BirkhäuserBoston • • Berlin

JJC. Constanda

Editors

Basel

Science and Engineering Integral Methods in

Volume 2Computational Methods

M.E. Pérez

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Mathematics Subject Classification (2000): 34-06, 35-06, 40-06, 40C10, 45-06, 65-06, 74-06, 76-06

ISBN 978-0-8176-4896-1 e-ISBN 978-0-8176-4897-8DOI 10.1007/978-0-8176-4897-8

Cover design: Joseph Sherman

Library of Congress Control Number: 2009939427

Editors

University of Tulsa

C. Constanda

Tulsa, OK 74104

and Computer Sciences

USA

Department of Mathematical

christian-constanda@utulsa.edu

800 South Tucker Drive

M.E. Pérez Departamento de Matemática Aplicada y Ciencias de la Computación Universidad de Cantabria Avenida de los Castros s/n 39005 Santander Spain meperez@unican.es

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Error Bounds for L1 Galerkin Approximations of WeaklySingular Integral OperatorsM. Ahues, F.D. d’Almeida, and R. Fernandes . . . . . . . . . . . . . . . . . . . . . . 1

2 Construction of Solutions of the Hamburger–Lowner MixedInterpolation Problem for Nevanlinna Class FunctionsJ.A. Alcober, I.M. Tkachenko, M. Urrea . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 A Three-Dimensional Eutrophication Model: Analysis andControlL.J. Alvarez-Vazquez, F.J. Fernandez, and R. Munoz-Sola . . . . . . . . . . . 21

4 An Analytical Solution for the Transient Two-DimensionalAdvection–Diffusion Equation with Non-Fickian Closure inCartesian Geometry by the Generalized Integral TransformTechniqueD. Buske, M.T. Vilhena, D. Moreira, and T. Tirabassi . . . . . . . . . . . . . . 33

5 A Numerical Solution of the Dispersion Equation of GuidedWave Propagation in N-Layered MediaJ. Cardona, P. Tabuenca, and A. Samartin . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Discretization of Coefficient Control Problems with aNonlinear Cost in the GradientJ. Casado-Dıaz, J. Couce-Calvo, M. Luna-Laynez, J.D. Martın-Gomez 55

7 Optimal Control and Vanishing Viscosity for the BurgersEquationC. Castro, F. Palacios, and E. Zuazua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

vi Contents

8 A High-Order Finite Volume Method for NonconservativeProblems and Its Application to Model Submarine AvalanchesM.J. Castro Dıaz, E.D. Fernandez-Nieto, J.M. Gonzalez-Vida,A. Mangeney, and C. Pares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9 Convolution Quadrature Galerkin Method for the ExteriorNeumann Problem of the Wave EquationD.J. Chappell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

10 Solution Estimates in Classical Bending of PlatesI. Chudinovich, C. Constanda, D. Doty, and A. Koshchii . . . . . . . . . . . . 113

11 Modified Newton’s Methods for Systems of NonlinearEquationsA. Cordero, J.R. Torregrosa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

12 Classification of Some Penalty MethodsA. Correia, J. Matias, P. Mestre, and C. Serodio . . . . . . . . . . . . . . . . . . . 131

13 A Closed-Form Formulation for Pollutant Dispersion inthe AtmosphereC.P. Costa, M.T. Vilhena, and T. Tirabassi . . . . . . . . . . . . . . . . . . . . . . . . 141

14 High-Order Methods for Weakly Singular VolterraIntegro-Differential EquationsT. Diogo, M. Kolk, P. Lima, and A. Pedas . . . . . . . . . . . . . . . . . . . . . . . . . 151

15 Numerical Solution of a Class of Integral EquationsArising in a Biological Laboratory ProcedureD.A. French, C.W. Groetsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

16 A Mixed Two-Grid Method Applied to a FredholmEquation of the Second KindL. Grammont . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

17 Homogenized Models of Radiation Transfer in MultiphaseMediaA.V. Gusarov, I. Smurov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

18 A Porous Finite Element Model of the Motion of theSpinal CordP.J. Harris, C. Hardwidge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

