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).