Course Catalogue

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The course introduces basic classes of linear and nonlinear partial differential equations. Different methods of solving the equations are considered. Relevant applications where these equations may arise are also
discussed.

34 x 1 hr lectures and 12 x 1 hr tutorials in a semester

Mandatory Entry Requirements

3H Mathematical Methods (MATHS4075) 

Recommended Entry Requirements

Assessment

90% Examination, 10% Coursework.

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

The main aim of this course is to introduce concepts and techniques for the solution of partial differential equations (PDEs). In particular, the course aims are to present the method of characteristics for first-order quasilinear PDEs, the theory of shocks and pplications to traffic flow; separation of variables for Laplace's and the linear wave equation cylindrical and spherical polar coordinates; Green's function methods for the one-dimensional heat equation and two- and three-dimensional waves.

By the end of this course students will be able to:

a) Formulate and solve the Monge equations for a first order quasi-linear PDE in 2D;

b)  Calculate the earliest time at which solutions become multi-valued;

c) Construct shock solutions and study their behaviour, in particular calculate shock speeds;

d) Apply the theory to model specific problems in traffic flow (such as traffic congestion, road blockages, traffic lights);

e) Apply separation of variables to construct solutions to Laplace's equation and the wave equation in cylindrical and spherical polar coordinates in terms of Bessel functions and Legendre polynomials;

f) Derive properties of Bessel and Legendre polynomial (symmetry, recurrence, orthogonality) from their definitions and by making use of generating functions;

g) Find Fourier-Bessel or Fourier-Legendre expansion to match initial or boundary conditions;

h) Apply the Fourier transform method to find the Green's function for the heat equation and to use this to solve initial value problems;

i) Formulate and solve initial value problems for the wave equation in 1D, 2D and 3D.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.