19 Boundary Hybrid Galerkin Method for Elliptic and WavePropagation Problems in R3 over Planar StructuresC. Jerez-Hanckes, J.-C. Nedelec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Contents vii

20 Boundary Integral Solution of the Time-FractionalDiffusion EquationJ. Kemppainen, K. Ruotsalainen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

21 Boundary Element Collocation Method for Time-Fractional Diffusion EquationsJ. Kemppainen, K. Ruotsalainen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

22 Wavelet-Based Holder Regularity Analysis in ConditionMonitoringV. Kotila, S. Lahdelma, K. Ruotsalainen . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

23 Integral Equation Technique for Finding the CurrentDistribution of Strip Antennas in a Gyrotropic MediumA.V. Kudrin, E.Yu. Petrov, and T.M. Zaboronkova . . . . . . . . . . . . . . . . . 243

24 A Two-Grid Method for a Second Kind Integral Equationwith Green’s KernelR.P. Kulkarni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

25 A Brief Overview of Plate Finite Element MethodsC. Lovadina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

26 Influence of a Weak Aerodynamics/Structure Interactionon the Aerodynamical Global Optimization of ShapeA. Nastase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

27 Multiscale Investigation of Solutions of the Wave EquationM. Perel, M. Sidorenko, and E. Gorodnitskiy . . . . . . . . . . . . . . . . . . . . . . . 291

28 The Laplace Transform Method for the Albedo BoundaryConditions in Neutron Diffusion Eigenvalue ProblemsC.Z. Petersen, M.T. Vilhena, D. Moreira, and R.C. Barros . . . . . . . . . . 301

29 Solution of the Fokker–Planck Pencil Beam Equation forElectrons by the Laplace Transform TechniqueB. Rodriguez, M.T. Vilhena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

30 Nonlinear Functional Parabolic EquationsL. Simon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

31 Grid Computing for Multi-Spectral TomographicReconstruction of Chlorophyll Concentration in Ocean WaterR.P. Souto, H.F. de Campos Velho, F.F. Paes, S. Stephany, P.O.A.Navaux, A.S. Charao, and J.K. Vizzotto . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

32 Long-Time Solution of the Wave Equation Using NonlinearDissipative StructuresJ. Steinhoff, S. Chitta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

viii Contents

33 High-Performance Computing for Spectral ApproximationsP.B. Vasconcelos, O. Marques, and J.E. Roman . . . . . . . . . . . . . . . . . . . . 351

34 An Analytical Solution for the General PerturbedDiffusion Equation by an Integral Transform TechniqueM.T. Vilhena, B.E.J. Bodmann, I.R. Heinen . . . . . . . . . . . . . . . . . . . . . . . 361

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Preface

The international conferences on Integral Methods in Science and Engineering(IMSE) are biennial opportunities for academics and other researchers whosework makes essential use of analytic or numerical integration methods todiscuss their latest results and exchange views on the development of noveltechniques of this type.

The first two conferences in the series, IMSE1985 and IMSE1990, werehosted by the University of Texas–Arlington. At the latter, the IMSE con-sortium was created and charged with organizing these conferences under theguidance of an International Steering Committee. Subsequently, IMSE1993took place at Tohoku University, Sendai, Japan, IMSE1996 at the Universityof Oulu, Finland, IMSE1998 at Michigan Technological University, Houghton,MI, USA, IMSE2000 in Banff, AB, Canada, IMSE2002 at the University ofSaint-Etienne, France, IMSE2004 at the University of Central Florida, Or-lando, FL, USA, and IMSE2006 at Niagara Falls, ON, Canada. The IMSEconferences are now recognized as an important forum where scientists andengineers working with integral methods express their views about, and inter-act to extend the practical applicability of, a very elegant and powerful classof mathematical procedures.

