Topics in Applied Mathematics (Multiple Scales and Singular Perturbation Methods)

Instructor:

Course Description

Multiple Scales and Singular Perturbation Methods, Kevorkian and Cole

Textbook

1. , J. Kevorkian and J. Cole, Springer,

Mathematical Concepts to be covered

1. Introduction to Asymptotics
2. Linear Oscillator
3. Singular perturbation methods for nonlinear problems
4. Singular boundary problems
5. Method of Multiple scales for ODEs
6. Strained coordinates
7. Two scale expansions for non-linear oscillators
8. Systems of first order equations

Homeworks

Learning Objectives and Instructor Expectations

Various physical problems are characterized by the presence of a small disturbance which because it is active over a long period of time, has a non-negligible cumulative effect. An example would be that of a satalite which is orbiting the Earth. Perturbation methods , first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually every branch of science. The aim of the course will be to survey perturbation methods as currently used in various physical, medical and engineering applications. Topics will be introduced by means of simple illustrative examples and then built up to consider more challenging problems. For a brief review of topics we will consider

1) Limit process expansions for ordinary differential equations both linear and non-linear problems.

2) The method of multiple scales for ODE's, via the method of strained coordinates and two scale expansions for the weakly non-linear autonomous oscillator. Also this method will be applied to general non-linear oscillators and systems of first order equations.

3) Limit process expansions for PDE's such as the ones used in studying viscous incompressible flow.

Grading