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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
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 [email protected]
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, [email protected]
Juan A. AlcoberUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, [email protected]
Filomena D. d’AlmeidaUniversidade do PortoRua Roberto FriasPorto 4200-465, [email protected]
Lino J. Alvarez-VazquezUniversidad de VigoETSI TelecomunicacionVigo 36310, [email protected]
Ricardo C. de BarrosUniversidade do Estado do Rio
de JaneiroRua Alberto Rangel s.n.Nova Friburgo, RJ 28630-050, [email protected]
Bardo E.J. BodmannUniversidade Federal do Rio Grande
do SulAv. Osvaldo Aranha 99/4Porto Alegre, RS 90035-190, [email protected]
Daniela BuskeUniversidade Federal de PelotasCampus Capao do LeaoCaixa Postal 354Pelotas, RS 96010-900, [email protected]
Haroldo F. de Campos VelhoInstituto Nacional de
Pesquisas EspaciaisP.O. Box 515Sao Jose dos Campos, SP 12245-970,[email protected]
Juan CardonaUniversidad de CantabriaDique de Gamazo, 1Santander 39004, [email protected]
Juan Casado-DıazUniversidad de SevillaC/ Tarfia s/nSevilla 41012, [email protected]
xiv List of Contributors
Carlos CastroUniversidad Politecnica de MadridProfessor ArangurenMadrid 28040, [email protected]
Manuel J. Castro DıazUniversidad de MalagaCampus de TeatinosMalaga 29071, [email protected]
David J. ChappellUniversity of NottinghamUniversity ParkNottingham NG7 2RD, [email protected]
Andrea S. CharaoUniversidade Federal de Santa MariaAvenida Roraima, 1000Santa Maria, RS 97105-900, [email protected]
Subhashini ChittaFlow Analysis Inc.411 B.H. Goethert ParkwayTullahoma, TN 37388, [email protected]
Igor ChudinovichUniversity of Tulsa800 S. Tucker Drive,Tulsa, OK 74104, [email protected]
Christian ConstandaUniversity of Tulsa800 S. Tucker Drive,Tulsa, OK 74104, [email protected]
Alicia CorderoUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, [email protected]
Aldina I.A. CorreiaInstituto Politecnico do Portoand Universidade de Tras-os-Montes
e Alto DouroRua do Curral, Casa do CurralMargaride, 4610-156 Felgueiras,[email protected]
Camila Pinto da CostaUniversidade Federal de PelotasCampus Capao do LeaoCaixa Postal 354Pelotas, RS 96010-900, [email protected]
Julio Couce-CalvoUniversidad de SevillaC/ Tarfia s/nSevilla 41012, [email protected]
Teresa DiogoInstituto Superior TecnicoAv. Rovisco Pais, 1Lisbon 1049-001, [email protected]
Dale R. DotyUniversity of Tulsa800 S. Tucker DriveTulsa, OK 74104, [email protected]
Rosario FernandesUniversidade do MinhoCampus de GualtarBraga 4710-057, [email protected]
Francisco J. FernandezUniversidad de Santiago de
CompostelaCampus Universitario SurSantiago de Compostela 15782 [email protected]
List of Contributors xv
Enrique D. Fernandez-NietoUniversidad de SevillaAvda. Reina Mercedes, 2Sevilla 41012, [email protected]
Donald A. FrenchUniversity of Cincinnati2855 Campus WayCincinnati, OH 45221-0025, [email protected]
Jose M. Gonzalez-VidaUniversidad de MalagaCampus de TeatinosMalaga 29071, [email protected]
Evgeniy GorodnitskiySt. Petersburg UniversityUlyanovskaya 1-1, PetrodvoretsSt. Petersburg 198904, [email protected]
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, [email protected]
Andrey V. GusarovEcole Nationale d’Ingenieurs de
Saint-Etienne58 rue Jean ParotSaint-Etienne 42023, [email protected]
Carl HardwidgePrincess Royal HospitalLewes RoadHaywards Heath RH16 4EX, [email protected]
Paul J. HarrisUniversity of BrightonLewes RoadBrighton BN2 4GJ, [email protected]
Ismael R. HeinenUniversidade Federal do Rio Grande
do SulAv. Osvaldo Aranha 99/4Porto Alegre, RS 90046-900, [email protected]
Carlos F. Jerez-HanckesETH ZurichRamistrasse 101Zurich 8092, [email protected]
