AMS 501: Differential Equations and BVPs I (Ordinary Differential Equations)
Fall 2011

Time: Monday & Wednesday 3:50pm - 5:10pm
Location: Harriman Hall 116


Instructor: Prof. Xiangmin (Jim) Jiao
Email:  Phone: 631-632-4408
Office hours: Mon. & Fri., 2:20pm-3:40pm
Office: Math Tower 1-115

TA: Vladimir Dyedov
Email:
Office hours: Tue. & Thur. 3pm-4pm
Office: Math Tower S-250



[ Course Description | Course Outline | Course Policy | Homework and Sample Tests | Class Schedule ]

Announcements (back to top)

  • Class for 8/29 was cancelled due to Hurricane Irene.

Course Description (back to top)

From course catalog: Examples of initial and boundary value problems in which differential equations arise. Existence and uniqueness of solutions, systems of linear differential equations, and the fundamental solution matrix. Power series solutions, Sturm-Liouville theory, eigenfunction expansion, Green's functions.

Prerequisite: Prior knowledge of linear algebra and calculus (at the level of AMS 510).

Required Textbooks

  • Differential Equations and Boundary Value Problems: Computing and Modeling, 4th edition, by C. Henry Edwards and David E. Penney, Prentice Hall 2008. ISBN 0131-56107-3. (Earlier editions are OK.)

Supplementary Textbooks

  • Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, by Carl M. Bender and Steven A. Orszag. Springer, 1999. ISBN 0-387-98931-5.

Course Outline (back to top)


The course contains three parts:

  1. Linear Differential Equations

    • Introduction to differential equations (1 lecture)

    • Linear first-order equations (1 lecture)

    • Homogeneous linear equations of higher order (1 lecture)

    • Nonhomogeneous linear equations (2 lecture)

    • Systems of linear differential equations (2 lectures)

    • Laplace transform methods (if time permits)

  2. Nonlinear Differential Equations and Boundary Value Problems

    • First-order nonlinear differential equations (1 lecture)

    • High-order differential equations (1 lecture)

    • Nonhomogeneous BVPs (1 lecture)

    • Eigenvalue problems and Sturm-Liouville theory (1-2 lectures)

  3. Singularities and Power Series Methods

    • Classification of singularities (0.5-1 lecture)

    • Power series methods (1-2 lectures)

    • Approximate solutions near ordinary and regular singular points (3 lectures)

    • Analysis and approximate solutions near irregular singular points (2 lectures)

    • Spontaneous singularities (1 lecture)

    • Introduction to perturbation theory (if time permits)

Course Policy (back to top)


Assignments

Homework assignments are due in class typically two weeks after they are assigned. You are allowed to discuss course materials and homework problems in small groups, but limited to discussion of general ideas only. You must write your solutions completely independently. Under no circumstances may you copy solutions from any source, including but not limited to other students solutions, official solutions distributed in past terms, and solutions from courses taught at other universities. Violation of these rules may result in disciplinary actions.

Exams

The exams (including two tests and the final exam) are closed-book, but you are allowed to bring a one-sided, one-page, letter-size cheat sheet, which you must prepare by yourself.

Attendance

All students are expected to attend all the lectures and exams.

Grading

  • Assignments: 30%

  • Two tests: 40%

  • Final exam: 30%

Homework and Sample Tests (back to top)


Assignments

Sample Tests

Class Schedule (back to top)

 

  • Important: All schedules are tentative and are subject to change.

In the reading lists, E&P means Edwards and Penney, and B&O means Bender and Orszag.

Week

Date

Topic

Reading

Notes

1

Mon 08/29

Class cancelled due to Hurricane Irene




Wed 08/31

Course overview; Introduction to DEs

E&P §1.1-2,
B&O §1.1-2

HW1 out 

2

Mon 09/05

No class (Labor day observed)




Wed 09/07

Existence and uniqueness of IVPs;
Linear first-order equations

E&P §A.1, notes
E&P §1.4,1.5


3

Mon 09/12

General solutions of linear equations

E&P §3.1-2



Wed 09/14

Homogeneous linear equations

E&P §3.3, B&O §1.4

HW1 due

4

Mon 09/19

Nonhomogeneous linear equations

E&P §3.5, B&O §1.5

HW2 out


Wed 09/21

Green's functions for nonhomogeneous equations

B&O §1.5


5

Mon 09/26

More on Green's functions

notes



Wed 09/28

No class (Correction day for Friday)



6

Mon 10/03

Systems of linear differential equations

E&P §4.1,5.2



Wed 10/05

Systems of linear differential equations with non-diagonalizable matrices

E&P §5.4-5

HW2 due

7

Mon 10/10

First mid-term (covers material till 10/05)


HW3 out


Wed 10/12

Nonhomogeneous systems of equations

E&P §5.6; notes


8

Mon 10/17

First-order nonlinear differential equations

B&O §1.6



Wed 10/19

Higher-order nonlinear differential equations

B&O §1.7; notes


9

Mon 10/24

Eigenvalue proble;  Sturm-Liouville theory

E&P §3.8, B&O §1.8

HW3 due; HW4 out 


Wed 10/26

Eigenfunction expansions

E&P §10.1-2


10

Mon 10/31

Classification of singularities

B&O §3.1



Wed 11/2

Introduction to power series

E&P §10.1


11

Mon 11/7

More on local behavior near ordinary points; Review

E&P §8.2, B&O §3.2; slides

HW4 due


Wed 11/9

Second mid-term (covers material till 11/07)


HW5 out

12

Mon 11/14

Local behavior near regular singular points

E&P §8.3, B&O §3.3



Wed 11/16

Methods of Frobenius: exceptional cases

E&P §8.4


13

Mon 11/21

Gamma function and Bessel's equation

E&P §8.5



Wed 11/23

No class (Thanksgiving Break)



14

Mon 11/28

Local behavior at irregular singular points

B&O §3.4

HW5 due; HW6 out 


Wed 11/30

More on local behavior at irregular singular points


 

15

Mon 12/05

Irregular singular points at infinity

B&O §3.5



Wed 12/07

Local analysis of inhomogeneous linear  equations

B&O §3.6


16

Mon 12/12

Review

slides

HW6 due

17

Mon. 12/19

Final exam (17:15--19:45, Harriman Hall 116)





University Policies and Academic Integrity (back to top)

Americans with Disabilities Act

If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

Academic Integrity

Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instances of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at http://www.stonybrook.edu/uaa/academicjudiciary/

Critical Incident Management

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.