A distinguishing characteristic of all the IMSE meetings is their generalatmosphere—a blend of utmost professionalism and a strong collegial-socialcomponent. IMSE2008, organized at the University of Cantabria, Spain, andattended by delegates from 27 countries on 5 continents, maintained this tra-dition, marking another unqualified success in the history of the IMSE con-sortium. For the smoothness and detail-perfect arrangements throughout theconference, the participants and the Steering Committee would like to expresstheir special thanks to the Local Organizing Committee:

M. Eugenia Perez (Departamento de Matematica Aplicada y Ciencias dela Computacion, ETSI Caminos, Canales y Puertos), Chairman;

Miguel Lobo (Departamento de Matematicas, Estadıstica y Computacion,Facultad de Ciencias);

x Preface

Delfina Gomez (Departamento de Matematicas, Estadıstica y Computa-cion, Facultad de Ciencias).

The Local Organizing Committee and the Steering Committee also wishto acknowledge the financial support received from the following institutions:

Universidad de Cantabria (in particular, Vicerrectorado de Investigaciony Transferencia del Conocimiento, Facultad de Ciencias, ETSI Caminos,Canales y Puertos, Departamento de Matematicas, Estadıstica y Computa-cion, and Departamento de Matematica Aplicada y Ciencias de la Com-putacion);

Ministerio de Ciencia e Innovacion (Ref. MTM2007-30182-E);Sociedad Regional Cantabra de I+D+i (IDICAN. Ref. 25-2-2007);i-MATH Consolider (MEC, Ref. C3-0087);Caja de Burgos;Consejerıa de Cultura, Turismo y Deporte del Gobierno de Cantabria;Ayuntamiento de Santander;Sociedad Espanola de Matematica Aplicada (SeMA).

Last but not least, they would like to express their thanks to MICINN(MTM2005-07720) for partial support, to Antonio Jose Gonzalez for his workon the graphical design of the conference, to the colleagues involved in thecoordination of the monographic sessions, and to all the participants, whosepresence and scientific activity in Santander ensured the success of this meet-ing.

The next IMSE conference will be held in July 2010 in Brighton, UK.Details concerning this event are posted on the conference web page,

http://www.cmis.brighton.ac.uk/imse2010

This volume contains 2 invited papers and 32 contributed peer-reviewedpapers, arranged in alphabetical order by (first) author’s name. The editorswould like to thank the staff at Birkhauser-Boston for their efficient handlingof the publication process.

Tulsa, Oklahoma, USA Christian Constanda, IMSE Chairman

The International Steering Committee of IMSE:

C. Constanda (University of Tulsa), ChairmanM. Ahues (University of Saint-Etienne)B. Bodmann (Federal University of Rio Grande do Sul)I. Chudinovich (University of Tulsa)H. de Campos Velho (INPE, Sao Jose dos Campos)

Preface xi

P. Harris (University of Brighton)A. Largillier (University of Saint-Etienne)S. Mikhailov (Brunel University)A. Mioduchowski (University of Alberta)D. Mitrea (University of Missouri-Columbia)Z. Nashed (University of Central Florida)A. Nastase (Rhein.-Westf. Technische Hochschule, Aachen)M.E. Perez (University of Cantabria)S. Potapenko (University of Waterloo)K. Ruotsalainen (University of Oulu)S. Seikkala (University of Oulu)O. Shoham (University of Tulsa)

List of Contributors

Mario AhuesUniversite de Saint-Etienne23 rue du Dr. Paul MichelonSaint-Etienne 42023, cedex 2, Francemario.ahues@univ-st-etienne.fr

Juan A. AlcoberUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, Spainjuaalau@mat.upv.es

Filomena D. d’AlmeidaUniversidade do PortoRua Roberto FriasPorto 4200-465, Portugalfalmeida@fe.up.pt

Lino J. Alvarez-VazquezUniversidad de VigoETSI TelecomunicacionVigo 36310, Spainlino@dma.uvigo.es

Ricardo C. de BarrosUniversidade do Estado do Rio

de JaneiroRua Alberto Rangel s.n.Nova Friburgo, RJ 28630-050, Brazilrcbarros@pq.cnpq.br

Bardo E.J. BodmannUniversidade Federal do Rio Grande

do SulAv. Osvaldo Aranha 99/4Porto Alegre, RS 90035-190, Brazilbardo.bodmann@ufrgs.br

Daniela BuskeUniversidade Federal de PelotasCampus Capao do LeaoCaixa Postal 354Pelotas, RS 96010-900, Brazildanielabuske@gmail.com