Jukka KemppainenUniversity of OuluPO Box 4500Oulu 90014, [email protected]
Marek KolkUniversity of TartuLiivi 2Tartu 50409, [email protected]
Alexander F. KoshchiiInternational Solomon UniversityGrazhdanskaya 22/26Kharkiv 61057, [email protected]
Vesa KotilaUniversity of OuluPO Box 4500Oulu 90014, [email protected]
xvi List of Contributors
Alexander V. KudrinUniversity of Nizhny Novgorod23 Gagarin AvenueNizhny Novgorod 603950, [email protected]
Rekha P. KulkarniIndian Institute of Technology
BombayPowaiMumbai 400076, [email protected]
Sulo LahdelmaUniversity of OuluPO Box 4200Oulu 90014, [email protected]
Pedro LimaInstituto Superior TecnicoAv. Rovisco Pais, 1Lisbon 1049-001, [email protected]
Carlo LovadinaUniversita di PaviaVia Ferrata 1Pavia 27100, [email protected]
Manuel Luna-LaynezUniversidad de SevillaC/ Tarfia s/nSevilla 41012, [email protected]
Anne MangeneyInstitut de Physique du Globe
de Paris4, place JussieuParis 75252, cedex 05, [email protected]
Osni MarquesLawrence Berkeley National
Laboratory1 Cyclotron Road, MS 50F-1650Berkeley, CA 94720-8139, [email protected]
J.D. Martın-GomezUniversidad de Sevillac/Tartia s/nSevilla 41011, [email protected]
Joao L.H. MatiasUniversidade de Tras-os-Montes
e Alto DouroEdifıcio das Ciencias FlorestaisQuinta de Prados5001-801 Vila Real, Portugalj [email protected]
Pedro M.M.A. MestreUniversidade de Tras-os-Montes
e Alto DouroEdifıcio Engenharias II5001-801 Vila Real, [email protected]
Davidson M. MoreiraUniversidade Federal do PampaRua Carlos Barbosa s/nB. Getulio VargasBage, RS 96412-420, [email protected]
Rafael Munoz-SolaUniversidad de Santiago de
CompostelaCampus Universitario SurSantiago de Compostela [email protected]
Adriana NastaseRhein.-Westf. Technische HochschuleTemplergraben 55Aachen 52062, [email protected]
List of Contributors xvii
Phillippe O.A. NavauxUniversidade Federal do Rio Grande
do SulAv. Bento Gonalves, 9500Porto Alegre 91501-970, [email protected]
Jean-Claude NedelecEcole PolytechniqueRoute de SaclayPalaiseau 91128, [email protected]
Fabiana F. PaesInstituto Nacional de
Pesquisas EspaciaisP.O. Box 515Sao Jose dos Campos, SP [email protected]
Francisco PalaciosUniversidad Politecnica de MadridAvda ComplutenseMadrid 28040, [email protected]
Carlos Pares MadronalUniversidad de MalagaCampus de TeatinosMalaga 29071, [email protected]
Arvet PedasUniversity of TartuLiivi 2Tartu 50409, [email protected]
Maria PerelSt. Petersburg UniversityUlyanovskaya 1-1, PetrodvoretsSt. Petersburg 198904, Russiaand Ioffe Physical-Technical InstitutePolitekhnicheskaya 26St. Petersburg 194021, [email protected]
Claudio Z. PetersenUniversidade Federal do Rio Grande
do SulSarmento Leite, 425/3Porto Alegre, RS 90046-900, [email protected]
Evgeny Yu. PetrovUniversity of Nizhny Novgorod23 Gagarin AvenueNizhny Novgorod 603950, [email protected]
Barbara D.A. RodriguezUniversidade Federal do Rio GrandeAvenida Italia km 8Campus CarreirosRio Grande, RS 96201-900, [email protected]
Jose E. RomanUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, [email protected]
Keijo RuotsalainenUniversity of OuluPO Box 4500Oulu 90014, [email protected]
Avelino SamartinUniversidad Politecnica de MadridCiudad Universitaria s/nMadrid 28040, [email protected]
Carlos M.J.A. SerodioUniversidade de Tras-os-Montes
e Alto DouroEdifıcio Engenharias II5001-801 Vila Real, [email protected]
xviii List of Contributors
Mikhail SidorenkoSt. Petersburg UniversityUlyanovskaya 1-1, Petrodvorets198904 St. Petersburg, [email protected]
Laszlo SimonL. Eotvos UniversityPazmany P. setany 1/CBudapest 1117, [email protected]
Igor SmurovEcole Nationale d’Ingenieurs58 rue Jean ParotSaint-Etienne 42023, [email protected]
Roberto P. SoutoInstituto Nacional de
