Lecturer: Ben Sharp
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 lectures
Assessment: Exam 100%
Prerequisites: This module uses material from many of the Core 1st and 2nd year modules, particularly MA231 Vector Analysis, MA244 Analysis III and MA250 Introduction to Partial Differential Equations. A student taking this module will benefit from having taken MA222 Metric Spaces but this not a formal prerequisite.
Leads To: MA4L3 Large Deviation theory
The important and pervasive role played by pdes in both pure and applied mathematics is described in MA250 Introduction to Partial Differential Equations. In this module I will introduce methods for solving (or at least establishing the existence of a solution!) various types of pdes. Unlike odes, the domain on which a pde is to be solved plays an important role. In the second year course MA250, most pdes were solved on domains with symmetry (eg round disk or square) by using special methods (like separation of variables) which are not applicable on general domains. You will see in this module the essential role that much of the analysis you have been taught in the first two years plays in the general theory of pdes. You will also see how advanced topics in analysis, such as MA3G7 Functional Analysis I, grew out of an abstract formulation of pdes. Topics in this module include:
- Method of characteristics for first order PDEs.
- Fundamental solution of Laplace equation, Green's function.
- Harmonic functions and their properties, including compactness and regularity.
- Comparison and maximum principles.
- The Gaussian heat kernel, diffusion equations.
- Basics of wave equation (time permitting).
The aim of this course is to introduce students to general questions of existence, uniqueness and properties of solutions to partial differential equations.
Students who have successfully taken this module should be aware of several different types of pdes, have a knowledge of some of the methods that are used for discussing existence and uniqueness of solutions to the Dirichlet problem for the Laplacian, have a knowledge of properties of harmonic functions, have a rudimentary knowledge of solutions of parabolic and wave equations.
Fritz John, Partial Differential Equations, Springer, 1982.
L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
E. Dibenedetto, Partial Differential Equations, Birkhauser, 2010.
D.Gilbarg - N.Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001 (more advanced, optional).
More detailed advice on books will be given during lectures.