001 (Fall 2025)

Detailed Syllabus

The following table contains a tentative 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 08/25 - 08/29     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 09/01     No class! (Labor Day)
09/02 - 09/05     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 09/08 - 09/12     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 09/15 - 09/19     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 09/22 - 09/26     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 09/29 - 10/03     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 10/06 - 10/10     3. Regular Perturbations of Boundary Value Problems 77-79 [M] 2.5
10/08     Midterm Exam, 4:30pm-5:45pm
8 10/13     No class! (Fall Break)
10/14 - 10/17     4. Oscillatory Problems and Secular Terms 80-83 [L] 3.1.2, [M] 4.1
9 10/20 - 10/24     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 10/27 - 10/31     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 11/03 - 11/07     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 11/10 - 11/14     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 11/17 - 11/21     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 11/24 - 11/25     3. Michaelis-Menten Enzyme Kinetics 149-156 [H] 3.3.1, 3.6, [L] 2.5.2
11/26     No class! (Thanksgiving Break)
15 12/01 - 12/05     4. The SIR Model for Epidemics 157-161 [H] 3.1.3, [L] 2.6.3
16 12/08     5. Epidemics with Reinfection and Vaccination 162-167 [H] 3.3.2, 3.5.3, [L] 2.6.1
12/10     Final Exam, 4:30pm-7:15pm

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