Haroldo F. de Campos VelhoInstituto Nacional de

Pesquisas EspaciaisP.O. Box 515Sao Jose dos Campos, SP 12245-970,Brazilharoldo@lac.inpe.br

Juan CardonaUniversidad de CantabriaDique de Gamazo, 1Santander 39004, Spaincardonaj@unican.es

Juan Casado-DıazUniversidad de SevillaC/ Tarfia s/nSevilla 41012, Spainjcasadod@us.es

xiv List of Contributors

Carlos CastroUniversidad Politecnica de MadridProfessor ArangurenMadrid 28040, Spaincarlos.castro@upm.es

Manuel J. Castro DıazUniversidad de MalagaCampus de TeatinosMalaga 29071, Spaincastro@anamat.cie.uma.es

David J. ChappellUniversity of NottinghamUniversity ParkNottingham NG7 2RD, UKdavid.chappell@nottingham.ac.uk

Andrea S. CharaoUniversidade Federal de Santa MariaAvenida Roraima, 1000Santa Maria, RS 97105-900, Brazilandrea@inf.ufsm.br

Subhashini ChittaFlow Analysis Inc.411 B.H. Goethert ParkwayTullahoma, TN 37388, USAsubha@flowanalysis.com

Igor ChudinovichUniversity of Tulsa800 S. Tucker Drive,Tulsa, OK 74104, USAigor-chudinovych@utulsa.edu

Christian ConstandaUniversity of Tulsa800 S. Tucker Drive,Tulsa, OK 74104, USAchristian-constanda@utulsa.edu

Alicia CorderoUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, Spainacordero@mat.upv.es

Aldina I.A. CorreiaInstituto Politecnico do Portoand Universidade de Tras-os-Montes

e Alto DouroRua do Curral, Casa do CurralMargaride, 4610-156 Felgueiras,Portugalaldinacorreia@eu.ipp.pt

Camila Pinto da CostaUniversidade Federal de PelotasCampus Capao do LeaoCaixa Postal 354Pelotas, RS 96010-900, Brazilcamiladacosta@gmail.com

Julio Couce-CalvoUniversidad de SevillaC/ Tarfia s/nSevilla 41012, Spaincouce@us.es

Teresa DiogoInstituto Superior TecnicoAv. Rovisco Pais, 1Lisbon 1049-001, Portugaltdiogo@math.ist.utl.pt

Dale R. DotyUniversity of Tulsa800 S. Tucker DriveTulsa, OK 74104, USAdale-doty@utulsa.edu

Rosario FernandesUniversidade do MinhoCampus de GualtarBraga 4710-057, Portugalrosario@math.uminho.pt

Francisco J. FernandezUniversidad de Santiago de

CompostelaCampus Universitario SurSantiago de Compostela 15782 Spainfranfdz@usc.es

List of Contributors xv

Enrique D. Fernandez-NietoUniversidad de SevillaAvda. Reina Mercedes, 2Sevilla 41012, Spainedofer@us.es

Donald A. FrenchUniversity of Cincinnati2855 Campus WayCincinnati, OH 45221-0025, USAfrench@math.uc.edu

Jose M. Gonzalez-VidaUniversidad de MalagaCampus de TeatinosMalaga 29071, Spainjgv@uma.es

Evgeniy GorodnitskiySt. Petersburg UniversityUlyanovskaya 1-1, PetrodvoretsSt. Petersburg 198904, Russiaeugy@yandex.ru

Laurence GrammontUniversite de Saint-Etienne23, rue du Dr. Paul MichelonSaint-Etienne 42023, cedex 2, Francelaurence.grammont

@univ-st-etienne.fr

Charles W. GroetschThe Citadel171 Moultrie St.Charleston, SC 29409-6420, USAcharles.groetsch@citadel.edu

Andrey V. GusarovEcole Nationale d’Ingenieurs de

Saint-Etienne58 rue Jean ParotSaint-Etienne 42023, Francegusarov@enise.fr

Carl HardwidgePrincess Royal HospitalLewes RoadHaywards Heath RH16 4EX, UKcarl.hardwidge@bsuh.nhs.uk

