The main majority of physical processes and phenomena can be described by partial differential equations, i.e. equations involving partial derivatives (e.g. the Navier-Stokes equations of Fluid Dynamics, Maxwell's equations of Electromagnetism, Schrodinger equation in Quantum Mechanics, Einstein equations in General relativity). This module considers the properties of the most common first and second order PDEs, the mathematical concepts behind them and analytical methods of solution for such equations. The main difference from the case of ordinary differential equations is that here there is no analogue of existence and uniqueness theorem for a generic PDE. Instead of this, there is a variety of initial and boundary value problems one can impose for finding solutions of wide classes of equations. In addition, some important classes of nonlinear PDEs are also considered.
In this course we shall concentrate on second order linear PDEs (known as Equations of Mathematical Physics): elliptic equations (Laplace's equation), parabolic equations (heat equations) and hyperbolic equations (wave equations). We will study also various topics from real and complex analysis used for solving such equations: the Sturm-Liouville problem, maximum principle for harmonic functions, Fourier series, generalised functions (distributions).
The module aims to provide a general understanding of the theory of linear PDEs and methods of solution the most important types of such equations arising in applications to Physics and Geometry.
On completion of the course students should be able to (learning outcomes):
1. use the method of characteristics to solve first-order partial differential equations;
2. classify a second order PDE as elliptic, parabolic or hyperbolic;
3. use separation of variables for suitable boundary value problems for second order linear PDE;
4. have a basic knowledge and understanding of the theory of Fourier transforms and distributions and applying them to obtain fundamental solutions of Sturm-Liouville problems;
5. use Green's functions to solve elliptic equations;
SYLLABUS:
1. Linear Differential Operators
2. Method of Characteristics
3. The one-dimensional wave equation
4. The Sturm-Liouville problem
5. Fourier transforms and distributions (generalised functions)
6. Parabolic equations
7. Hyperbolic equations
8. Elliptic equations
- Module Supervisor: Georgios Papamikos