This module is not currently running in 2024 to 2025.
The lectures will introduce students to asymptotic and perturbation methods for the approximate evaluation of integrals and to obtaining approximations for solutions of ordinary differential equations. These methods are widely used in the study of physically significant differential equations which arise in Applied Mathematics, Physics and Engineering. The material is chosen so as to demonstrate a range of the Mathematical techniques available and to illustrate some different applications which are amenable to such analysis.
The indicative syllabus is:
• Asymptotics. Ordering symbols. Asymptotic sequences, expansions and series. Differentiation and integration of asymptotic expansions. Dominant balance. Solution of algebraic and transcendental equations.
• Asymptotic evaluation of integrals. Integration by parts. Laplace's method and Watson's lemma. Method of stationary phase.
• Approximate solution of linear differential equations. Classification of singular points. Local behaviour at irregular singular points. Asymptotic expansions in the complex plane. Stokes phenomena: Stokes and anti-Stokes lines, dominance and sub-dominance. Connections between sectors of validity. Airy functions.
• Matched asymptotic expansions. Regular and singular perturbation problems. Asymptotic matching. Boundary layer theory: inner, outer and intermediate expansions and limits.
• WKB method. Schrödinger equation and Sturm-Liouville problems. Turning points.
• Multiple scales analysis and related methods. Secular terms. Multiple scales method. Method of strained coordinates (Lindstedt--Poincaré method).
Total contact hours: 42
Private study hours: 108
Total study hours: 150
80% Examination, 20% Coursework
C M Bender and S A Orszag, "Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory", Springer-Verlag, New York (1999)
J D Murray, "Asymptotic Analysis", Springer-Verlag, New York (1997)
M H Holmes, "Introduction to Perturbation Methods”, Second Edition, Springer, New York (2013)
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes. On successfully completing the level 6 module students will be able to:
1 demonstrate a familiarity with the use of asymptotic techniques in the study of integrals and differential equations;
2 obtain asymptotic approximations of various types of integrals and determine approximate solutions of linear differential equations;
3 generate matched asymptotic expansions for singular perturbation and boundary layer problems
4 use WKB (Wentzel-Kramers-Brillouin), multiple scales and related methods to obtain asymptotic expansions of solutions of some differential equations.
The intended generic learning outcomes. On successfully completing the level 6 module students will be able to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make competent use of information technology skills such as online resources (Moodle);
7 demonstrate an increased level of skill in numeracy and computation.
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