Paul J. HarrisUniversity of BrightonLewes RoadBrighton BN2 4GJ, UKp.j.harris@brighton.ac.uk

Ismael R. HeinenUniversidade Federal do Rio Grande

do SulAv. Osvaldo Aranha 99/4Porto Alegre, RS 90046-900, Brazilheire@bol.com.br

Carlos F. Jerez-HanckesETH ZurichRamistrasse 101Zurich 8092, Switzerlandcjerez@math.ethz.ch

Jukka KemppainenUniversity of OuluPO Box 4500Oulu 90014, Finlandjukemppa@paju.oulu.fi

Marek KolkUniversity of TartuLiivi 2Tartu 50409, Estoniamarek.kolk@ut.ee

Alexander F. KoshchiiInternational Solomon UniversityGrazhdanskaya 22/26Kharkiv 61057, Ukraineaf-koshchii@rambler.ru

Vesa KotilaUniversity of OuluPO Box 4500Oulu 90014, Finlandvesa.kotila@oulu.fi

xvi List of Contributors

Alexander V. KudrinUniversity of Nizhny Novgorod23 Gagarin AvenueNizhny Novgorod 603950, Russiakud@rf.unn.ru

Rekha P. KulkarniIndian Institute of Technology

BombayPowaiMumbai 400076, Indiarpk@math.iitb.ac.in

Sulo LahdelmaUniversity of OuluPO Box 4200Oulu 90014, Finlandsulo.lahdelma@oulu.fi

Pedro LimaInstituto Superior TecnicoAv. Rovisco Pais, 1Lisbon 1049-001, Portugalplima@math.ist.utl.pt

Carlo LovadinaUniversita di PaviaVia Ferrata 1Pavia 27100, Italycarlo.lovadina@unipv.it

Manuel Luna-LaynezUniversidad de SevillaC/ Tarfia s/nSevilla 41012, Spainmllaynez@us.es

Anne MangeneyInstitut de Physique du Globe

de Paris4, place JussieuParis 75252, cedex 05, Francemangeney@ipgp.jussieu.fr

Osni MarquesLawrence Berkeley National

Laboratory1 Cyclotron Road, MS 50F-1650Berkeley, CA 94720-8139, USAoamarques@lbl.gov

J.D. Martın-GomezUniversidad de Sevillac/Tartia s/nSevilla 41011, Spainjdmartin@us.es

Joao L.H. MatiasUniversidade de Tras-os-Montes

e Alto DouroEdifıcio das Ciencias FlorestaisQuinta de Prados5001-801 Vila Real, Portugalj matias@utad.pt

Pedro M.M.A. MestreUniversidade de Tras-os-Montes

e Alto DouroEdifıcio Engenharias II5001-801 Vila Real, Portugalpmestre@utad.pt

Davidson M. MoreiraUniversidade Federal do PampaRua Carlos Barbosa s/nB. Getulio VargasBage, RS 96412-420, Brazildavidson@pq.cnpq.br

Rafael Munoz-SolaUniversidad de Santiago de

CompostelaCampus Universitario SurSantiago de Compostela 15782Spainrafams@usc.es

Adriana NastaseRhein.-Westf. Technische HochschuleTemplergraben 55Aachen 52062, Germanynastase@lafaero.rwth-aachen.de

List of Contributors xvii

Phillippe O.A. NavauxUniversidade Federal do Rio Grande

do SulAv. Bento Gonalves, 9500Porto Alegre 91501-970, Brazilnavaux@inf.ufrgs.br

Jean-Claude NedelecEcole PolytechniqueRoute de SaclayPalaiseau 91128, Francenedelec@cmapx.polytechnique.fr

Fabiana F. PaesInstituto Nacional de

Pesquisas EspaciaisP.O. Box 515Sao Jose dos Campos, SP 12245-970Brazilfabiana.paes@lac.inpe.br

Francisco PalaciosUniversidad Politecnica de MadridAvda ComplutenseMadrid 28040, Spainfpalacios@gmail.com

