Introductory description

In this module, methods for solving (or at least establishing the existence of a solution!) various types of pdes will be introduced. Unlike odes, the domain on which a pde is to be solved plays an important role. This module will show an essential role where much of the analysis which has 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.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• 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).

Learning outcomes

By the end of the module, students should be able to:

• 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 (including energy estimates) of solutions of parabolic equations in arbitrary dimensions.

Subject specific skills

Students who have successfully taken the module will have strong foundations to pursue further study on partial differential equations, as well as other branches of analysis. They will obtain important background on the structure of solutions to well-known and relevant equations; ranging from a qualitative description, to the comparison principles and the inequalities that relate to each of them. They will be left with solid background for further development on pure and applied mathematics at graduate level. However, the repertoire of techniques discussed in the curse will also serve students who lean towards applied sciences, engineering and financing careers.

Transferable skills

Students who have successfully taken the module will have strong foundations to pursue further study on partial differential equations, as well as other branches of analysis. They will obtain important background on the structure of solutions to well-known and relevant equations; ranging from a qualitative description, to the comparison principles and the inequalities that relate to each of them. They will be left with solid background for further development on pure and applied mathematics at graduate level. However, the repertoire of techniques discussed in the curse will also serve students who lean towards applied sciences, engineering and financing careers.