Pesquisas EspaciaisSao Jose dos Campos, SP [email protected]
John SteinhoffUniversity of Tennessee411 B.H. Goethert ParkwayTullahoma, TN 37388, [email protected]
Stephan StephanyInstituto Nacional de
Pesquisas EspaciaisSao Jose dos Campos, SP [email protected]
Pedro TabuencaUniversidad de CantabriaDique de Gamazo, 1Santander 39004, [email protected]
Tiziano TirabassiIstituto di Scienze dell’Atmosfera e
del Clima–CNRVia P. Gobetti 101Bologna 40129, [email protected]
Igor M. TkachenkoUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, [email protected]
Juan R. TorregrosaUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, [email protected]
Marcel UrreaUniversidad Politecnica de ValenciaCamino de Vera s/nValencia 46022, [email protected]
Paulo B. VasconcelosUniversidade do PortoRua Dr. Roberto FriasPorto 4200-464, [email protected]
Marco T.M.B. de VilhenaUniversidade Federal do Rio Grande
do SulRua Sarmento Leite, 425/3Porto Alegre, RS [email protected]
Juliana K. VizzottoCentro Universitario FranciscanoRua dos Andradas, 1614Santa Maria, RS 97010-032, [email protected]
List of Contributors xix
Tatyana M. ZaboronkovaTechnical University of Nizhny
Novgorod24 Minin StreetNizhny Novgorod 603950,[email protected]
Enrique ZuazuaBasque Center for Applied
MathematicsBizkaia Technology ParkZamudio (Bilbao) 48170,Basque Country, [email protected]
1
Error Bounds for L1 Galerkin Approximationsof Weakly Singular Integral Operators
M. Ahues,1 F.D. d’Almeida,2 and R. Fernandes3
1 Universite de Lyon, Laboratoire de Mathematiques de l’Universite deSaint-Etienne, France; [email protected]
2 Universidade do Porto, Portugal; [email protected] Universidade do Minho, Portugal; [email protected]
1.1 Introduction
From all standard projection approximations of a bounded linear opera-tor in a Banach space, a general (i.e., not necessarily orthogonal) Galerkinscheme ([At97] and [ALL01]) is the simplest one from a computational pointof view. In this chapter, we give an upper bound of the relative error in termsof the mesh size of the underlying discretization grid on which no regular-ity assumptions are made. A weakly singular second kind Fredholm integralequation is used as an application to illustrate the actual sharpness of theerror estimates. As is usual in the case of weakly singular error bounds, thesharpness of our bound is rather poor compared with practical results.
We consider the space L1([0, τ∗],C) of complex-valued Lebesgue-integrable(classes of) functions on [0, τ∗]. For x ∈ L1([0, τ∗],C), define
(Tx)(s) :=∫ τ∗
0g(|s− t|)x(t) dt, s ∈ [0, τ∗], (1.1)
where g :]0,∞[→ R is a weakly singular function at 0 in the following sense:
g(0+) = ∞, g ∈ L1([0,∞[,R) ∩ C0(]0,∞[,R), g ≥ 0, g ↘ in ]0,∞[. (1.2)
It can be checked that Tx ∈ L1([0, τ∗],C), and that T is compact as anoperator on L1([0, τ∗],C) (see [ALL01]). Let z ∈ re(T ), the resolvent set of T ,so T − zI is bijective and has a bounded inverse. Since T is compact, z �= 0.This implies that, for any f ∈ L1([0, τ∗],C), the Fredholm integral equationof the second kind
(T − zI)ξ = f (1.3)
has a unique solution ξ ∈ L1([0, τ∗],C).
1
© Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2010 Volume 2: Computational Methods, DOI 10.1007/978-0-8176-4897-8_1,C. Constanda and M.E. Pérez (eds.), Integral Methods in Science and Engineering,
2 M. Ahues, F.D. d’Almeida, and R. Fernandes
The resolvent operator R(z) := (T − zI)−1 allows us to write this solutionas ξ = R(z)f .
Concerning the derivative of Tx we have the following theorem provedin [AAF09]:
Theorem 1. For any x ∈ L1([0, τ∗],C) such that x′ ∈ L1([0, τ∗],C), Tx is adifferentiable function at all s ∈ ]0, τ∗[, and its derivative is given by
(Tx)′(s) = x(0)g(s) − x(τ∗)g(τ∗ − s) + (Tx′)(s), s ∈ ]0, τ∗[.
