Module Code - Title:

MS4407 - PERTURBATION TECHNIQUES AND ASYMPTOTICS

Year Last Offered:

2026/7

Hours Per Week:

Lecture

2

Lab

1

Tutorial

1

Other

0

Private

6

Credits

6

Grading Type:

N

Prerequisite Modules:

MS4403

MS4404

Rationale and Purpose of the Module:

To learn the basic concepts and techniques of asymptotic and perturbation methods.

Syllabus:

Non-dimensionalisation, scaling, ordering, definition of asymptotic series, algebraic equations, integrals, LaplaceÆs method, method of steepest descent, regular and singular perturbations, multiple scales, strained coordinates, boundary layer techniques.

Learning Outcomes:

Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)

1. Demonstrate an understanding of scaling asymptotic scaling of differential equations and the principles of asymptotic approximations. 2. Solve a regular perturbation differential or algebraic equation. 3. Demonstrate an understanding of the concept of a singular perturbation in algebraic equations and boundary layers (matched asymptotic expansions) in differential equations. 4. Demonstrate an understanding of multiple scale techniques for differential equations. 5. Demonstrate an understanding of WKB techniques. 6. Demonstrate an understanding of asymptotic techniques for approximation of integrals (LaplaceÆs method, steepest descent).

Affective (Attitudes and Values)

N/A

Psychomotor (Physical Skills)

N/A

How the Module will be Taught and what will be the Learning Experiences of the Students:

Normal mathematics lectures. Students will have homework (not for credit), a mid-term text worth 5%, 5% for participation in tutorial problem solving sessions and a final exam worth 90%.

Prime Texts:

J. Kevorkian, J.D. Cole (1981) Perturbation methods in applied mathematics , Springer-Verlag

M.H. Holmes (1995) Introdcution to perturbation methods , Springer

J. Hinch (1991) Perturbation methods , CUP

A.H. Nayfeh (1981) Introduction to perturbatio techniques , Wiley

Other Relevant Texts:

J. Keener (1995) Principles of applied mathematics , Addison-Wesley

J. David Logan (1996) Applied mathematics , Wiley

J. Murdock (1991) Perturbations , Wiley

C.A. Bender, S.A. Orszag (1987) Advanced mathematical methods for scientists and engineers , Mcgraw-Hill

Programme(s) in which this Module is Offered:

BACHELOR OF SCIENCE IN ECONOMICS AND MATHEMATICAL SCIENCES

BACHELOR OF SCIENCE IN ECONOMICS AND MATHEMATICS

BACHELOR OF SCIENCE IN FINANCIAL MATHEMATICS

BACHELOR OF SCIENCE IN MATHEMATICAL SCIENCES

BACHELOR OF SCIENCE IN MATHEMATICS AND PHYSICS

MASTER OF SCIENCE IN MATHEMATICAL MODELLING

Semester(s) Module is Offered:

Module Leader:

doireann.okiely@ul.ie