Approximation Theory and Numerical Analysis PDF
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Abstract and Applied Analysis Approximation Theory and Numerical Analysis Guest Editors: Sofiya Ostrovska, Elena Berdysheva, Grzegorz Nowak, and Ahmet Yas¸ ar Özban Approximation Theory and Numerical Analysis Abstract and Applied Analysis Approximation Theory and Numerical Analysis Guest Editors: Sofiya Ostrovska, Elena Berdysheva, ¨ Grzegorz Nowak, and Ahmet Ya¸sar Ozban Copyright©2014HindawiPublishingCorporation.Allrightsreserved. Thisisaspecialissuepublishedin“AbstractandAppliedAnalysis.”AllarticlesareopenaccessarticlesdistributedundertheCreative CommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginal workisproperlycited. Editorial Board RaviP.Agarwal,USA JaeyoungChung,Korea LorenzoGiacomelli,Italy BashirAhmad,SaudiArabia SilviaCingolani,Italy JaumeGine´,Spain M.O.Ahmedou,Germany JeanM.Combes,France ValeryY.Glizer,Israel NicholasD.Alikakos,Greece MonicaConti,Italy LaurentGosse,Italy DeboraAmadori,Italy DiegoCo´rdoba,Spain JeanP.Gossez,Belgium PabloAmster,Argentina JuanC.Corte´s,Spain JoseL.Gracia,Spain DouglasR.Anderson,USA GrazianoCrasta,Italy MaurizioGrasselli,Italy JanAndres,CzechRepublic BernardDacorogna,Switzerland QianGuo,China GiovanniAnello,Italy VladimirDanilov,Russia YuxiaGuo,China StanislavAntontsev,Portugal MohammadT.Darvishi,Iran ChaitanP.Gupta,USA MohamedKamalAouf,Egypt LuisF.PinheirodeCastro,Portugal UnoHa¨marik,Estonia NarcisaC.Apreutesei,Romania T.Diagana,USA FerencHartung,Hungary NatigM.Atakishiyev,Mexico Jesu´sI.D´ıaz,Spain BehnamHashemi,Iran FerhanM.Atici,USA JosefDibl´ık,CzechRepublic NorimichiHirano,Japan IvanG.Avramidi,USA FasmaDiele,Italy JiaxinHu,China SoohyunBae,Korea TomasDominguez,Spain ChengmingHuang,China ZhanbingBai,China AlexanderI.Domoshnitsky,Israel ZhongyiHuang,China ChuanzhiBai,China MarcoDonatelli,Italy GennaroInfante,Italy DumitruBaleanu,Turkey Bo-QingDong,China IvanIvanov,Bulgaria Jo´zefBana´s,Poland OndˇrejDoˇsly´,CzechRepublic HosseinJafari,Iran GerassimosBarbatis,Greece Wei-ShihDu,Taiwan JaanJanno,Estonia MartinoBardi,Italy LuizDuarte,Brazil ArefJeribi,Tunisia RobertoBarrio,Spain RomanDwilewicz,USA UnC.Ji,Korea FeyziBa¸sar,Turkey PaulW.Eloe,USA ZhongxiaoJia,China AbdelghaniBellouquid,Morocco AhmedEl-Sayed,Egypt L.Jdar,Spain DanieleBertaccini,Italy LucaEsposito,Italy JongSooJung,RepublicofKorea MichielBertsch,Italy JoseA.Ezquerro,Spain HenrikKalisch,Norway LucioBoccardo,Italy KhalilEzzinbi,Morocco HamidRezaKarimi,Norway IgorBoglaev,NewZealand DashanFan,USA SatyanadKichenassamy,France MartinJ.Bohner,USA AngeloFavini,Italy TeroKilpelinen,Finland JulianF.Bonder,Argentina MarciaFederson,Brazil SungGuenKim,RepublicofKorea GeraldoBotelho,Brazil StathisFilippas,EquatorialGuinea LjubisaKocinac,Serbia ElenaBraverman,Canada AlbertoFiorenza,Italy AndreiKorobeinikov,Spain RomeoBrunetti,Italy ToreFla˚tten,Norway PekkaKoskela,Finland JanuszBrzdek,Poland IlariaFragala,Italy VictorKovtunenko,Austria DetlevBuchholz,Germany BrunoFranchi,Italy Ren-JiehKuo,Taiwan Sun-SigByun,Korea XianlongFu,China