Carlos Pares MadronalUniversidad de MalagaCampus de TeatinosMalaga 29071, Spainpares@anamat.cie.uma.es

Arvet PedasUniversity of TartuLiivi 2Tartu 50409, Estoniaarvet.pedas@ut.ee

Maria PerelSt. Petersburg UniversityUlyanovskaya 1-1, PetrodvoretsSt. Petersburg 198904, Russiaand Ioffe Physical-Technical InstitutePolitekhnicheskaya 26St. Petersburg 194021, Russiaperel@mph.phys.spbu.ru

Claudio Z. PetersenUniversidade Federal do Rio Grande

do SulSarmento Leite, 425/3Porto Alegre, RS 90046-900, Brazilclaudiopetersen@yahoo.com.br

Evgeny Yu. PetrovUniversity of Nizhny Novgorod23 Gagarin AvenueNizhny Novgorod 603950, Russiaepetrov@rf.unn.ru

Barbara D.A. RodriguezUniversidade Federal do Rio GrandeAvenida Italia km 8Campus CarreirosRio Grande, RS 96201-900, Brazilbarbara.rodriguez@gmail.com

Jose E. RomanUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, Spainjroman@dsic.upv.es

Keijo RuotsalainenUniversity of OuluPO Box 4500Oulu 90014, Finlandkeijo.ruotsalainen@ee.oulu.fi

Avelino SamartinUniversidad Politecnica de MadridCiudad Universitaria s/nMadrid 28040, Spainavelino.samartin@upm.es

Carlos M.J.A. SerodioUniversidade de Tras-os-Montes

e Alto DouroEdifıcio Engenharias II5001-801 Vila Real, Portugalcserodio@utad.pt

xviii List of Contributors

Mikhail SidorenkoSt. Petersburg UniversityUlyanovskaya 1-1, Petrodvorets198904 St. Petersburg, Russiam-sidorenko@yandex.ru

Laszlo SimonL. Eotvos UniversityPazmany P. setany 1/CBudapest 1117, Hungarysimonl@ludens.elte.hu

Igor SmurovEcole Nationale d’Ingenieurs58 rue Jean ParotSaint-Etienne 42023, Francesmurov@enise.fr

Roberto P. SoutoInstituto Nacional de

Pesquisas EspaciaisSao Jose dos Campos, SP 12245-970Brazilrpsouto@gmail.com

John SteinhoffUniversity of Tennessee411 B.H. Goethert ParkwayTullahoma, TN 37388, USAjsteinho@utsi.edu

Stephan StephanyInstituto Nacional de

Pesquisas EspaciaisSao Jose dos Campos, SP 12245-970Brazilstephan@lac.inpe.br

Pedro TabuencaUniversidad de CantabriaDique de Gamazo, 1Santander 39004, Spaintabuencp@orange.es

Tiziano TirabassiIstituto di Scienze dell’Atmosfera e

del Clima–CNRVia P. Gobetti 101Bologna 40129, Italyt.tirabassi@isac.cnr.it

Igor M. TkachenkoUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, Spainimtk@mat.upv.es

Juan R. TorregrosaUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, Spainjrtorre@mat.upv.es

Marcel UrreaUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, Spainmurrea@mat.upv.es

Paulo B. VasconcelosUniversidade do PortoRua Dr. Roberto FriasPorto 4200-464, Portugalpjv@fep.up.pt

Marco T.M.B. de VilhenaUniversidade Federal do Rio Grande

do SulRua Sarmento Leite, 425/3Porto Alegre, RS 90046-900Brazilvilhena@pq.cnpq.br

Juliana K. VizzottoCentro Universitario FranciscanoRua dos Andradas, 1614Santa Maria, RS 97010-032, Braziljuvizzotto@gmail.com

List of Contributors xix

Tatyana M. ZaboronkovaTechnical University of Nizhny

Novgorod24 Minin StreetNizhny Novgorod 603950,Russiazabr@nirfi.sci-nnov.ru

Enrique ZuazuaBasque Center for Applied

MathematicsBizkaia Technology ParkZamudio (Bilbao) 48170,Basque Country, Spainzuazua@bcamath.org