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:

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]