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 | Dr. Misha Rudnev |

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 |

**Lecturer: **Misha Rudnev

**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 practical methods for solving partial differential equations.

The course starts by characterising first and second-order PDEs, 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 PDEs (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 plane 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 Green's 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 PDEs illustrating their power for solving PDEs 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.

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.

The unit will be taught through a combination of

- synchronous online and, if subsequently possible, face-to-face lectures
- asynchronous online materials, including narrated presentations and worked examples
- guided asynchronous independent activities such as problem sheets and/or other exercises
- synchronous weekly group problem/example classes, workshops and/or tutorials
- synchronous weekly group tutorials
- synchronous weekly office hours

90% Timed, open-book examination

10% Coursework

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.

**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