PavelKurasov,Sweden FabioM.Camilli,Italy MassimoFuri,Italy MiroslawLachowicz,Poland JindeCao,China GiovanniP.Galdi,USA KunquanLan,Canada AnnaCapietto,Italy IsaacGarcia,Spain RuedigerLandes,USA JianqingChen,China Jesu´sGarc´ıaFalset,Spain IrenaLasiecka,USA Wing-SumCheung,HongKong Jose´A.Garc´ıa-Rodr´ıguez,Spain MattiLassas,Finland MichelChipot,Switzerland LeszekGasinski,Poland Chun-KongLaw,Taiwan ChangbumChun,Korea Gyo¨rgyGa´t,Hungary Ming-YiLee,Taiwan SoonY.Chung,Korea VladimirGeorgiev,Italy GongbaoLi,China ElenaLitsyn,Israel SomyotPlubtieng,Thailand JuanJ.Trujillo,Spain YanshengLiu,China MilanPokorny,CzechRepublic GabrielTurinici,France ShengqiangLiu,China SergioPolidoro,Italy MilanTvrdy,CzechRepublic CarlosLizama,Chile ZiemowitPopowicz,Poland MehmetUnal,Turkey MiltonC.LopesFilho,Brazil MariaM.Porzio,Italy CsabaVarga,Romania JulianLo´pez-Go´mez,Spain EnricoPriola,Italy CarlosVazquez,Spain GuozhenLu,USA VladimirS.Rabinovich,Mexico GianmariaVerzini,Italy JinhuLu¨,China IrenaRachu˙nkova´,CzechRepublic JesusVigo-Aguiar,Spain GrzegorzLukaszewicz,Poland MariaAlessandraRagusa,Italy Qing-WenWang,China ShiwangMa,China SimeonReich,Israel YushunWang,China WanbiaoMa,China AbdelazizRhandi,Italy ShawnX.Wang,Canada EberhardMalkowsky,Turkey HassanRiahi,Malaysia YouyuWang,China SalvatoreA.Marano,Italy JuanP.Rinco´n-Zapatero,Spain JingPingWang,UK CristinaMarcelli,Italy LuigiRodino,Italy PeixuanWeng,China PaoloMarcellini,Italy YuriyV.Rogovchenko,Norway NoemiWolanski,Argentina Jesu´sMar´ın-Solano,Spain JulioD.Rossi,Argentina Ngai-ChingWong,Taiwan JoseM.Martell,Spain WolfgangRuess,Germany PatriciaJ.Y.Wong,Singapore MieczysławMastyło,Poland BernhardRuf,Italy ZiliWu,China MingMei,Canada MarcoSabatini,Italy YongHongWu,Australia TarasMel’nyk,Ukraine SatitSaejung,Thailand ShanheWu,China AnnaMercaldo,Italy StefanG.Samko,Portugal Tie-chengXia,China ChangxingMiao,China MartinSchechter,USA XuXian,China StanislawMigorski,Poland JavierSegura,Spain YanniXiao,China MihaiMihaˇilescu,Romania SigmundSelberg,Norway FudingXie,China FelizMinho´s,Portugal ValerySerov,Finland GonangXie,China DumitruMotreanu,France NaseerShahzad,SaudiArabia NaihuaXiu,China RobertaMusina,Italy AndreyShishkov,Ukraine DaoyiXu,China G.M.N’Gue´re´kata,USA StefanSiegmund,Germany ZhenyaYan,China MariaGraziaNaso,Italy Abdel-MaksoudA.Soliman,Egypt XiaodongYan,USA SylviaNovo,Spain PierpaoloSoravia,Italy NorioYoshida,Japan MicahOsilike,Nigeria MarcoSquassina,Italy BeongInYun,Korea MitsuharuOˆtani,Japan SvatoslavStaneˇk,CzechRepublic VjacheslavYurko,Russia TurgutO¨zi¸s,Turkey StevoStevic,Serbia AgacikZafer,Turkey FilomenaPacella,Italy AntonioSua´rez,Spain SergeyV.Zelik,UK NikolaosS.Papageorgiou,Greece WenchangSun,China JianmingZhan,China SehiePark,Korea RobertSzalai,UK MeirongZhang,China AlbertoParmeggiani,Italy SanyiTang,China WeinianZhang,China KailashC.Patidar,SouthAfrica Chun-LeiTang,China ChengjianZhang,China KevinR.Payne,Italy YoushanTao,China ZengqinZhao,China AdemirFernandoPazoto,Brazil GabriellaTarantello,Italy SiningZheng,China JosipE.Pecˇaric´,Croatia Nasser-eddineTatar,SaudiArabia TianshouZhou,China