Since the solution ξ of (1.3) satisfies ξ =1z(Tξ− f), we may expect boundary
layers for ξ at the end points and at points where f has a discontinuity.Boundary layers lead us to decompose the interval [0, τ∗] into subdomains.Those including the boundary layers will be discretized with finer grids thanthe ones used elsewhere.
1.2 Numerical Approximations
Let us consider the operator (1.1) in an arbitrary interval [a, b] and let theunderlying complex Banach space be X := L1([a, b],C).
(Tx)(s) :=∫ b
a
g(|s− t|)x(t) dt, s ∈ [a, b], x ∈ X,
where g :]0,+∞[→ R satisfies (1.2). We describe the general Galerkin scheme.To compute a numerical solution ϕn of the exact solution ϕ of the equation
(T − zI)ϕ = f (1.4)
we use a sequence of bounded projections (πn)n≥1 each one having finite rank,and the corresponding sequence of operators (Tn)n≥1 given by Tn := πnTπnand we assume that re(T ) ⊆ re(Tn). We replace the exact equation (1.4) withthe approximate problem of solving exactly the following equation for ϕn:
(Tn − zI)ϕn = πnf. (1.5)
The approximate resolvent Rn(z) := (Tn−zI)−1 allows us to write the uniquesolution of the approximate equation as ϕn := Rn(z)πnf . The second resol-vent identities,
Rn(z) −R(z) = Rn(z)(T − Tn)R(z) = R(z)(T − Tn)Rn(z),
will be useful in the sequel.
1 Galerkin Approximations for Weakly Singular Operators 3
Proposition 1 (See [ALL01], [Ch83]). If (πn)n≥1 is pointwise convergentto I, then there exists n0 such that
β := |z| supn≥n0
‖(πnT − zI)−1‖
is finite, and there exists a constant α > 0 such that, for large enough n,
α‖(I − πn)ϕ‖ ≤ ‖ϕn − ϕ‖ ≤ β‖(I − πn)ϕ‖.
Theorem 2. For f �= 0, the solution of the Galerkin approximation satisfies
‖ϕn − ϕ‖‖ϕ‖ ≤ C(‖(I − πn)T‖ +
‖(I − πn)f‖‖ϕ‖ ), (1.6)
for n large enough and any C ≥ supn
‖(πnT − zI)−1‖.
The proof can be found in [AAF09].Let us consider a general grid Gn := (τj)nj=0 such that
τ0 := a, τn := b, hj := τj − τj−1 > 0, hmax := max1≤j≤n
hj , hmin := min1≤j≤n
hj .
We associate to this grid the local mean functionals e∗j defined by
〈x , e∗j 〉 :=
1hj
∫ τj
τj−1
x(t) dt,
and the piecewise constant canonical functions ej given by
ej(s) :{
1 for s ∈ ]τj−1, τj ],0 otherwise.
Since the families (ej)nj=1 and
(e∗j
)nj=1
are adjoint to one another, these linearlyindependent families lead to a sequence of projections with finite rank n:
πnx :=n∑
j=1
〈x , e∗j 〉ej for x ∈ X.
Recall that the oscillation of x ∈ X is given by
ω1(x, δ) := sup0≤h≤δ
∫ b−h
a
|x(s + h) − x(s)| ds.
Theorem 3. For all x ∈ X, ‖(I − πn)x‖ ≤ 2n∑
j=1ω1(x∣∣[τj−1,τj ]
, hj).
4 M. Ahues, F.D. d’Almeida, and R. Fernandes
A proof of this estimate can be found in [AAF09], and a proof of a similarbound can be found in [AAL09].
Theorem 3 with x = f gives a bound on one part of (1.6):
‖(I − πn)f‖ ≤ 2n∑
j=1
ω1(f∣∣[τj−1,τj ], hj). (1.7)
The following theorem establishes a bound on the other part of (1.6).
Theorem 4. If g satisfies (1.2) then
‖(I − πn)T‖ ≤ 2hmax(g(hmin/2) + g(hmin) − 2g(b− a))
+ 4∫ hmax/2
0g(σ)dσ + 4
∫ hmax
0g(σ)dσ + 4
∫ 3hmax/2
0g(σ)dσ (1.8)
in the subordinated operator norm.
Proof. If we write the bound of Theorem 3 with the definition of ω1 andperform the change of variable τ = αhj , dτ = hjdα, we get
‖(I − πn)x‖ ≤ 2n∑
j=1
∫ 1
0
∫ τj−αhj
τj−1
|x(s + αhj) − x(s)| ds dα.
