Approximation Theory and Numerical Analysis

192 Pages·2014·13.62 MB·English

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
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StanislavAntontsev,Portugal MohammadT.Darvishi,Iran ChaitanP.Gupta,USA
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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
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DanieleBertaccini,Italy LucaEsposito,Italy JongSooJung,RepublicofKorea
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ElenaBraverman,Canada AlbertoFiorenza,Italy AndreiKorobeinikov,Spain
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Sun-SigByun,Korea XianlongFu,China PavelKurasov,Sweden
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AnnaCapietto,Italy IsaacGarcia,Spain RuedigerLandes,USA
JianqingChen,China Jesu´sGarc´ıaFalset,Spain IrenaLasiecka,USA
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ShiwangMa,China SimeonReich,Israel YushunWang,China
WanbiaoMa,China AbdelazizRhandi,Italy ShawnX.Wang,Canada
EberhardMalkowsky,Turkey HassanRiahi,Malaysia YouyuWang,China
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JoseM.Martell,Spain WolfgangRuess,Germany PatriciaJ.Y.Wong,Singapore
MieczysławMastyło,Poland BernhardRuf,Italy ZiliWu,China
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TarasMel’nyk,Ukraine SatitSaejung,Thailand ShanheWu,China
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ChangxingMiao,China MartinSchechter,USA XuXian,China
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SylviaNovo,Spain PierpaoloSoravia,Italy NorioYoshida,Japan
MicahOsilike,Nigeria MarcoSquassina,Italy BeongInYun,Korea
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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
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KevinR.Payne,Italy YoushanTao,China ZengqinZhao,China
AdemirFernandoPazoto,Brazil GabriellaTarantello,Italy SiningZheng,China
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SergeiV.Pereverzyev,Austria GerdTeschke,Germany QijiJ.Zhu,USA
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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