MA40203: Theory of Partial Differential Equations

This applied unit is a prerequisite for MA40203. It introduces three classic PDEs: the Laplace equation, the heat equation and the wave equation. Moreover, it shown how to solve these equations in (extremely) special domains using separation of variables.

Potential topics include ‘special surfaces’ such as minimal surfaces and surfaces of constant mean or Gauss curvature. These are closely related to PDEs such as the minimal surface equation.

This applied unit covers quasilinear first-order PDEs, the Cauchy–Kovalevskaya theorem, classification of equations as elliptic/parabolic/hyperbolic and d’Alembert’s solution for the one-dimensional wave equation.

This pure unit is a prerequisite for MA40203. It shows that \(C^k([a,b])\) is a Banach space, defines the Hilbert space \(L^2(a,b)\) as a completion, and proves the Arzelà–Ascoli and Weierstrass approximation theorems.

Introduces a variety of PDEs, including Euler’s equations for inviscid fluids, and studies them from an applied point of view.

Introduces variational and weak forms of elliptic PDEs, and how to solve them numerically using finite element methods.

Covers numerical methods for parabolic and hyperbolic PDEs.

An applied unit which often covers more advanced techniques for the Laplace and heat equations, for instance Green’s functions and separation of variables for inhomogeneous systems. May also cover the calculus of variations, which is an important source of PDEs.

This applied unit concerns the modelling and analysis of PDEs arising in biology.

This pure unit studies linear mappings between Hilbert spaces, which is an abstract setting often used to study PDEs. Potential topics include the Lax–Milgram theorem, which is a powerful tool to show the existence of solutions to certain PDEs. (MA40256 and MA40057 are offered in alternating years.)

Another applied unit with many connections to PDEs.

Provides a detailed description of the function spaces \(L^p\) for \(1 \le p \le \infty\text{,}\) which are very useful in the study of PDEs.

This pure unit studies linear operators between Banach spaces, which is another commonly-used abstract setting for PDEs. Covers weak and weak-* convergence, which are often used to show existence of solutions to PDEs. (MA40256 and MA40057 are offered in alternating years.)

There are also connections to some of the higher-level probability units through the Feynman–Kac formula, Fokker–Plank equations, and of course stochastic PDEs.

There are also connections to the units on ordinary differential equations and dynamical systems, especially some of the higher-level units with a focus rigorous existence and/or qualitative properties of solutions.

We will study PDEs from a pure rather than applied point of view. We will not do any modelling, nor indeed spend much time at all motivating the PDEs we study. Instead, we will spend a lot of time working through rigorous proofs with \(\varepsilon\)s, \(\delta\)s, and convergent subsequences of functions.

On the other hand we will not use tools from Functional analysis, Hilbert space theory or Measure theory & integration.

We will consider second-order PDEs on quite general domains and with quite general coefficients. Such equations cannot be solved explicitly using separation of variables or Green’s functions techniques. Instead, we focus on qualitative properties such as existence, uniqueness and symmetry.

Rather than attempt a broad survey, we will focus almost exclusively on a single tool, the maximum principle, and develop it to a high level of sophistication.