Skip to main content

Unit information: Mathematical Methods in 2019/20

Please note: Due to alternative arrangements for teaching and assessment in place from 18 March 2020 to mitigate against the restrictions in place due to COVID-19, information shown for 2019/20 may not always be accurate.

Please note: you are viewing unit and programme information for a past academic year. Please see the current academic year for up to date information.

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. Porter
Open unit status Not open
Pre-requisites

MATH20402 Applied Partial Differential Equations 2

Co-requisites

None

School/department School of Mathematics
Faculty Faculty of Science

Description

Unit 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: the wave equation; the diffusion equation; Laplace's equation.

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.

Unit Description

This unit is concerned with analytical methods in mathematics. They have considerable intrinsic interest, but their importance for applications is the driving motive behind this lecture course, in which we will derive many pratcical methods for solving partial differential equations.omains.

The course starts by characterising first and second-order partial differential equations (PDE's), including classification of equations. The method of characteristics and their existence for different types of equations will be examined as will the use of characteristics for solving equations.

Fourier transforms, the natural extension of Fourier Series to an infinite domain, come next. They correspond to the 'spectrum' of physical signals such as light. However, we give more emphasis to the way they can be used as a tool for simplifying partial differential equations that lead to elegant methods for solving partial differential equations on infinite and semi-infinite domains with certain boundary conditions.

Following Fourier Transforms are Laplace Transforms which are shown to be particularly useful for solving certain initial-value PDE's (arising in many physical applications) for which Fourier Transforms are not well-suited. Their use is also shown to extend to solutions of ODE's. Evaluating inverse Laplace and Fourier transforms may entail integration in the complex plance and this course will develop techniques of contour integration for this purpose.

The wider insight that transforms give of a function's behaviour leads to the idea of generalised functions. The best known of these is Dirac's delta function: infinite at the origin and zero elsewhere - but that description is insufficient for a definition.

Green's function representations follow naturally, and their power is glimpsed as we interpret them as the inverses of differential operators, on both infinite and bounded domains. First Greens functions are developed for ODE's along with the theory behind their application to the solution of ODE's. In the final part of the course, Green's functions are introduced for PDE's illustrating their power for solving PDE's on unbounded domains in terms of arbitrary initial and boundary conditions.

Finally similarity solutions to partial differential equations will be introduced, showing how they emerge as exact solutions and how they often represent the long term behaviour of systems.

Relation to Other Units

This unit is a natural progression from Applied Partial Differential Equations 2 and develops methods useful in a wide range of applied mathematics topics. The techniques introduced in this course are developed further in the Asymptotics unit, and are used in Advanced Fluid Dynamics.

Intended learning outcomes

Learning Objectives

At the end of the unit, the students should

  • be able to classify simple 2nd order PDEs, recognise the types of boundary conditions and/or initial conditions a linear PDE requires for solution, identify appropriate techniques, and be able to use the method of characteristics to solve simple problems;
  • be familiar with the definitions, simple inversions and convolutions of the Fourier transform, the Fourier cosine transform, the Fourier sine transform and the Laplace transform, and their derivatives;
  • be able to use transform methods for the solution of second order linear o.d.e.s and p.d.e.s;
  • be familiar with the definitions of the simpler generalized functions and be able to manipulate, differentiate and integrate these functions;
  • be familiar with heuristic definitions and properties of both one-dimensional and multi-dimensional free space Green's functions, and the method of images;
  • be able to solve simple inhomogeneous o.d.e.s and p.d.e.s using Green's functions on both bounded and unbounded domains.

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.

Teaching details

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.

Assessment Details

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

Recommended

  • R. F. Hoskins, Generalized Functions, Ellis Horwood,1979
  • James P. Keener, Principles of Applied Mathematics: Transformation and Approximation, Addison Wesley, 1988
  • G. F. Roach, Green's Functions, Van Nostrand, 1970
  • Donald W. Trim, Applied Partial Differential Equations, PWS-KENT, 1990
  • W.E.Williams, Partial Differential Equations, Oxford, 1980

Further

  • Erich Zauderer, Partial Differential Equations of Applied Mathematics, Wiley, 1989
  • Richard Haberman, Elementary Applied Partial Differential Equations, Prentice Hall, 1998
  • George B. Arfken, Mathematical Methods for Physicists, Academic, 2001

Feedback