Many physical processes such as vibrating strings, diffusion of heat and fluid flows are well modelled by partial differential equations and/or integral equations. This course provides an introduction to methods for solving and analysing standard partial differential equations and integral equations, including an introduction to complex analytic techniques.

The course consists out of two main modules: Complex Analysis and Partial Differential Equations. Complex Analysis: differentiability; analytic continuation; conformal mapping; complex integration; Cauchy integral theorems; residue theorem; applications to real integration. Laplace transform: properties, Watson's lemma, the inversion integral, inversions involving residues and branch cuts, asymptotics, application to ODE's and PDE's Partial Differential Equations; classification of second order partial differential equations into elliptic, parabolic and hyperbolic types; elliptic equations; integral formulae, maximum principle; parabolic equations; diffusion; representation by a kernel (Green's functions); hyperbolic equations; d'Alembert solution and the method of characteristics; analytic methods; separation of variables; orthogonal expansions; Fourier series; Distributions, Transforms, Complex Analysis and applications; Distributions: definition, convergence of distributions, derivative. Fourier transform: definition, properties, application to Green's functions.

On satisfying the requirements of this course, students will have the knowledge and skills to:

• Assignments (30%; LO 1-4)
• Final exam (70%; LO 1-4)