Replacing x with Tx, for all x ∈ L1([a, b],C), and changing the order ofthe integrals, we can easily prove that
‖(I − πn)Tx‖ ≤ 2∫ b
a
n∑j=1
∫ 1
0
∫ τj−αhj−t
τj−1−t
|g(|τ + αhj |) − g(|τ |)|dτdα |x(t)| dt
≤ 2 ‖x‖ supt∈[a,b]
∫ 1
0
n∑j=1
∫ τj−αhj−t
τj−1−t
|g(|τ + αhj |) − g(|τ |)|dτdα.
Let
Aj(t) :=∫ 1
0
∫ tj−αhj
tj−1
|g(|τ + αhj |) − g(|τ |)|dτdα
and tj := τj − t, t ∈ [a, b]. We estimate an upper bound of supt∈[a,b]
n∑j=1
Aj(t).
This proof is based on the geometry of the underlying discretization gridand it includes the dependence on the possible subdomains [a, b] of the interval[0, τ�].
Any t in [a, b] belongs to a certain subinterval of the grid, say [τk−1, τk] andit may be located in the second half of it—case (A), or in the first—case (B).
(A) In this case tk−1 k
j:≤−h /2 (see Figure 1.1) and we have four subcases for
1 Galerkin Approximations for Weakly Singular Operators 5
Fig. 1.1. Location of t, tk,−tk−1 and tk−1.
(A1) For all j such that τj ≤ τk−1, that is, j ≤ k − 1 (see Figure 1.1),tj , tj−1, tj − αhj , and τ are negative. So −tj < −tj + αhj < −tj−1.As τ is negative, |τ + αhj | < |τ | and since g is a decreasing function,
|g(|τ + αhj |)− g(|τ |)| = g(|τ + αhj |)− g(|τ |) = g(−τ −αhj)− g(−τ).
Replacing in the first term g(−τ − αhj) with a larger value, g(−tj +αhj − αhj), and in the second term g(−τ) with a smaller value,g(−tj−1), we have a larger value for the integral:
Aj(t) ≤∫ 1
0
∫ tj−αhj
tj−1
(g(−tj) − g(−tj−1))dτdα,
and enlarging the interval for τ to [tj−1, tj ], we have
Aj(t) ≤ hj(g(−tj) − g(−tj−1)) ≤ hmax(g(−tj) − g(−tj−1)).
Hence,
k−1∑j=1
Aj(t) ≤ hmax(g(−tk−1) − g(−t0)) ≤ hmax(g(hk/2) − g(b− a)),
because tk−1 ≤ −hk/2 implies that −tk−1 ≥ hk/2 and g is a decreasingfunction.
(A2) For all j such that τj−1 > τk, i.e., j > k + 1, tj and tj−1 are positive.Also, tj − αhj ≥ τj−1 is positive, τ is positive, and so is τ + αhj . Asg is a decreasing function,
|g(|τ + αhj |) − g(|τ |)| = g(|τ |) − g(|τ + αhj |).
Using the same arguments as in the previous case, we get
n∑j=k+2
Aj(t) ≤ hmax(g(tk+1) − g(tn)) ≤ hmax(g(hmin) − g(b− a)),
because tk ≥ 0 implies tk+1 ≥ hk+1 ≥ hmin, and g is a decreasingfunction.
6 M. Ahues, F.D. d’Almeida, and R. Fernandes
(A3) For the interval [τk−1, τk], τk−1 + hk/2 ≤ t ≤ τk and we consider that
|g(|τ + αhk|) − g(|τ |)| ≤ |g(|τ + αhk|)| + |g(|τ |)|.
We decompose, accordingly, Ak into two integrals:
Ak(t) ≤∫ 1
0
∫ tk−αhk
tk−1
g(|τ + αhk|)dτdα +∫ 1
0
∫ tk−αhk
tk−1
g(|τ |)dτdα.
With the change of variable σ = τ + αhk in the first integral, andenlarging all the intervals of τ to [tk−1, tk], we get
Ak(t) ≤∫ 1
0
∫ tk
tk−1+αhk
g(|σ|)dσdα +∫ 1
0
∫ tk
tk−1
g(|τ |)dτdα
≤ 2∫ 1
0
∫ tk
tk−1
g(|σ|)dσdα ≤ 2∫ 1
0
∫ hk/2
−hk
g(|σ|)dσdα,
since 0 ≤ tk ≤ hk/2 implies that −hk ≤ tk−hk = tk−1 and tk ≤ hk/2.Hence,
Ak(t) ≤ 2(∫ hk/2
0g(σ)dσ +
∫ hk
0g(σ)dσ
).