ShuangjiePeng,China SusannaTerracini,Italy YongZhou,China SergeiV.Pereverzyev,Austria GerdTeschke,Germany QijiJ.Zhu,USA MariaEugeniaPerez,Spain AlbertoTesei,Italy Chun-GangZhu,China JosefinaPerles,Spain BevanThompson,Australia MalisaR.Zizovic,Serbia AllanPeterson,USA SergeyTikhonov,Spain WenmingZou,China AndrewPickering,Spain ClaudiaTimofte,Romania CristinaPignotti,Italy ThanhTran,Australia Contents ApproximationTheoryandNumericalAnalysis,SofiyaOstrovska,ElenaBerdysheva,GrzegorzNowak, andAhmetYa¸sarO¨zban Volume2014,ArticleID390983,2pages AnEfficientApproachtoNumericalStudyoftheMRLWEquationwithB-SplineCollocationMethod, SeydiBattalGaziKarakoc¸,TurgutAk,andHalilZeybek Volume2014,ArticleID596406,15pages OnExtremalRanksandLeastSquaresSolutionsSubjecttoaRankRestriction,HongxingWangand YeguoSun Volume2014,ArticleID457298,7pages AnIterativeSchemeforSolvingSystemsofNonlinearFredholmIntegrodifferentialEquations, M.I.Berenguer,D.Ga´mez,andA.J.Lo´pezLinares Volume2014,ArticleID401541,9pages TheEHInterpolationSplineandItsApproximation,JinXieandXiaoyanLiu Volume2014,ArticleID745765,6pages ANewApproachtoGeneralInterpolationFormulaeforBivariateInterpolation,LeZouandShuoTang Volume2014,ArticleID421635,11pages NumericalSolutionforEllipticInterfaceProblemsUsingSpectralElementCollocationMethod, PeymanHessari,SangDongKim,andByeong-ChunShin Volume2014,ArticleID780769,11pages SuperclosenessResultofHigherOrderFEM/LDGCoupledMethodforSolvingSingularlyPerturbed ProblemonS-TypeMesh,ShenglanXie,HuonianTu,andPengZhu Volume2014,ArticleID260840,11pages OptimalBackwardPerturbationAnalysisfortheBlockSkewCirculantLinearSystemswithSkew CirculantBlocks,ZhaolinJiang,JuanLi,andJianweiZhou Volume2014,ArticleID523102,8pages GeneralizedPreconditionedMHSSMethodforaClassofComplexSymmetricLinearSystems, Cui-XiaLi,Yan-JunLiang,andShi-LiangWu Volume2014,ArticleID206821,9pages OnNewGeneralizedOstrowskiTypeIntegralInequalities,A.Qayyum,M.Shoaib,A.E.Matouk, andM.A.Latif Volume2014,ArticleID275806,8pages TheConstantsinAPosterioriErrorIndicatorforState-ConstrainedOptimalControlProblemswith SpectralMethods,JianweiZhou Volume2014,ArticleID946026,8pages Two-LevelBrezzi-Pitka¨rantaStabilizedFiniteElementMethodsfortheIncompressibleFlows, RongAnandXianWang Volume2014,ArticleID698354,14pages PolynomialandRationalApproximationsandtheLinkbetweenSchro¨der’sProcessesoftheFirstand SecondKind,Franc¸oisDubeau Volume2014,ArticleID719846,5pages Shape-PreservingandConvergencePropertiesfor the𝑞-Sza´sz-MirakjanOperatorsforFixed𝑞∈(0,1),HepingWang,FaguiPu,andKaiWang Volume2014,ArticleID563613,8pages TheDerivationofStructuralPropertiesofLM-gSplinesbyNecessaryConditionofOptimalControl, XiongweiLiu,XinjianZhang,andLizhiCheng Volume2014,ArticleID641435,5pages OnaClassof𝑞-Bernoulli,𝑞-Euler,and𝑞-GenocchiPolynomials,N.I.MahmudovandM.Momenzadeh Volume2014,ArticleID696454,10pages AMatrixIterationforFindingDrazinInversewithNinth-OrderConvergence,A.S.Al-Fhaid,S.Shateyi, M.ZakaUllah,andF.Soleymani Volume2014,ArticleID137486,7pages TheRateofConvergenceofLupas𝑞-AnalogueoftheBernsteinOperators,HepingWangand YanboZhang