Lecturer: Thomasina Ball

Term(s): Term 2

Status for Mathematics students: List A (MA269 is for 2nd years, MA369 for 3rd years provided MA269 has not been taken in a previous year).

Commitment: 30 lectures

Assessment: 100% by 2 hour examination

Formal registration prerequisites: None

Assumed knowledge:

• MA146 Methods of Mathematical Modelling 1 or MA147 Mathematical Methods and Modelling 1 or MA133 Differential Equations. Methods for solving ODEs (particularly 2nd order linear ODEs)
• MA270 Analysis 3 or MA271 Mathematical Analysis 3: Complex differentiability, and complex contour integration (including calculating residues)

Useful background:

• MA144 Methods of Mathematical Modelling 2 or MA145 Mathematical Methods and Modelling 2: Provides background and intuition for some of the example models used in this course.

Synergies: The following modules go well together with this module:

• MA250 Introduction to Partial Differential Equations: Fourier transforms can be used to solve many linear PDEs on infinite domains, including all the ones covered in MA250
• MA254 Theory of ODEs: Asymptotics can be seen as a generalization of the concept of linearization near critical points covered in this course
• MA2K4 Numercial Methods and Computing: Asymptotics are used to simplify complex mathematical models, and numerical Fourier transforms are extensively used to speed up numerics

Leads to:

• MA3B8 Complex Analysis: Experience of using complex contour integration will help prepare for the more rigorous treatment in Complex Analysis
• MA3D1 Fluid Dynamics: Many of the techniques and concepts covered in this course are used in fluid dynamics, including both asymptotics and Fourier transforms, and several examples in this course originate in fluid dynamics
• MA3G1 Theory of Partial Differential Equations: Many of the techniques covered here can be used to solve, or understand the solutions of, many forms of PDEs

Aims: A two-part course covering an introduction to asymptotics, and an introduction to integral transforms, focusing on their properties and their applications, with proofs to come in later courses (although these may be hinted at by the lecturer). The course covers standard techniques that are of widespread use throughout applied mathematics, physics, and engineering.

Content: We will cover the following topics:

Asymptotics:

• Formal definition of an asymptotic series, with examples (e.g. erf(z)). Discussion of the origins of small parameters (e.g. dimensionless parameters, stability analysis)
• Asymptotics of algebraic equations, with examples (e.g. solutions of nearly-linear quadratic equations)
• Asymptotics of integrals, with examples (e.g. Stirling's formula, computing oscillatory integrals)
• Asymptotics of differential equations, with examples (e.g. boundary layers).

Integral Transforms:

• Definition of an Integral Transform, with examples including Fourier (superposition of musical notes) and Radon (CAT scans) transforms
• Fourier Transforms, and their applications to linear ODEs and PDEs, with examples including waves in a waveguide
• Laplace transforms, and their use in solving initial-value problems for ODEs
• A brief tour of other integral transforms, including Mellin, Z, and Radon transforms

If time permits, we may also touch on Green’s Functions, Discrete Fourier Transforms or Half-Range Fourier Transforms.

Objectives:

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
• Use various techniques to construct asymptotic series
• Be aware of a range of integral transforms and their interpretations
• Be able to calculate Fourier and Laplace transforms and understand their similarities and differences

Additional Resources