# Mathematical Methods

## 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 Asymptotics, 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.

## Syllabus

First order PDEs: characteristics. Second order PDEs 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 ODEs. Laplace transform methods for solutions of PDEs 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 ODEs 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 PDEs on bounded domains.

Similarity solutions to linear and nonlinear partial differential equations.

## Reading and References

A good basic text, which covers most of the course, and has a lot of examples is:

- D. W. Trim,
*Applied Partial Differential Equations*, PWS-KENT (1990) QA 391 TRI

More advanced and comprehensive texts are:

- E. Zauderer,
*Partial Differential Equations of Applied Mathematics*, Wiley (1989) QA 377 ZAU - R.Haberman,
*Elementary Applied Partial Differential Equations*, Prentice Hall (1998) QA 377 HAB - G.B. Arfken,
*Mathematical Methods for Physicists*, Academic (2001) 02.00 ARF

Various parts of the course are covered in:

- W.E.Williams,
*Partial differential equations*, Oxford. (1980) QA 374 WIL - J. P. Keener,
*Principles of Applied Mathematics: Transformation and Approximation*, Addison Wesley (1988) QA 401 KEE - R. F. Hoskins,
*Generalized Functions*, Ellis Horwood (1979) QA 324 HOS - G. F. Roach,
*Green's Functions*, Van Nostrand (1970) QA 372 ROA

**Unit code:** MATH30800

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 2 (13-24)

**Lecturers:** Dr Silke Henkes and Dr Richard Porter

## Pre-requisites

MATH20402 Applied Partial Differential Equations 2

## Co-requisites

None

## Methods of teaching

Lectures supported by problem classes and problem and solution sheets.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 3 hour exam in May/June

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.