Volume2014,ArticleID521709,6pages ApproximationbyGenuine𝑞-Bernstein-DurrmeyerPolynomialsinCompactDisksintheCase𝑞>1, NazimI.Mahmudov Volume2014,ArticleID959586,11pages Approximationby𝑞-BernsteinPolynomialsintheCase𝑞 → 1+,XuezhiWu Volume2014,ArticleID259491,6pages TheTruncated𝑞-BernsteinPolynomialsintheCase𝑞>1,MehmetTuran Volume2014,ArticleID126319,7pages Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 390983, 2 pages http://dx.doi.org/10.1155/2014/390983 Editorial Approximation Theory and Numerical Analysis SofiyaOstrovska,1ElenaBerdysheva,2GrzegorzNowak,3andAhmetYaGarÖzban1 1DepartmentofMathematics,AtilimUniversity,Incek,06836Ankara,Turkey 2DepartmentofMathematicsandScience,GermanUniversityofTechnologyinOman,P.O.Box1816,130Muscat,Oman 3TheGreatPolandUniversityofSocialandEconomicsinSrodaWielkopolska,63000SrodaWielkopolska,Poland CorrespondenceshouldbeaddressedtoSofiyaOstrovska;[email protected] Received24September2014;Accepted24September2014;Published22December2014 Copyright©2014SofiyaOstrovskaetal. This is an open access article distributed under the Creative Commons Attribution License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly cited. Approximation Theory and Numerical Analysis are closely bothinrealandcomplexcases,shape-preservingproperties, related areas of mathematics. ApproximationTheory lies in and discuss possible generalizations of the classical results. thecrossroadsofpureandappliedmathematics.Itincludes Moreover,thepaperbyX.Wupresentsacompletesolution a wide spectrum of areas ranging from abstract problems to the long-standing open problem on the approximation in real, complex, and functional analysis to direct applica- of all continuous functions on [0,1] by the 𝑞-Bernstein tionsinengineeringandindustry.Therefore,Approximation polynomialsinthecase𝑞 → 1+. Theoryemploysagreatvarietyofmethods,whichoriginate Furthermore, the issue includes a paper on inequalities in analysis, operator theory, harmonic analysis, quantum for real functions. To be specific, the subject of the paper calculus,algorithms,probabilitytheory,andfurtherareasof of A. Qayyum et al. is Ostrowski type inequalities, that is, mathematics. inequalities giving bounds for the deviation of a function ThisspecialissuewaslaunchedinNovember2013aiming from its integral mean. The authors obtain bounds for the atbringingoutnewdevelopmentsinthesesubjectswithmain deviationofafunctionfromacombinationofintegralmeans focus on the interaction between Approximation Theory over two subintervals covering the entire interval in terms and Numerical Analysis. All papers submitted to the issue of the 𝐿𝑝-norms of the second derivative of the function, have been refereed by experts in their respective fields 1≤𝑝≤∞. andexaminedbytheacademiceditors.Thoroughlyselected Theissuecontainsapaperonthebivariateinterpolation papersfallingintothescopeoftheissuehavebeenpublished, and some related topics. Particularly, the paper by L. Zou andhopefullytheywillbeofinterestforthereader. and S. Tang studies the interpolation theorem, algorithms, The issue contains a number of papers related