Department of Mathematics & Computer Science

MC434 Time Dependent Partial Differential Equations

Explanation of Pre-requisites

The essential prerequisites for this course are a knowledge of ordinary differential equations and the calculus of functions of several variables.

Course Description

Partial differential equations arise in the mathematical modelling of many physical, chemical and biological phenomena. Indeed, they play a crucial role in many diverse subject areas, such as fluid dynamics, electromagnetism, material science, astrophysics and financial modelling, for example. The aim of this module is to provide a first course on the mathematical theory of partial differential equations. In particular, we discuss analytical solution techniques for both first and second-order equations; typical examples will include the linear transport equation, Poisson's equation and the heat equation. The final part of the course will be devoted to nonlinear first-order partial differential equations and the formation of shock waves.

Aims

The aim of this course is to introduce the basic theory of partial differential equations and provide a good basis for those students who wish to pursue the study of more advanced topics.

Objectives

To learn about the basics of the modern theory of partial differential equations.

To understand and apply the method of characteristics.

To classify second-order partial differential equations as elliptic, parabolic and hyperbolic.

To provide analytical techniques for solving partial differential equations of each type.

To present an introduction to the theory of first-order nonlinear partial differential equations.

Transferable Skills

The ability to apply the techniques developed in this course to a large variety of partial differential equations.

The importance of partial differential equations in all areas of science and engineering makes this course an essential prerequisite for any student wishing to pursue a career in applied mathematics.

Syllabus

Introduction to partial differential equations; definition of linear, semilinear, quasilinear and nonlinear partial differential equations.

First-order linear partial differential equations; method of characteristics; extensions to systems.

Second-order partial differential equations; classification of type to elliptic, parabolic and hyperbolic.

Elliptic equations; Poisson's equation and Helmholtz equation; maximum principles; Green's functions in two dimensions.

Parabolic equations; heat equation; maximum principles, similarity solutions.

First-order, nonlinear hyperbolic problems; Burgers' equation; Rankine Hugoniot condition; shock waves; rarefaction waves.

Reading list

Essential:

J. Ockendon, S. Howison, A. Lacey and A. Movchan, Applied Partial Differential Equations, Oxford University Press, 1999 (£25).

Recommended:

M. G. Smith, Introduction to the Theory of Partial Differential Equations, Van Nostrand, 1967.

E. C. Zachmanoglu and D. W. Thoe, Introduction to Partial Differential Equations with Applications, Dover, 1986.

E. Zauderer, Partial Differential Equations of Applied Mathematics, Wiley, 1989.

Background:

D. J. Logan, An Introduction to Nonlinear Partial Differential Equations, Wiley, 1994.

Details of Assessment

Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 2001-09-20
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