(A4) For the interval [τk, τk+1], τk+hk/2 ≤ t ≤ τk+1. We consider a similardecomposition of Ak+1 into two integrals:
Ak+1(t) ≤∫ 1
0
∫ tk+1−αhk+1
tk
g(|τ + αhk+1|)dτdα +∫ 1
0
∫ tk+1−αhk+1
tk
g(|τ |)dτdα.
With the change of variable σ = τ + αhk+1 in the first integral, andenlarging the intervals of τ to [tk, tk+1], we have
Ak+1(t) ≤∫ 1
0
∫ tk+1
tk+αhk+1
g(|σ|)dσdα +∫ 1
0
∫ tk+1−αhk+1
tk
g(|τ |)dτdα
≤ 2∫ 1
0
∫ tk+1
tk
g(|σ|)dσdα ≤ 2∫ 1
0
∫ hk+1+hk/2
0g(σ)dσdα,
since 0 ≤ tk and tk+1 ≤ hk/2 + hk+1; hence,
Ak+1(t) ≤ 2∫ 3hmax/2
0g(σ)dσ.
(B) The case τk−1+ hk/2 ≥ t ≥ τk−1, that is tk−1 ≥ −hk/2, gives the samepartial bounds.
So for all cases of t we have
1 Galerkin Approximations for Weakly Singular Operators 7
n∑j=1
Aj(t) ≤ hmax(g(hmin/2) + g(hmin) − 2g(b− a))
+ 2∫ hmax/2
0g(σ)dσ + 2
∫ hmax
0g(σ)dσ + 2
∫ 3hmax/2
0g(σ)dσ, (1.9)
and the bound (1.8) follows by considering the supremum of (1.9) when t ∈[a, b] and by multiplying it by 2.
1.3 Computational Experiments
We consider the function g(s) = − ln(s/2), s ∈]0, 2], z = 4 and the followingright-hand side function:
f(s) :={
−1 if 0 ≤ s ≤ 1,0 if 1 < s ≤ 2.
In this example we will compute the Galerkin approximate solution withuniform grids of 501 and 1001 points, respectively.
As we do not know the exact solution, we will take as reference solutionthe one obtained with a uniform grid of 4001 nodes, in Figure 1.2, and useit in the computation of the absolute errors of solutions corresponding tothe two, much coarser, grids built with n = 500 and n = 1000 subintervals,respectively.
Figure 1.2, the reference solution, and Figure 1.3, the approximate coarserone, look very similar, and so the error with respect to this reference solutionis plotted in Figure 1.4 for a uniform grid with 501 nodes. In Figure 1.5 weplot the error corresponding to an approximation with a uniform grid with1001 nodes.
As we can see, the error reduces by a factor of approximately 2, when wedouble the number of subintervals in the grid. We can also see that the erroris larger where the kernel has a logarithmic discontinuity (near 0) and wherethe right-hand side function f has a discontinuity (near 1).
Elementary computations and [ALL01], [AALT05], and [ALT01] showthat, in Theorem 2,
0 < C ≤ 12 − 2 ln 2
< 1.63 and (I − πn)f = 0,
and so the error bound in Theorem 4 can be computed explicitly as given inTable 1.1. This table also contains the values of the L1-norm of the relativeerror (using the reference solution) and, as expected, it shows that the boundis somewhat pessimistic, in this example. It also shows that doubling thenumber of subintervals, the error bound reduces correspondingly.
8 M. Ahues, F.D. d’Almeida, and R. Fernandes
0 0.5 1 1.5 20.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 1.2. Reference solution with uniform grid of 4001 nodes.
0 0.5 1 1.5 20.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 1.3. Approximate solution with uniform grid of 501 nodes.
1.4 Bibliographical Comments and Conclusions
The Galerkin approximation to a compact integral operator is the cheapestone among projection discretizations (see [At97], [ALL01], and [Ch83]). TheL1 class of functions is the largest space among the Lebesgue ones. Weaklysingular kernels define the most general integral operators among the com-
1 Galerkin Approximations for Weakly Singular Operators 9
0 0.5 1 1.5 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2x 10
−3
Fig. 1.4. Error of a 501-node-uniform-grid solution with respect to the referencesolution.