to the and dual interpolation. It also provides many kinds of Bernstein type operators based on the 𝑞-integers. After 𝑞- interpolationschemes. analoguesoftheBernsteinpolynomialshadbeenintroduced A classical topic—solution of nonlinear equations—is by A. Lupa¸s in 1987 and G. Phillips in 1997, the study of alsorepresentedintheissue.F.Dubeau’spaperisdevotedto various𝑞-analoguesoftheclassicaloperatorsbecameanarea studyingSchro¨der’sprocesseswhicharefixedpointprocesses of intensive research. The present special issue supplies a for finding simple roots of nonlinear equations. The author selectionofpapersonthe𝑞-operatorsincludingthosewritten shows that Schro¨der’s processes of the first kind and of bywell-knownspecialistsintheareasuchasN.I.Mahmudov, the second kind are related by polynomial and rational H.Wang,andX.Wu.Inthoseworks,distinctmodificationsof approximations, thus giving an answer to a question raised the𝑞-Bernsteinpolynomialshavebeenstudied,forexample, by M. Petkovic´ to find and explain a possible link between 𝑞-Bernstein-Durrmeyer polynomials, 𝑞-Lupa¸s operator, 𝑞- thetwoprocesses. Sza´sz-Mirakjanoperator,andtruncated𝑞-Bernsteinpolyno- ThepapersoutlinedbelowregardtheareaofNumerical mials.Theauthorsdealwiththeconvergenceoftheoperators, Analysis.Thefirstdealswith𝐿𝑀−𝑔splines.𝐿𝑀−𝑔splines 2 AbstractandAppliedAnalysis had been introduced by R. J. P. de Figueiredo in 1977 as a system. Then, based on this decomposition, they perform generalizationofthe𝐿−𝑔splinesintermsoftwodifferential structuredbackwardperturbationanalysisfortheblockskew operators𝐿and𝑀,andtheyreduceto𝐿−𝑔splinesif𝑀isthe circulantlinearsystems. identityoperator.Intheirpaper,X.Liuetal.investigatethe M. I. Berenguer et al. introduce a new iterative method structuralpropertiesofthe𝐿𝑀−𝑔splinesbyoptimization forthenumericalsolutionofsystemsofnonlinearFredholm andoptimalcontroltheoryaswellastherelationshipbetween integrodifferentialequationsofthesecondkindbyemploying 𝐿𝑀−𝑔splinesand𝐿−𝑔splines. Banachfixed-pointtheoremandSchauderbasisandperform Besides,anewinterpolationsplinewithtwoparameters, theconvergenceanalysisoftheirmethods. calledanEHinterpolationspline,whichistheextensionof The paper by P. Hessari et al. aims to solve an elliptic thestandardcubicHermiteinterpolationspline,ispresented interface problem with a discontinuous coefficient and a in the paper by J. Xie and X. Liu. The authors demonstrate singularsourcetermbythespectralcollocationmethod.An propertiesandadvantagesofthissplineforaclassofinterpo- algorithmforthisproblemhasbeendesigned.Afterwards,its lationproblems. efficiencyhasbeendemonstrated. In the paper by R. An and X. Wang, new models are Anothertopiccoveredbytheissueistheapplicationsof considered which help to understand the behavior of nu- numericalanalysisinoptimalcontroltheory.Namely,inthe merical methods in presence of layers in more complex paper of J. Zhou, Legendre-Galerkin spectral methods are problemsliketheNavier-Stokesequationsinfluiddynamics employed to solve state-constrained optimal control prob- orconvectiondiffusionequationsinchemicalreactionpro- lems. Explicit