# Methods in Applied Math -- 2950

## Textbook

Principles of applied mathematics, James Keener.

• Homework 60%
• Exam(s) 40%

## Schedule

• August 28: Chapt 1.1,1.2

Homwwork (all with respect to the new book), due Sept 13: 1.1:1,2,3,6,9a; 1.2:1,3,4;1.4:4

• Sept 11,13:1.3,1.4,little bit 1.5; miscellaneous linear algebra

HW 1.3:2a,1.4:2,3,4 This handout All Due sept 20

• Sep 18,20: 2.1,some of 2.2.

HW:2.1:1,3,8,11; 2.2:2a,b,3,4,8b,9,14; Due Sep 27

Consider the equation:

x''(t) = a(t) x(t)

Assume a(t) if periodic. Prove that either both Floquet multipliers are on the unit circle or that they are both real with one inside and one outside the unit circle. (Hint: write as a system and use the fundamental matrix).

Note that since I did not get to that much of 2.2, you can hand in the HW from section 2.2 next Friday instead of Wednesday

• Week of Sept 25:3.1-3.2 ; A bit of homework Due Oct 4
• Week of Oct 2: 3.3-3.5: HW: 3.2:3a,b,c; 3.3:1; 3.4:1,2,3,4 (Due Oct 11)
• Week of Oct 9: 4.1-4.2: HW: 4.1: 1a,2,3,5,6,9 ; 4.2:1,2,3,4,7 (Due Oct20)

Suppose that f(0)=0 and f'(0) is nonzero. What is delta(f(x)) ?

• Week of Oct 16: 4.2-4.3; HW: 4.3:1,3,5,9,10 (Due Oct 27)
• Week of Oct 23: 5.1; HW:1,2,9,10,12 (Due Nov 3)
• Week of Oct 30: 5.2-5.4; Some other variational stuff.
• Week of Nov 5th: 11.1-3, HW Due Nov 10
• Week of Nov 12th: 11.3-5. HW:11.2 1,2;11.3:2,5,6,7a ,11.4:3a,5,

Find parameters A,B,C such that there is more than one periodic solution to :

u'' + epsilon u' [ A + B u^2 + C u^4] + u = 0

All due Nov 20th

• Week of Nov 19-end of term: chapt 12 Hw Due dec 8 11.5: 1,3 (Only find conditions for the Hopf - dont do the calculation) 12.1: 2,6,7a,12.3:2,11a

Use the method of averaging to find determine the range of the parameter b such that there are periodic solutions to this problem when eps is small:

x' = 1 + eps sin(x)(1-cos(y))^2

y' = 1 + eps sin(y)(1-cos(x))^2 + eps b