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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 second type; the Asymptotic and Analytical Methods module deals with the first.
- Numerical methods for ODEs, including implicit, explicit and multistep methods (lectures 1 and 2; Dugald Duncan).
- Numerical methods for stochastic DEs (lectures 3 and 4; Des Higham).
- Numerical methods for PDEs, in particular finite-difference methods/finite-element methods (lectures 5 to 8; Ping Lin).
- Numerical linear algebra (lectures 9 and 10; Sebastien Loisel).
The module will be assessed by two written assignments with provisional deadlines shown.
- Assignment 1 (lectures 1–4): to be submitted by late February 2023 (dates to be confirmed).
- Assignment 2 (lectures 5–10): to be submitted by end March/early April 2023.
Assignments may include both “paper and pencil” and computer work, and will be set at least two weeks before the deadline.
This module assumes basic undergraduate knowledge of ordinary differential equations (in particular first order separable and first- and second-order linear equations); single- and multivariable calculus; Taylor’s theorem; and linear algebra.
Some exercises and assignment questions in this course use the Matlab mathematical software, which is commonly used in undergraduate mathematics programmes. Most universities have licences for this that you can use. There is a very good guide: An Introduction to Matlab (by David F. Griffiths, University of Dundee).
An alternative (nearly equivalent) to Matlab is the free software Octave https://www.gnu.org/software/octave/