Asymptotic and Analytical Methods
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Description
Analytical and Asymptotic Methods
In applied mathematics, physical and other problems are often modelled by differential equations. It is extremely rare that one can obtain exact solutions to the differential equations that may occur in, for example, fluid dynamics, mathematical biology or magnetohydrodynamics. Additionally, the problems may involve the evaluation of integrals which arise, for example, through contour integration or Fourier or Laplace transform methods for solving ODEs. Thus, in many cases we are forced to employ some kind of approximation in order to make progress with our problem. Hence, we must obtain an approximate solution rather than the exact solution.
In essence there are two main types of approximation: analytical approximations and numerical approximations. This module deals with the first type; the Numerical Methods module deals with the second.
Topics
- Asymptotic methods for differential equations, including the methods of multiple scales and matched asymptotics (lectures 1 to 5; Thomas Neukirch, David Pritchard, and Antonia Wilmot-Smith).
- Contour integral methods for differential equations, including the method of steepest descents (lectures 6 to 8; Alex Wray).
- Further applications of asymptotics (lectures 9 and 10; David Pritchard).
Delivery
Some lectures will be delivered in “flipped” format, with material to read and exercises to complete before each class. It is essential that you carry out this work in order to be prepared for the class. Pre-class material will appear at least a couple of days beforehand, but full notes will appear only after each class.
Assessment
The module will be assessed by two written assignments with provisional deadlines shown.
- Assignment 1 (lectures 1–5): to be submitted by 24 November 2023.
- Assignment 2 (lectures 6–10): to be submitted by 15 January 2024.
Assignments may include both “paper and pencil” and computer work, and will be set at least two weeks before the deadline.
Prerequisites
This module assumes basic undergraduate knowledge of: basic ODEs (first-order separable and first- and second-order linear equations); single- and multivariable calculus; Taylor’s theorem; linear algebra; contour integration including Cauchy's theorem. "Lecture 0" notes that give a brief recap of some of these topics will be provided in advance, along with some associated exercises.