MA3A4 Asymptotic Methods

Introductory description

Asymptotics has wide ranging applications, not only to mathematical physics, but to analysis and number theory, for example. In many cases there is a need to approximate a function in a specific limit. This function may be a solution to an algebraic equation, an integral or a solution of a differential equation. Asymptotic expansions can give an excellent approximation in these limits, despite often being divergent series. Asymptotics also give approximate analytical solutions to problems with no exact analytical solution, often helping in understanding not only what happens in a particular limit, but also why.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Introduction to asymptotics: Recap of big-O, little-o notation. Formal definition of an asymptotic series. Examples, including Erf(z).

Asymptotics of algebraic equations: Regular and singular perturbations. Examples including solutions of quadratic equations.

Asymptotics of integrals: Local and non-local contributions. Exponentially dominated integrals. Watson's Lemma and Laplace’s method. Examples, including Stirling's formula.

Asymptotics of differential equations:

• Matched asymptotic expansions: Origin of small parameters (e.g. dimensionless parameters). Regular and singular perturbations. Van Dyke's matching rule. Examples of ODEs and PDEs. Logarithms and matching via intermediate variable.
• WKB Theory: WKB expansion, validity of WKB. Applications to quantum mechanics.
• Method of Multiple Scales: Resonance and secular behaviour. Multiple scale analysis. Examples include van der Pol oscillator, Mathieu equation, advection-diffusion equation.

Learning outcomes

By the end of the module, students should be able to:

• Understand the formal definition of asymptotic series and their uses.
• Be able to identify both regular and singular perturbations.
• Understand and be able to use standard techniques to construct asymptotic series for simple perturbation problems.
• Understand and be able to use more advanced techniques to construct asymptotic series in some more complicated perturbation problems, involving matched asymptotic expansions with logarithms and intermediate variables.
• Understand and be able to use WKB theory for singular differential equations.
• Understand and be able to use the method of multiple scales for differential equations with processes at different scales.

Indicative reading list

Reading lists can be found in Talis

Subject specific skills

• Ability to describe the limiting behavior of functions using "big O" and "little o" notation.
• Ability to analyze equations and identify which terms are dominant and which are negligible in a specific limit.
• Ability to derive approximate analytical solutions to problems with no exact analytical solution.
• Ability to recognise the limitations of asymptotic solutions and their region of validity.

Transferable skills

Time management. Independent Study. Logical and systematic thinking. Problem solving. Adapting a theoretical framework to real-world problems.