## MATH3552 Perturbation Methods

### 10 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

(MATH1031 and MATH1200 and MATH1921 or MATH1931) or (MATH1050 and MATH1400) or equivalent.

This module is approved as an Elective

### Objectives

To give an introduction to approximate methods of solution of both ordinary and partial differential equations with a small parameter, since these represent the most important techniques for deriving analytical solutions when modelling real phenomena. Examples will be drawn from many areas of science and engineering. On completion of this module, students should be able to: (a) manipulate gauge functions and construct asymptotic sequences; (b) quantify the practical advantages of divergent asymptotic series over convergent power series; (c) derive approximate solutions to a wide variety of algebraic and transcendental equations which include a small parameter; (d) rescale variables and subsequently apply this to the method of matched asymptotic expansions, and thereafter to be able to apply Prandtl?s and Van Dyke?s matching rules; (e) locate and classify types of boundary layers occurring in second-order boundary value problems; (f) apply the following methods for the approximate solution of perturbed periodic systems; Linstedt-Poincari; Lighthill?s; multiple scales; Krylov-Bogoliubov; (g) construct Domb-Sykes plots in order to determine the characteristics of the dominant singularity of an expansion; (h) construct Euler?s transformation and singularity subtraction for accelerating the convergence of an asymptotic series; (i) construct Padi approximants and Shanks transformations.

### Syllabus

The governing equations of mathematical models often involve features which make it impossible to obtain their exact solution, e.g. the occurrence of a complicated algebraic equation; the occurrence of a complicated integral; varying coefficients in a differential equation; an awkwardly shaped boundary; a non-linear term in a differential equation. When a large or small parameter occurs in a mathematical model of a process there are various methods of constructing perturbation expansions for the solution of the governing equations. Often the terms in the perturbation expansions are governed by simpler equations for which exact solution techniques are available. Even if exact solutions cannot be obtained, the numerical methods used to solve the perturbation equations approximately are often easier to construct than the numerical approximation for the original governing equations. Moreover, analytic perturbation approximations often constitute a powerful validation of any numerical model which might be employed. In summary, this module aims to describe how precise approximations - i.e. ones in which the error is both understood and controllable - can be obtained using analytical (rather than numerical) techniques. Topics covered include: Asymptotic approximations: Convergence and asymptoticness, definitions, uniqueness, parameteric expansions. Algebraic equations: Iteration and expansion, singular perturbations, rescaling, non-integral powers, logarithms, convergence. Matched asymptotic expansions: Inner and outer expansions, matching, stretching, examples of applications. Strained co-ordinates: Linstedt-Poincari method; Lighthill?s method: Strained co-ordinates and applications. Multiple scales: Multiple scales, method of Averaging. Accelerated convergence: Euler transformation, Shanks transformation. Padi approximants.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures: 20 hours. 6 example classes.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

One 2 hour examination at end of semester (100%).