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Up: Year 4
Previous: MC431 Classical Approximation Theory
MC434 Time Dependent Partial Differential Equations
| Credits: 20 |
Convenor: Dr P. Houston |
Semester: 2 |
| Prerequisites: |
essential: MC127,MC224 |
desirable: MC248,MC380 |
| Assessment: |
Coursework: 10% |
Three hour exam: 90% |
| Lectures: |
36 |
Classes: |
10 |
| Tutorials: |
0 |
Private Study: |
104 |
| Labs: |
none |
Seminars: |
none |
| Project: |
none |
Other: |
none |
| Total: |
150 |
|
|
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
The final assessment of this module will consist of 10% coursework
and 90% from a three hour examination during the Summer exam
period. The 10% coursework contribution will be determined by students'
solutions to coursework problems. The examination paper will contain
6 questions with full marks on the paper obtainable from 4 complete answers.
Next: MC435 General Relativity
Up: Year 4
Previous: MC431 Classical Approximation Theory
S. J. Ambler
11/20/1999