| Unit name | Mathematical Methods |
|---|---|
| Unit code | MATH30800 |
| Credit points | 20 |
| Level of study | H/6 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Professor. Hogg |
| Open unit status | Not open |
| Pre-requisites |
MATH20100, MATH33000 |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
The theme of this unit can be though of as a continuation of Fourier Series: using various alternative ways to represent functions. This leads to new types of functions, to practical methods of solving differential equations, and interpreting signals. Fourier transforms, the extension of Fourier series to an infinite domain, come first. They correspond to the 'spectrum' of physical signals such as light. However, we give more emphasis to the way they can be used to simplify partial differential equations. They lead to the idea of generalised functions, such as Dirac's delta function. Green function representations follow naturally. Finally we examine some general aspects of partial differential equations.
Aims
The unit aims to provide a variety of analytical tools to solve linear partial differential equations (PDEs) arising from problems in physics and engineering, in particular:
Throughout the course, physical interpretations of the mathematical solutions found will be stressed as much as possible. Through this physical emphasis, the unit aims to foster the students' ability to model and solve mathematically problems of physical significance.
Syllabus
First order p.d.e.s: characteristics. Second order p.d.e.s and real and imaginary characteristics. Classification of equations of hyperbolic, parabolic and elliptic types. Hyperbolic equations, initial conditions, domain of dependence, zone of influence and boundary conditions. Examples of elliptic and parabolic equations, and their appropriate boundary conditions.
Definition of Fourier transforms on infinite domains. Sine and cosine transforms. Properties of the transforms and their derivatives. Convolution and inversion. Application to the solution of initial boundary value problems. Generalization to multi-dimensions.
Definition of the Laplace transform. Properties of the transform and its derivatives. Application to the solution of initial value o.d.e.s. Laplace transform methods for solutions of p.d.e.s on unbounded and bounded domains. The complex inversion integral. Inversion of Laplace transforms using complex residues.
Definition of the Heaviside function, and the Dirac delta function, and its derivatives. Properties of generalized functions. Fourier transforms of generalized functions.
Definition of a Green's function in one dimension. Mathematical interpretation as a representation of an inverse differential operator. Physical interpretation in terms of forcing. Discussion of significance of boundary conditions. Green's identity and Lagrange's identity. Application to the solution of inhomogeneous o.d.e.s on bounded and unbounded domains, with both homogeneous and inhomogeneous boundary conditions.
Generalization to multi-dimensions of the delta function, and Green's functions. Definition of the free-space Green's functions for the wave equation and heat equation. Integral representations of solutions in terms of Green's functions incorporating initial and boundary conditions. The method of images. Application of Green's functions to the solution of p.d.e.s on bounded domains.
Similarity solutions to linear and nonlinear partial differential equations.
Relation to Other Units
This unit is a natural progression from Level 2 Applied Partial Differential Equations and develops methods useful in a wide range of applied mathematics topics. The techniques introduced in this course are developed further in the Level M Asymptotics unit, and are used in Level M unit Advanced Fluid Dynamics.
At the end of the unit, the students should:
Transferable Skills:
Clear logical thinking, problem solving, modelling skills, i.e. the ability to transform a real physical problem into a tractable and understandable form.
The primary content of the course is taught using lectures, with some distributed notes. Extensive use of example sheets, problems classes and some computer software illustrates the techniques of problem solving developed in the course.
The course will be supported by a web site (www.maths.bris.ac.uk/~maajh/methods.html).
The assessment mark for Mathematical Methods is calculated from a 3-hour written examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted be used in this examination.
A good basic text, which covers most of the course, and has a lot of examples is:
More advanced and comprehensive texts are:
Various parts of the course are covered in:
