# Practical Asymptotics

#### Lecturers

Dr Michael Chen, The University of Adelaide

#### Synopsis

Differential equation models of real world problems are often very complex. Perturbation methods and asymptotic techniques can be used to systematically derive simpler versions of these models by exploiting the presence of small (or large) parameters; the idea being that the new model is mathematically tractable and still describes the behaviour of the original. This is useful, for example, in problems which involve slender geometries, or for situations where both small and large time scales are important.

This course is a broad introduction to asymptotic techniques and their application. Topics covered include: asymptotic evaluation of integrals; perturbation methods; boundary-layer theory; asymptotic matching; multi-scale analysis and asymptotics beyond all orders. Case studies will be used to demonstrate the utility of these techniques for problems from fluid mechanics, biology and industry.

#### Course Overview

Introduction to asymptotics

• notation
• local behaviour of ODEs

Regular perturbation methods

• examples from fluid mechanics and biology

Boundary layer theory and matching

• singular perturbation problems
• boundary layers in ODEs/PDEs

Multiple scales

• oscillators
• homogenisation

Evaluation of integrals

• Laplace’s method
• Method of steepest descent

#### Prerequisites

• (Essential) Knowledge of differential equations from undergraduate maths courses.
• (Helpful) Some knowledge of complex variables (especially countour integrals).
• (Helpful) Some experience with MATLAB. We’ll use this for things like numerical/computer algebra solution of ODEs and making plots. Codes will be provided for you to adapt.

#### Assessment

• Quizzes and assignments (50%)
• Final exam (50%)

Lecture notes will be provided during the course.

(Optional) Some books in this area include:

• T. Witelski, M. Bowen, Methods of Mathematical Modelling: Continuous Systems and Di fferential Equations, Springer, 2015.
• C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer, 1999.