# Introduction to Asymptotic Analysis and Singular Perturbations

## Matthias K. Gobbert, Weijia Kuang, and Thomas I. Seidman

Scores and grades will be posted here beginning after the midterm.

## Overview

Asymptotic Analysis includes a wide range of techniques in applied mathematics used to analyze or simplify problems involving multiple scales (in length or time) whose ratio appears as a `small' parameter.

The first half of this course will be devoted to a short but thorough introduction with a particular emphasis on boundary value problems in ordinary differential equations. These serve as prototype examples of applications with boundary layers, which will be studied in more detail in the second half of the semester in problems arising in fluid mechanics.

Hopefully, there will be time to highlight some other areas of asymptotic analysis, for instance, multi-scale analyses and homogenization techniques.

This course will require familiarity with differential equations (Math 225) and some formal background (Math 301) and is therefore accessible to incoming graduate students, senior undergraduates, as well as students from application areas in the physical sciences and engineering.

In practice, asymptotic analysis has to be used in concert with other techniques to extract as much information from a given model as possible; accordingly, this course will also use other aspects of mathematical modeling and analysis, and familiarity with some mathematical software package is recommended, but not required.

## Basic Information

• Instructors:
• Matthias K. Gobbert,
Math/Psyc 416, (410) 455-2404, gobbert@math.umbc.edu,
office hours: TTh 04:00-05:00 or by appointment
• Weijia Kuang,
Math/Psyc 407, (410) 455-2407, wkuang@math.umbc.edu,
office hours: TTh 04:30-05:30 or by appointment
• Thomas I. Seidman,
Math/Psyc 438, (410) 455-2438, seidman@math.umbc.edu,
office hours: MW 04:00-05:00 or by appointment
• Lectures: MW 02:30-03:45 p.m., Math/Psyc 401
• Prerequisite: Math 225 and Math 301, or instructor approval
• Textbook: C.C. Lin and L.A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, Philadelphia, 1988. A copy of the textbook is on reserve in the library.
 Homework Midterm Final 40% 20% 40%
Generally, there will be weekly homework during the first half of the course, then maybe less homework with a stronger project orientation during the second half. The date and format of the exams are to be arranged in class.

## Recommended Literature for Asymptotic Analysis

• Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, 1978.
• Ulrich Hornung, Homogenization and Porous Media, Springer, 1997.
• J. Kevorkian and J.D. Cole, Multiple Scale and Singular Perturbation Methods, Applied Mathematical Sciences series, volume 114, Springer, 1996.
• C.C. Lin and L.A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, Philadelphia, 1988.
• Robert E. O'Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Sciences series, volume 89, Springer, 1991.

## Recommended Literature for Fluid Mechanics

• Harvey P. Greenspan, The Theory of Rotating Fluids, Breukelen Press, 1990.
• P.G. Drazin and W.H. Reid, Hydrodynamic Stability, Cambridge University Press, 1987.