OverviewThe 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).
Level 7 Students will study selected topics in greater depth than level 6 students.
This module appears in:
Method of assessment
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)
The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate a systematic understanding of the use of asymptotic techniques in the study of integrals and differential equations;
2 critically apply the techniques to obtain asymptotic approximations of various types of integrals and approximate solutions of linear differential equation in complex situations;
3 demonstrate a good understanding of the techniques of matched asymptotic expansions for singular perturbation and boundary layer problems;
4 make effective use of 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 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
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 effective use of information technology skills such as using online resources (Moodle);
7 demonstrate an increased level of skill in numeracy and computation.