Advanced Techniques in Applied Mathematics

Shaun Bullett, Tom Fearn

• Publisher: WSPC
• Year: 2016
• ISBN:
  9781786340245
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This book is a guide to advanced techniques used widely in applied mathematical sciences research. Chapter by chapter, readers will be led from a foundation level understanding to advanced level understanding. This is the perfect text for graduate or PhD mathematical-science students looking for support in techniques such as practical analytical methods, finite elements and symmetry methods for differential equations.

Advanced Techniques in Applied Mathematics is the first volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Edited by the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas.

Contents:

• Practical Analytical Methods for Partial Differential Equations (Helen J Wilson)
• Resonances in Wave Scattering (Dmitry V Savin)
• Modelling — What is it Good For? (Oliver S Kerr)
• Finite Elements (Matthias Maischak)
• Introduction to Random Matrix Theory (Igor E Smolyarenko)
• Symmetry Methods for Differential Equations (Peter A Clarkson)

Readership: Researchers, graduate or PhD mathematical-science students who require a reference book that covers advanced techniques used in applied mathematics research.
Key Features:

• Gives advanced technique used in applied mathematics research
• Each chapter is written by a leading lecturer in the field
• Concise and versatile
• Can be used as a masters level teaching support or a reference handbook for researchers

Chapter 1

Practical Analytical Methods for Partial Differential Equations

Helen J. Wilson

Department of Mathematics, University College London,
Gower Street, London WC1E 6BT, UK
[email protected]

This chapter runs through some techniques that can be used to tackle partial differential equations (PDEs) in practice. It is not a theoretical work — there will be no proofs — instead I will demonstrate a range of tools that you might want to try. We begin with first-order PDEs and the method of characteristics; classification of second-order PDEs and solution of the wave equation; and separation of variables. Finally, there is a section on perturbation methods which can be applicable to both ordinary differential equations (ODEs) and PDEs of any order as long as there is a small parameter.

1.Introduction

We will see a variety of techniques for solving, or approximating the solution of, differential equations. Each is illustrated by means of a simple example. In many cases, these examples are so simple that they could have been solved by simpler methods; but it is instructive to see new methods applied without having to wrestle with technical difficulties at the same time.

Section 2 deals with first-order equations. In Section 3, we classify second-order partial differential equations (PDEs) into hyperbolic, parabolic, and elliptic; then hyperbolic equations are tackled in Section 4 and we briefly discuss elliptic equations in Section 5. Section 6 reviews the well-known theory of separation of variables. Finally, in Section 7 we develop the theory of matched asymptotic expansions, suitable for use in PDEs having a small parameter.

The principal text for most of the chapter is by Weinberger [1]; though the book is out of print the full text is freely available online. In the later material on asymptotic expansions, there are several relevant texts, including those by Bender and Orszag [2], Kevorkian and Cole [3], and Van Dyke [4]. My presentation is most similar to that by Hinch [5].

2.First-order PDEs

First-order partial differential equations can be tackled with the method of characteristics. We will develop the method from the simplest case first: a constant-coefficient linear equation.

2.1.Wave equation with constant speed

The first-order wave equation with constant speed:

responds well to a change of variables:

The extended chain rule gives us

and so the wave equation is equivalent to

Integrating gives the general solution u = F(η), u = F(x − ct).

2.2.Characteristics

Where did we get the change of variables from? We can see that, in our choice of variables, only the definition of η is important. Any ξ (independent of η) would be fine as the other variable; since u is a function of η, differentiating while holding η ...