formulae of the constants in the a posteriori cesses.Moreover,anewstabilizedfiniteelementmethodfor errorindicatorareinvestigated. incompressible flows based on Brezzi-Pitka¨ranta stabilized The paper authored by A. S. Al-Fhaid et al. aims to method is presented. The stability and error estimates of constructamatrixiterationforfindingapproximateinverses finite element solutions are derived for the classical one- of nonsingular square matrices and also apply the new levelmethod.AuthorsalsoproposeanewNewtoncorrection method for computing the Drazin inverse. It is proved that scheme based on the above two-level iteration methods. In themethodpossessestheconvergenceratenine.Numerical addition,somenumericalexperimentsaregiventosupport experimentsareperformedtosupportthefindings. thetheoreticalresultsandtochecktheefficiencyofthesetwo- Onthewhole,theissuecomprisespaperswhichcovera leveliterationmethods. varietyofdifferenttopicswithinthescopeofApproximation AfirstsuperclosenessanalysisforhigherorderFEM/LDG TheoryandNumericalAnalysis.Hopefully,theywillbeuse- coupledmethodforsolvingsingularlyperturbedconvection- fulandinterestingforthereaderworkinginthosesubjects. diffusion problem is presented by S. Xie et al. Based on piecewise polynomial approximations of degree 𝑘 (𝑘 ≥ 1), Acknowledgments a supercloseness property of order 𝑘 + 1/2 in DG norm is established on an S-type mesh. Numerical experiments Theguesteditorswouldliketoexpresstheirdeepestgratitude complementthetheoreticalresults. toallcontributorswhochosethisspecialissuetopublishfind- S.B.G.Karakoc¸etal.employaseptic𝐵-splinecollocation ingsoftheirresearches.Ourappreciationgoestoallreviewers methodtofindnumericalsolutionsofthemodifiedregular- whose precious comments and professional judgments are izedlongwave(MRLW)equationforthemotionofasingle of the paramount importance for both the authors and the solitarywave,theinteractionoftwoandthreesolitarywaves, editors. andthedevelopmentoftheMaxwellianinitialconditioninto solitarywaves.Moreover,theygivealinearstabilityanalysis SofiyaOstrovska oftheirmethods. ElenaBerdysheva The article by C.-X. Li et al. considers the numerical GrzegorzNowak solution of complex symmetric linear systems. They intro- AhmetYa¸sarO¨zban duceageneralizedpreconditionedmodifiedHermitianand skew Hermitian splitting (GPMHSS) method based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods. Numerical testresultsdiscussingtheefficiencyofGPMHSSandinexact GPMHSSareprovided. The paper by H. Wang and Y. Sun discusses a feasible interval of the parameter 𝑘 and a general expression of the matrix𝑋whichsatisfiestherankequation𝑟(𝐴−𝐵𝑋𝐶) = 𝑘. The authors study the problems to determine the maximal andminimalranksunderthisrankconstraint,aswellasto derivetheleastsquaressolutionsof‖𝐵𝑋𝐶−𝐴‖𝐹 =min. A backward perturbation analysis for the block skew circulantlinearsystemswithskewcirculantblockshasbeen carried out by Z. Jiang et al. First, they give a block style spectraldecompositionofthecoefficientmatrixofthelinear
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