Table 1.1. L1-norm relative errors in uniform grids.
n hmax Error Bound L1-norm Relative Error500 0.004 0.729 0.0007361000 0.002 0.401 0.000251
pact ones. Hence, the framework of this paper is as general and as weak aspossible in the domain of numerical resolution of Fredholm integral equationsof the second kind. The main theoretical result is Theorem 4, in which arelative error bound is produced. Other efforts in this sense have been accom-plished in [AALT05] and [ALT01], where the condition of quasi-uniformity isimposed to the underlying grid, and in [AAL09] where other Banach spacesand other projection-type discretizations are considered. The investigation ofthe existence of possible boundary layers in the solution thus deserving gridrefinements has been studied in [AAL09] and [AAF09]. The numerical exper-iments presented in this paper are done with uniform grids and show thatfor a kernel with a logarithmic singularity and an equation whose right-handside is a piecewise constant discontinuous function the Galerkin discretizationstudied in this chapter gives significantly better approximations than the onesexpected by theory. The shape of the relative error function shows that thepredicted boundary layers have occurred in practice and that the numericalsolution is less accurate in those subdomains.
10 M. Ahues, F.D. d’Almeida, and R. Fernandes
0 0.5 1 1.5 2−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1x 10
−3
Fig. 1.5. Error of a 1001-node-uniform-grid solution with respect to the referencesolution.
Acknowledgement. This research was partially supported by FCT (Fundacao paraa Ciencia e a Tecnologia) through the doctoral scholarship SFRH/BD/30826/2006.
References
[At97] Atkinson, K.: The Numerical Solution of Integral Equations of the Sec-ond Kind, Cambridge University Press, London (1997).
[AAL09] Amosov, A., Ahues, M., Largillier, A.: Superconvergence of some projec-tion approximations for weakly singular integral equations using generalgrids, SIAM J. Numer. Anal., 47, 646–674 (2009).
[AALT05] Ahues, M., Amosov, A., Largillier, A., Titaud, O.: Lp error estimatesfor projection approximations. Appl. Math. Lett., 18, 381–386 (2005).
[AAF09] Ahues, M., D’Almeida, F., Fernandes, R.: Piecewise constant Galerkinapproximations of weakly singular integral equations, accepted for pub-lication in Internat. J. Pure Appl. Math. (to appear).
[ALL01] Ahues, M., Largillier, A., Limaye, B.V.: Spectral Computations withBounded Operators, Chapman & Hall/CRC, Boca Raton, FL (2001).
[ALT01] Ahues, M., Largillier, A., Titaud, O.: The roles of a weak singularity andthe grid uniformity in relative error bounds. Numer. Functional Anal.Optimization, 22, 789–814 (2001).
[Ch83] Chatelin, F.: Spectral Approximations of Linear Operators, AcademicPress, New York (1983).
2
Construction of Solutions of theHamburger–Lowner Mixed InterpolationProblem for Nevanlinna Class Functions
J.A. Alcober, I.M. Tkachenko, and M. Urrea
Universidad Politecnica de Valencia, Spain; [email protected],[email protected], [email protected]
2.1 Formulation of the Problem
By definition, a Nevanlinna class function ϕ ∈ R is holomorphic and has anonnegative imaginary part in the half-plane Im z > 0. In this chapter wealso consider Nevanlinna functions which belong to the subclass R0 ⊂ R
such that if ϕ (z) ∈ R0, limz→∞ (ϕ (z) /z) = 0, Im z > 0. Then, due to theRiesz–Herglotz theorem,
ϕ(z) =
∞∫−∞
dσ(t)t− z
, Im z > 0, (2.1)
where σ(t) is a nondecreasing function such that∫∞
−∞(1 + t2
)−1dσ(t) < ∞.
Consider the mixed Lowner–Nevanlinna problem [Low34, KrNu77, Akh65,KaSt66, CuFi91, CuFi96, AdTk00, UrTkFC01, AdAlTk03], see also[AdTk01(a)] and (for the matrix version of the problem) [AdTk01(b)].
Problem 1. Given a set of real numbers (c0, ..., c2n), a finite set of points(t1, ..., tp) on the real axis, and a set of complex numbers (w1, ..., wp) withnonnegative imaginary parts, find a function of the Nevanlinna class ϕ ∈ R0such that asymptotically, for z → ∞ inside any angle δ < arg z < π−δ, δ > 0,
ϕ(z) = −2n+1∑r=1
cr−1z−r + o
(|z|−2n−1
)(2.2)
possesses continuous boundary values in some vicinities of the points (t1, ..., tp)and
ϕ(ts + i0) = ws, s = 1, ..., p. (2.3)
Remark 1. Notice that by virtue of the representation (2.1) [Akh65], condi-tion (2.2) is equivalent to the moment conditions
© Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2010
11, DOI 10.1007/978-0-8176-4897-8_2, Volume 2: Computational Methods
C. Constanda and M.E. Pérez (eds.), Integral Methods in Science and Engineering,
12 J.A. Alcober, I.M. Tkachenko, and M. Urrea∫ ∞
−∞tkdσ(t) = ck, k = 0, 1, ..., 2n, (2.4)
for the generating distribution σ(t).
