Partial Differential Equations

COURSE OBJECTIVE
The majority of physical phenomena can be described by partial differential equations. This module discusses these equations and methods for their solution. For first order equations we discuss the method of characteristics and the solution by methods of ordinary differential equations. For second order equations, in particular for the heat and wave equation we discuss the method of separation of variables. This ties in with the remarkable result of Fourier that almost any periodic function can be represented as a sum of sines and cosines, called its Fourier series. An analogous representation for non- periodic functions is provided by the Fourier transform, to be discussed briefly in part 2, as well as some theoretical background for Fourier series. In Part 2 we discuss some of the background for generalised Fourier series: the role of eigenvalue problems and some basic spectral theory. Potential methods and fundamental solutions will be discussed for the standard examples: heat, wave and Poisson equation. Harmonic functions will be discussed in relation to mean value properties.

COURSE CONTENT
Part 1: - Classical examples - First order equations and characteristics - d'Alembert's solution for the wave equation - Separation of variables for second order equations - Fourier Series - Fundamental solutions for heat and wave equation in one spatial dimension - The Dirac delta- function.

Part 2 - Fourier theory - Laplace and Poisson equation through potential methods - Eigenvalue problems and some spectral theory - Special functions (Bessel functions) - Harmonic functions - Fundamental solutions in 2 and 3 spatial dimensions

TEACHING METHODS
Course and exercise class

TYPE OF ASSESSMENT
Two written exams and incidental homework

RECOMMENDED BACKGROUND KNOWLEDGE
Calculus, in particular vectorcalculus, Gauss divergence Theorem and Green's formulas