Delaware State University
Department of Applied Mathematics and Theoretical Physics
Dover, DE 19901
Advanced Perturbation Theory
60-835-00,
3 cr.
Text:
J. A. Murdock:
Perturbations: Theory and Methods
recommended:
E. J. Hinch:
B. K. Shivamoggi:
A. H. Nayfeh:
Perturbation Methods
Perturbation Methods for Differential Equations
Introduction to Perturbation Techniques
The aim of the course is to lay an introduction to the perturbation theory to solve ordinary
differential equations, partial differential equations as well as integral equations. Topics that will be
covered in this course are Regular perturbations; Error Estimates; Periodic solutions and Lindstedt
Series, Harmonic Resonance, Duffing’s equation, Multiple Scales, Struble’s Method, Averaging,
Krylov-Bogoliubov Method of Averaging, Krylov-Bogoliubov-Mitropoloski generalized method of
Averaging; Forced Duffing and Van der Pol’s equations, Wentzel–Kramer–Brillouin–Jeffreys (WKBJ)
Approximation, Fredholm’s Alternative, Latta’s method of composite expansion; Matched Asymptotic
Expansion. The emphasis in this course is on the adaptation of these mathematical methods and
techniques to their swift and effective application in solving advanced problems in applied mathematics
and theoretical physics.
Prerequisite: Partial Differential Equations (60-853) or equivalent. A successful student is
expected to gain a working knowledge of the covered material, so as to be able to (1) follow the
applications in the literature, (2) solve typical problems in the field, and (3) discuss adequately the termpaper subject.
Topical schedule:



Series Solution
Application: Ordinary and Partial Differential Equations
Lindstedt Series
Application: Ordinary and Partial Differential Equations
Multiple Scales
Application: Partial Differential Equations, nonlinear evolution equations, integral
equations



Struble’ Method
Application: nonlinear wave equations
WKBJ Approximations
Application: perturbation of nonlinear partial differential equations
Averaging
Application: Solving ordinary differential equations
CURRICULUM COURSE REVIEW:
Advanced Perturbation Theory
1. Course Title/Number:
Advanced Perturbation Theory / 60-835-00
2. Number of Credits:
3
3. Curriculum Program Title:
Ph.D. in Applied Mathematics and Theoretical Physics
4. Curriculum/Course is:
[X]
New
[
[
Required Course
[X]
]
]
Revised
Elective Course
5. List Prerequisites:
60-853 (Partial Differential Equations) or equivalent
6. List Courses Being Replaced or Changed:
This is a new course.
7. List Courses Being Deleted:
No courses are being deleted.
8. Needs Statement:
This course material is going to serve as a problem solving tool for problems in ordinary and partial
differential equations as well as integral equations and to a certain extent in functional equations. The
course frame work is structured around the following topics:
Regular perturbations; Error Estimates; Periodic solutions and Lindstedt Series, Harmonic Resonance,
Duffing’s equation, Multiple Scales, Struble’s Method, Averaging, Krylov-Bogoliubov Method of
Averaging, Krylov-Bogoliubov-Mitropoloski generalized method of Averaging; Forced Duffing and Van
der Pol’s equations, WKBJ Approximation, Fredholm’s Alternative, Latta’s method of composite
expansion; Matched Asymptotic Expansion.
A student finishing up this course will be knowledgeable in the basic concepts of perturbation and
asymptotic theory that will serve as an important and advanced technique to solve problems in nonlinear
differential equations.
CURRICULUM COURSE REVIEW:
Advanced Perturbation Theory
9. Catalog Description of the Course:
This course introduces the methods and means to solve nonlinear differential and integral
equations and functional equations, too. The emphasis is on problem solving techniques that arises in
various areas of Applied Mathematics and Theoretical Physics.
10. List of Objectives of the Course:
(1) To provide an introduction to the problem solving techniques of nonlinear differential and
integral equations that arises in applied mathematics and theoretical physics.
(2) To see how these techniques apply to the analysis of problems that arise in Fluid Dynamics,
Nonlinear Optics, Plasma Physics and other areas of Theoretical Physics.
(3) To learn how to identify those phenomena throughout theoretical physics, which are best
described using these methods.
(4) To develop the problem-solving skills associated with the application of these methods in
theoretical physics, and learn how to extract numerically verifiable information from such application.
11. Course Outline:
See the “Topical schedule” section in the attached brief syllabus.
12. Show how the proposed course fits into the curriculum or course sequence:
This course is an elective within the curriculum of the Ph.D. program in Applied Mathematics
and Theoretical Physics, and is indispensable for students focusing on Nonlinear Studies.
13. Are there comparable courses in other departments?
No.
14. How will the students be affected by this course change?
This course provides the students an opportunity to increase their integration with the research
program of the Department of Applied Mathematics and Theoretical Physics, by understanding the
mathematical underpinnings of the techniques that are used in theoretical physics. This course will
improve students’ professional competence, employability in technical fields and ability to pass
professional examinations. Neither this course nor its prerequisites increase the total number of semester
hours in this curriculum or the number of credit hours required for graduation.
15. What effect will this new course have on College resource?
None: this course will not require new or additional resources or staffing.
CURRICULUM COURSE REVIEW:
Advanced Perturbation Theory
16. How will the course benefit the College?
This course will address applications of nonlinear differential equations and integral equations
that arises in areas of Applied Mathematics and Theoretical Physics. In addition, this course will lay the
foundation stone for courses in Engineering (Optical Solitons) and Physics (Quantum Mechanics).
17. How will the change affect the program?
This course will introduce students to a few select topics in “higher” mathematics and their
application in various branches of physics. This course will be one of the electives specific to the Ph.D.
program in this department. Besides providing such a cross-disciplinary broadening of knowledge for the
students in this program.
18. Evaluation of Student Performance:
Homework Assignments
15 %
Two (2) in-term examinations
30 %
Term-paper
15 %
Final Examination
40 %
Sample homework assignments, in-term and final examination question-sheets, work sheets,
course notes, review sheets and term papers will be accessible on-line.
Course Structure: Three (3) 50-minute lectures per week.
References
1. J. A. Murdock: Perturbations: Theory and Methods
(SIAM Publishers, 1999; ISBN-10: 89871-443-5)
2. E. J. Hinch: Perturbation Methods
(Cambridge University Press, 1992; ISBN = 0-521-37897-4)
3. B. K. Shivamoggi: Perturbation Methods for Differential Equations
(Birkhauser, 2003; ISBN = 0-8176-4189-0)
4. A. H. Nayfeh: Introduction to Perturbation Techniques
(John Wiley and Sons, 1993; ISBN = 0-471-31013-1)
Submitted to Department of Applied Mathematics and Theoretical Physics
by: Anjan Biswas, on 25th of November, 2007