Advanced Techniques in Applied Mathematics

204 Pages·2016·3.342 MB·English

by  Shaun Bullett, Tom Fearn, Frank Smith (eds.)

Advanced Techniques in
Applied Mathematics
Q0007hc_9781786340214_tp.indd 1 20/4/16 3:02 PM
LTCC Advanced Mathematics Series
Series Editors: Shaun Bullett (Queen Mary University of London, UK)
Tom Fearn (University College London, UK)
Frank Smith (University College London, UK)
Published
Vol. 1 Advanced Techniques in Applied Mathematics
edited by Shaun Bullett, Tom Fearn & Frank Smith
Vol. 2 Fluid and Solid Mechanics
edited by Shaun Bullett, Tom Fearn & Frank Smith
Vol. 3 Algebra, Logic and Combinatorics
edited by Shaun Bullett, Tom Fearn & Frank Smith
Suraj - Advanced Techniques in Applied Mathematics.indd 1 5/4/2016 11:51:21 AM
LTCC
Advanced Mathematics Series - Volume 1
Advanced Techniques in
Applied Mathematics
Editors
Shaun Bullett
Queen Mary University of London, UK
Tom Fearn
University College London, UK
Frank Smith
University College London, UK
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
Q0007hc_9781786340214_tp.indd 2 20/4/16 3:02 PM
Published by
World Scientific Publishing Europe Ltd.
57 Shelton Street, Covent Garden, London WC2H 9HE
Head office: 5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
Library of Congress Cataloging-in-Publication Data
Names: Bullett, Shaun, 1967– | Fearn, T., 1949– | Smith, F. T. (Frank T.), 1948–
Title: Advanced techniques in applied mathematics / Shaun Bullett (Queen Mary
University of London, UK), Tom Fearn (University College London, UK) &
Frank Smith (University College London, UK).
Description: New Jersey : World Scientific, 2016. | Series: LTCC advanced
mathematics series ; vol. 1 | Includes bibliographical references and index.
Identifiers: LCCN 2015047092| ISBN 9781786340214 (hc : alk. paper) |
ISBN 9781786340221 (sc : alk. paper)
Subjects: LCSH: Numerical analysis. | Differential equations. | Differential equations, Partial. |
Finite element method. | Random matrices.
Classification: LCC QA300 .B83 2016 | DDC 518--dc23
LC record available at http://lccn.loc.gov/2015047092
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2016 by World Scientific Publishing Europe Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
Desk Editors: Suraj Kumar/Mary Simpson
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore
Suraj - Advanced Techniques in Applied Mathematics.indd 2 5/4/2016 11:51:21 AM
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Preface
The London Taught Course Centre (LTCC) for PhD students in the Math-
ematical Sciences has the objective of introducing research students to a
broad range of topics. For some students, some of these topics might be of
obvious relevance to their PhD projects, but the relevance of most will be
muchlessobviousorapparentlynon-existent.Howeverallofus involvedin
mathematical research have experienced that extraordinary moment when
thepennydropsandsometinygemofinformationfromoutsideone’simme-
diateresearchfieldturnsouttobethekeytounravellingaseeminglyinsol-
uble problem, or to opening up a new vista of mathematical structure. By
offering our students advanced introductions to a range of different areas
of mathematics, we hope to open their eyes to new possibilities that they
might not otherwise encounter.
Each volume in this series consists of chapters on a group of related
themes, based on modules taught at the LTCC by their authors. These
modules were already short (five two-hour lectures) and in most cases the
lecture notes here are even shorter, covering perhaps three-quarters of the
content of the original LTCC course. This brevity was quite deliberate on
the part of the editors — we asked the authors to confine themselves to
around35pagesineachchapter,inordertoallowasmanytopicsaspossible
to be included in each volume, while keeping the volumes digestible. The
chaptersare“advancedintroductions”,andreaderswhowishtolearnmore
are encouragedto continue elsewhere. There has been no attempt to make
the coverage of topics comprehensive. That would be impossible in any
case — any book or series of books which included all that a PhD student
in mathematics might need to know would be so large as to be totally
unreadable.Insteadwhatwepresentinthisseriesisacross-sectionofsome
ofthetopics,bothclassicalandnew,thathaveappearedinLTCCmodules
in the nine years since it was founded.
v
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vi Advanced Techniques inApplied Mathematics
Thepresentvolumeiswithintheareaofadvancedtechniquesinapplied
mathematics.Themainreadersarelikelytobegraduatestudentsandmore
experiencedresearchersin the mathematicalsciences,lookingfor introduc-
tions to areas with which they are unfamiliar. The mathematics presented
is intended to be accessible to first year PhD students, whatever their
specialisedareasofresearch.Whateveryourmathematicalbackground,we
encourage you to dive in, and we hope that you will enjoy the experience
of widening your mathematical knowledge by reading these concise intro-
ductory accounts written by experts at the forefront of current research.
Shaun Bullett, Tom Fearn, Frank Smith
April20,2016 14:36 AdvancedTechniquesinAppliedMathematics 9inx6in b2303-v1-fm pagevii
Contents
Preface v
1. Practical Analytical Methods for Partial Differential Equations 1
Helen J. Wilson
2. Resonances in Wave Scattering 35
Dmitry V. Savin
3. Modelling — What is it Good For? 69
Oliver S. Kerr
4. Finite Elements 107
Matthias Maischak
5. Introduction to Random Matrix Theory 139
Igor E. Smolyarenko
6. Symmetry Methods for Differential Equations 173
Peter A. Clarkson
vii

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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 —therewill benoproofs —instead I will demonstratearangeof
tools that you might want to try. We begin with first-order PDEs and
the method of characteristics; classification of second-order PDEs and
solutionofthewaveequation;andseparationofvariables.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
Wewillseeavarietyoftechniquesforsolving,orapproximatingthesolution
of,differentialequations.Eachis illustratedby means ofa simple example.
In many cases, these examples are so simple that they could have been
solvedbysimplermethods;butitisinstructivetoseenewmethodsapplied
without having to wrestle with technical difficulties at the same time.
Section 2 deals with first-order equations. In Section 3, we clas-
sify 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.
1
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2 Advanced Techniques inApplied Mathematics
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
materialonasymptoticexpansions,thereareseveralrelevanttexts,includ-
ing 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:
∂u/∂t+c∂u/∂x=0
responds well to a change of variables:
ξ =x+ct; η =x−ct.
The extended chain rule gives us
∂ ∂ξ ∂ ∂η ∂ ∂ ∂
= + = + ;
∂x ∂x∂ξ ∂x∂η ∂ξ ∂η
(cid:1) (cid:2)
∂ ∂ξ ∂ ∂η ∂ ∂ ∂
= + =c −
∂t ∂t∂ξ ∂t ∂η ∂ξ ∂η
and so the wave equation is equivalent to
2c∂u/∂ξ=0.
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 (indepen-
dent of η) would be fine as the other variable; since u is a function of η,
differentiating while holding η constant will always give zero.