Remark 2. The suggested Problem 1 is a mixture of the truncated Hamburgermoment problem [KrNu77, CuFi91, AdTk00] with the Lowner-type interpo-lation problem in the class of Nevanlinna functions [Low34].
We describe and test numerically an algorithm for finding irrational so-lutions of this problem. The rational solutions of a similar problem werediscussed in [AdAlTk03]. Errors of such approximations depending on thenumber and distribution of the interpolation nodes on the real axis will bediscussed elsewhere.
These kind of problems occur when a distribution density reconstructionfrom scarce experimental data is attempted. In other words, we are interestedin the possibility of solving the problem when only a very small number ofmoments and constraints (data at the interpolation nodes) is known.
The studies of convergence as the number of moments and/or interpolationnodes grows are out of the scope of this work. Untruncated moment problemsare solved in the classical theory of moments, see [KrNu77] and [Akh65]. Thebehavior of the problem solution when the number of interpolation nodesgrows is treated in [DeDy81].
2.2 The Mixed Problem Solution
2.2.1 Solvability and Contractive Functions
Recall that the truncated Hamburger moments problem is solvable [KrNu77,CuFi96, AdTk00] if and only if the block-Hankel matrix (ck+l)nk,l=0 is non-negative. If, in addition, we exclude from our consideration the nonnegativeblock-Hankel matrices like (
0 00 γ
), γ > 0,
which cannot be generated by power moments of nonnegative measures, and ifthe set (c0, ..., c2n) is positive definite, there exists an infinite set of nonnegativemeasures σ on the real axis satisfying (2.4).
Let (Dk(t))nk=0 be the finite set of polynomials constructed according to
the formulas
2 The Hamburger–Lowner Problem for Nevanlinna Functions 13
D0 =1
√c0
, Dk(t) =1√
Δk−1Δk
det
⎛⎜⎜⎜⎝c0 · · · ck−1 1c1 · · · ck t...
......
...ck · · · c2k−1 tk
⎞⎟⎟⎟⎠ ,
Δ−1 = 1 , Δ0 = c0 , Δk =det
⎛⎜⎝c0 · · · ck...
......
ck · · · c2k
⎞⎟⎠ , k = 1, 2, ..., n.(2.5)
Polynomials Dk form an orthogonal system with respect to each σ-measuresatisfying (2.4). Let
E0 ≡ 0, Ek(t) =∫ ∞
−∞
Dk(t) −Dk(s)t− s
dσ(s), k = 1, ..., n,
be the corresponding set of conjugate polynomials.Then the formula
ϕ(z) =∫ ∞
−∞
dσ(t)t− z
= −En(z)(ζ(z) + z) − En−1(z)Dn(z)(ζ(z) + z) −Dn−1(z)
, Im z > 0, n = 1, 2, . . .
(2.6)according to the Nevanlinna theorem [KrNu77, UrTkFC01], establishes a one-to-one correspondence between the set of all Nevanlinna functions ϕ(z) satis-fying (2.2) and the elements ζ(z) of the subclass R0.
Notice that the zeros of each orthogonal polynomial Dk(z) are real and byvirtue of the Schwarz–Christoffel identity [KrNu77]
Dn−1(z)En(z) −Dn(z)En−1(z) ≡ Ξn =Δn−1√Δn−2Δn
> 0 , n = 1, 2, . . . ,
(2.7)the zeros of Dn−1(z) alternate with the zeros of Dn(z) as well as with thezeros of En−1(z). Therefore, any function ϕ(z) given by the expression onthe right-hand side of (2.6) has a continuous boundary value on the real axisif and only if the corresponding Nevanlinna function ζ ∈ R0 is continuousin the closed upper half-plane and such that ζ(z) + z has no joint zeros withDn−1(z).
To meet the constraints (2.3) it suffices to substitute into the right-handside of (2.6) any function ζ(z) ∈ R0 which is continuous in the closed upperhalf-plane and satisfies the following conditions:
ξs = ζ (ts) = −ts +wsDn−1(ts) + En−1(ts)
wsDn(ts) + En(ts), s = 1, ..., p. (2.8)
Note that by (2.7), Im ξs = ΞnIm ws |wsDn(ts) + En(ts)|−2> 0, s = 1, ..., p.
Thus, Problem 1 reduces to the following.
