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

4H: Partial Differential Equations (MATHS4110)

Assessment

90% Examination, 10% Coursework.

Reassessment

Resit opportunities available for MSc students.

The main aim of this course is to introduce concepts and techniques for the solution of partial differential equations (PDEs). The second-order linear PDEs of elliptic, hyperbolic and parabolic types are considered as well as some nonlinear PDEs. Some particular aims are to present the method of characteristics for hyperbolic PDEs, separation of variables, the method of Green's functions and transform methods.

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 canonical examples in traffic flow;

e) State uniqueness and existence theorems for different types of initial and boundary value problems;
f) Apply separation of variables to construct solutions to Laplace's and Poisson's equation by making use of special functions and their properties, including associated Legendre polynomials and Bessel functions;
g) Apply the Fourier transform method to solve initial value problems for the heat equation;

h) Derive and the Cole-Hopf transformation appropriate to solving a Burgers' equation and use it to derive exact solutions or solve IVPs for this equation;
i) Formulate and solve initial value problems for the wave equation in 1D, 2D and 3D;
j) Derive and apply Green's functions for second order linear PDEs, including Laplace's equation.

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