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:

M.H. Holmes: Introduction to the Foundations of Applied Mathematics, Springer, 2019. [H]
C.C. Lin, L.A. Segel: Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, 1988. [LS]
J.D. Logan: Applied Mathematics, Wiley, 2013. [L]
J.A. Murdock: Perturbations, SIAM, 1999. [M]