Partial Differential Equations - 2024 entry

A PDE is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

In this module, you will learn how PDEs can be well-posed or ill-posed, and will find out about a range of analytical techniques used to solve PDEs. The module will strengthen your ability to interpret theoretical mathematical concepts, and acquire a deeper understanding of how mathematics relates to real world problems. The module builds on material in the Differential Equation module MTH2003, in particular separation of variables and Fourier series.

Partial differential equations (PDEs) form a central part of mathematics. The laws of physics are formulated in terms of PDEs, so the subject is of great practical importance. However, the range of application of PDEs goes beyond the physical world into the modelling of subjects as diverse as ecology and economics. This leads to interesting connections between subjects that at first seem unrelated. The purpose of this module is to develop some of the main analytical and numerical techniques used to solve PDEs, building on the work done in MTH2003 and MTH2004. We will illustrate the topic using a range of real world examples.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 demonstrate understanding of the classification of linear partial differential equations (PDEs) of first and second order;

2 apply a range of analytical techniques and a wider knowledge and appreciation of applications of PDEs in mathematics;

3 exhibit detailed knowledge of specific parabolic, elliptic and hyperbolic second order PDEs.

Discipline Specific Skills and Knowledge:

4 complete extended multi-step calculations using a variety of mathematical techniques;

5 translate unfamiliar problems into ones that can be tackled by familiar techniques;

6 show a knowledge of the relevance of PDEs in applications.

Personal and Key Transferable/ Employment Skills and Knowledge:

7 illustrate self-management and time management skills;

8 express complex abstract arguments in a logical and coherent manner;

9 use learning resources, including e-learning resources to extend their knowledge.

- Introduction. Examples of PDE models. First order PDEs. Linear, quasilinear and nonlinear cases;

- Second-order linear PDEs and their classification into elliptic, hyperbolic and parabolic classes. Domains, boundary conditions and well-posedness;

- Hyperbolic equations: method of characteristics, canonical form, wave equation in one and three dimensions, conservation of energy, D'Alembert and Kirchhoff formulas;

- Parabolic equations: canonical form, uniqueness and stability theorems, diffusion equation on finite and infinite domains, solution by transform methods;

- Elliptic equations: canonical form, uniqueness theorem, Laplace equation in finite and infinite domains, solution by transform methods;

- Green's function methods for solving non-homogeneous linear equations;

- Selected examples of solutions to nonlinear PDEs.

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.

ELE – http://vle.exeter.ac.uk

Reading list for this module: