Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications)

346 Pages·2010·1.86 MB·English

by  Vladimir Dorodnitsyn

C3090_FM.indd 1 10/22/10 2:40 PM
C3090_FM.indd 2 10/22/10 2:40 PM
C3090_FM.indd 3 10/22/10 2:40 PM
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2011 by Taylor and Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number: 978-1-4200-8309-5 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid-
ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this
form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may
rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti-
lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy-
ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the
publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
C3090_FM.indd 4 10/22/10 2:40 PM
Contents
Preface ix
Introduction xvii
0.1 BriefintroductiontoLiegroupanalysisofdifferential
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
0.2 Preliminaries: Heuristicapproachinexamples . . . . . . . . . . . lxxii
1 Finite Differences and Transformation Groups in Space of Discrete
Variables 1
1.1 TheTaylorgroupandfinite-differencederivatives . . . . . . . . . 2
1.2 DifferenceanalogoftheLeibnizrule . . . . . . . . . . . . . . . . 9
1.3 Invariantdifferencemeshes . . . . . . . . . . . . . . . . . . . . . 11
1.4 Transformationspreservingthegeometricmeaningoffinite-differ-
encederivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5 Newton’sgroupandLagrange’sformula . . . . . . . . . . . . . . 34
1.6 Commutation properties and factorization of group operators on
uniformdifferencemeshes . . . . . . . . . . . . . . . . . . . . . 39
1.7 Finite-differenceintegrationandprolongationofthemeshspaceto
nonlocalvariables . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.8 Changeofvariablesinthemeshspace . . . . . . . . . . . . . . . 50
2 InvarianceofFinite-DifferenceModels 55
2.1 Aninvariancecriterionforfinite-differenceequationsonthediffer-
encemesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Symmetry preservation in difference modeling: Method of finite-
differenceinvariants . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3 Examplesofconstructionofdifferencemodelspreservingthesym-
metryoftheoriginalcontinuousmodels . . . . . . . . . . . . . . 69
3 InvariantDifferenceModelsofOrdinaryDifferentialEquations 79
3.1 First-orderinvariantdifferenceequationsandlattices . . . . . . . 79
3.2 Invariantsecond-orderdifferenceequationsandlattices . . . . . . 82
v
vi CONTENTS
4 InvariantDifferenceModelsofPartialDifferentialEquations 101
4.1 Symmetry preserving difference schemes for the nonlinear heat
equationwithasource . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Symmetrypreservingdifferenceschemesforthelinearheat
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3 InvariantdifferencemodelsfortheBurgersequation . . . . . . . . 133
4.4 Invariantdifferencemodeloftheheatequationwithheatflux
relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.5 InvariantdifferencemodeloftheKorteweg–deVriesequation. . . 143
4.6 InvariantdifferencemodelofthenonlinearSchro¨dinger
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5 CombinedMathematicalModelsandSomeGeneralizations 151
5.1 Second-orderordinarydelaydifferentialequations . . . . . . . . . 151
5.2 Partialdelaydifferentialequations . . . . . . . . . . . . . . . . . 154
5.3 Symmetryofdifferential-differenceequations . . . . . . . . . . . 156
6 LagrangianFormalismforDifferenceEquations 161
6.1 DiscreterepresentationofEuler’soperator . . . . . . . . . . . . . 161
6.2 Criterionfortheinvarianceofdifferencefunctionals . . . . . . . . 165
6.3 InvarianceofdifferenceEulerequations . . . . . . . . . . . . . . 169
6.4 Variationofdifferencefunctionalandquasi-extremal
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.5 Invariance of global extremal equations and properties of quasi-
extremalequations . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6 Conservationlawsfordifferenceequations . . . . . . . . . . . . . 181
6.7 Noether-typeidentitiesanddifferenceanalogofNoether’s
theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.8 Necessary and sufficient conditions for global extremal equations
tobeinvariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.9 Applications of Lagrangian formalism to second-order difference
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.10 MovingmeshschemesforthenonlinearSchro¨dingerequation. . . 206
7 HamiltonianFormalismforDifferenceEquations: Symmetries
andFirstIntegrals 221
7.1 DiscreteLegendretransform . . . . . . . . . . . . . . . . . . . . 221
7.2 VariationalstatementofthedifferenceHamiltonianequations . . . 223
7.3 SymplecticityofdifferenceHamiltonianequations . . . . . . . . 225
7.4 InvarianceoftheHamiltonianaction . . . . . . . . . . . . . . . . 227
7.5 Difference Hamiltonian identity and Noether-type theorem for dif-
ferenceHamiltonianequations . . . . . . . . . . . . . . . . . . . 228
CONTENTS vii
7.6 InvarianceofdifferenceHamiltonianequations . . . . . . . . . . 229
7.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
8 DiscreteRepresentationofOrdinaryDifferentialEquations
withSymmetries 237
8.1 ThediscreterepresentationofODEasaseries . . . . . . . . . . . 238
8.2 Three-pointexactschemesfornonlinearODE . . . . . . . . . . . 243
Bibliography 251
Index 263

Preface
This book is dedicated to applications of Lie groups to finite-difference equations,
meshes, and difference functionals. The interest in continuous symmetries of dis-
creteequations(i.e.,Lietransformationgroupsadmittedbysuchequations)springs
from at least two sources. First, discrete equations serve as primary, fundamental
mathematical models in physics and mechanics. Cellular automata and neural nets
also clearly belong to the realm of discrete models. Their integrability and the
existence of exact solutions and conservation laws are undoubtedly related to the
presence of continuous symmetries. This raises the question of finding and using
thetransformationgroupadmittedbyagivendiscreteequation.
Second, modeling a given system of differential equations with the use of dif-
ference equations and meshes can also be based on symmetries. It is well known
that one and the same system of differential equations can be approximated by
infinitely many difference schemes. Hence finite-difference modeling always in-
volvestheproblemofselectingtheschemesthatareinsomerespectadvantageous.
The selection criteria are often given by fundamental physical principles present
in the original model, such as conservation laws, variational principles, the exis-
tence of physically meaningful exact solutions, etc. In this connection, qualitative
considerations play a significant role in the construction of numerical algorithms,
because they permit including the “physical meaning” of the object under study in
the numerical method used to analyze the mathematical model. This point of view
has led to the development of methods for constructing conservative difference
schemes, to the integro-interpolation approach to constructing numerical schemes,
tovariationalmethodsforconstructingschemes,tosymplecticnumericalmethods,
etc.
Theinvarianceofdifferentialequationsundercontinuoustransformationgroups
is certainly a fundamental property of these models and reflects the homogeneity
and isotropy of space–time, the Galilean principle, and other symmetry properties
thatareintuitively(orexperimentally)takenintoaccountbythecreatorsofphysical
models. Therefore,itisapparentlyimportantinthetheoryofdifferenceschemesto
preservethesymmetrypropertieswhenpassingtothefinite-differencemodel,thus
adequately representing the symmetry of the original differential model. This may
serveastheabove-mentionedselectioncriterion.
The author first became acquainted with the idea that the qualitative (physical)
ix