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.

On this module, you will learn which types of PDEs can be solved exactly, and which require a numerical approach. Furthermore, you will discover how PDEs can be well-posed or ill-posed, and will find out about a range of analytical and numerical techniques used to solve PDEs.

The module describes how computers can be utilised to model equations from physics. In addition, you will strengthen your ability to interpret theoretical mathematical concepts, and acquire a deeper understanding of how mathematics relates to real world problems.

Pre-requisite modules: “Differential Equations” (ECM2903) and “Vector Calculus and Applications” (ECM2908) or equivalent

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 ECM2902 and ECM2907. 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 and derivation of PDE models, revision of first PDEs methods, linear, quasilinear and nonlinear cases;

- Domains, boundary conditions and well-posedness;

- Green’s function to solve boundary value PDEs;

- Second-order linear PDEs and their classification into elliptic, hyperbolic and parabolic problems;

- Elliptic problems: uniqueness theorem for Dirichlet and Neumann problems;

- Laplace and Poisson equations;

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

- Hyperbolic problems: method of characteristics, wave and Helmholtz equations, inhomogeneous Helmholtz problems;

- Parabolic problems: diffusion equation on finite, infinite and semi-infinite domains, solution by transform methods;

- Examples from selected applications; examples of phenomena from nonlinear PDEs such as Burger’s (shock solutions) and Korteweg-deVries (solutions);

- Introduction to basic numerical methods for PDEs, finite difference and geometric integration methods.

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

Basic Reading:

ELE - http://vle.exeter.ac.uk

Reading list for this module: