MATH3291/4041 Partial Differential Equations III/IV
The topic of partial differential equations (PDEs) is central to mathematics. It is of fundamental importance not only in classical areas of applied mathematics, such as fluid dynamics and elasticity, but also in financial forecasting and in modelling biological systems, chemical reactions, traffic flow and blood flow in the heart. PDEs are also important in pure mathematics and played a fundamental role in Perelman's proof of the Poincaré conjecture.
In this module we are concerned with the theoretical analysis of PDEs (the numerical analysis of PDEs is covered in the course Numerical Differential Equations III/IV). We will study first-order nonlinear PDEs and second-order linear PDEs, including the classical examples of Laplace's equation, the heat equation and the wave equation. For some simple equations with appropriate boundary conditions we will find explicit solutions. For equations where this is not possible we will study existence and properties of solutions. We will see for example that solutions of the heat equation have very different properties to solutions of the wave equation.
Outline of Course
Aim: To develop a basic understanding of the theory and methods of solution for partial differential equations.
- Introduction: examples of important PDEs, notation, the concept of well-posedness.
- First-order PDEs and characteristics: the transport equation, general nonlinear first-order equations.
- Conservation laws: models of traffic flow and gases, shocks and rarefactions, systems of conservation laws.
- Second-order linear PDEs: examples and classification (elliptic, parabolic, hyperbolic).
- Poisson's equation: fundamental solution.
- Laplace's equation: mean value formula, properties of Harmonic functions, maximum principle, Green's functions, energy method.
- The heat equation: fundamental solution, maximum principle, energy method, infinite speed of propagation, properties of solutions.
- The wave equation: solution formulas, energy method, finite speed of propagation, properties of solutions.