Numerical Analysis on Time Scales

392 Pages·2022·5.932 MB·English

by  Svetlin G. Georgiev, İnci M. Erhan

SvetlinG.Georgiev,İnciM.Erhan
NumericalAnalysisonTimeScales
Also of Interest
IntegralInequalitiesonTimeScales
SvetlinG.Georgiev,2018
ISBN978-3-11-070550-8,e-ISBN(PDF)978-3-11-070555-3
FunctionalAnalysiswithApplications
SvetlinG.Georgiev,KhaledZennir,2019
ISBN978-3-11-065769-2,e-ISBN(PDF)978-3-11-065772-2
DataScience
TimeComplexity,InferentialUncertainty,andSpacekimeAnalytics
IvoD.Dinov,MilenVelchevVelev,2021
ISBN978-3-11-069780-3,e-ISBN(PDF)978-3-11-069782-7
NumericalAnalysis
AnIntroduction
TimoHeister,LeoG.Rebholz,FeiXue,2019
ISBN978-3-11-057330-5,e-ISBN(PDF)978-3-11-057332-9
RealAnalysis
MeasureandIntegration
MaratV.Markin,2019
ISBN978-3-11-060097-1,e-ISBN(PDF)978-3-11-060099-5
Svetlin G. Georgiev, İnci M. Erhan
Numerical
Analysis on Time
Scales
|
MathematicsSubjectClassification2010
Primary:34N05,39A10,41A05;Secondary:65D05,65L05
Authors
Prof.Dr.SvetlinG.Georgiev Prof.Dr.İnciM.Erhan
KlimentOhridskiUniversityofSofia AtılımUniversity
DepartmentofDifferentialEquations DepartmentofMathematics
FacultyofMathematicsandInformatics 06830Ankara
1126Sofia Turkey
Bulgaria [email protected]
[email protected]
ISBN978-3-11-078725-2
e-ISBN(PDF)978-3-11-078732-0
e-ISBN(EPUB)978-3-11-078734-4
LibraryofCongressControlNumber:2022936834
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2022WalterdeGruyterGmbH,Berlin/Boston
Coverimage:maxkabakov/iStock/GettyImagesPlus
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Preface
Numericalanalysisisanextremelyimportantfieldinmathematicsandothernatural
sciences.Almostallreallifeproblemsthataremodeledmathematicallydonothave
exactsolutions.Moreover,themathematicalmodelsoftenhaveanonlinearstructure
whichmakesthemevenmoredifficulttosolveanalytically.Inthissense,thedevel-
opmentandconstructionofefficientnumericalmethodsgainabigsignificance.Mo-
tivatedbythisfact,thestudiesrelatedtothedevelopmentofnewpowerfulnumerical
methodsorimprovementoftheexistingonesarestillcontinuing.
Inparticular,numericalsolutionsofdifferentialequationsareofgreatimportance
sincemanyprocessesinnaturearetimedependentandtheirmathematicalmodelsare
usuallydescribedbypartialorordinarydifferentialequationsandoften,bydifference
equations.Thetheoryoftimescalesanddynamicequations,ontheotherhand,uni-
fiesthecontinuousanddiscretemodels,thusprovidingamoregeneralviewtothe
subject.
Dynamic equations, which describe how quantities change across the time or
space, arise in any field of study where measurements can be taken. Most realistic
mathematicalmodelscannotbesolvedusingthetraditionalanalyticalmethodsfor
dynamicequationsontimescales.Theymustbehandledwithcomputationalmeth-
odsthatdeliverapproximatesolutions.
Untilrecently,therewereveryfewstudiesrelatedtonumericalmethodsontime
scales.Inthelastfewyears,someinitialresultsonthesubjecthavebeenpublished,
whichinitiatedthedevelopmentofnumericalanalysisontimescales.
Thisbookisdevotedtodesigning,analyzing,andapplyingcomputationaltech-
niquesfordynamicequationsontimescales.Thebookprovidesmaterialforatypical
firstcourse.Thisbookisanintroductiontonumericalmethodsforinitialvalueprob-
lemsfordynamicequationsontimescales.
Thebookcontains12chapters.InChapter1,theLagrange,σ-Lagrange,Hermite,
andσ-Hermitepolynomialinterpolationsareintroduced.Fromtheseinterpolations,
approximationsforthedeltaderivativeofcontinuouslydelta-differentiablefunctions
arededucted.InChapter2,formulaefornumericalintegrationontimescalesarede-
rivedandtheassociatedapproximationerrorsareestimated.InChapter3,linearin-
terpolatingsplines,linearinterpolatingσ-splines,cubicandHermitesplinesarein-
troduced. Chapter 4 is presented as a study of the Euler method. Chapters 5 and 6
considertheTaylorseriesmethodsoforder-2andorder-pandanalyzeconvergenceof
thesemethods.LinearmultistepmethodsareinvestigatedinChapter7.Chapter8con-
tainstheanalysisofRunge–Kuttamethods.Chapter9dealswiththeseriessolution
methodforfractionaldynamicequationsanddynamicequationsontimescales.The
AdomianpolynomialsmethodisinvestigatedinChapter10.Chapter11isdevotedto
weaksolutionsandvariationalmethodsforsomeclassesoflineardynamicequations
ontimescales.Nonlineardynamicequationsandvariationalmethodsareinvestigated
inChapter12.
https://doi.org/10.1515/9783110787320-201
VI | Preface
Wepresumethatthereadersarefamiliarwiththebasicnotionsontimescales
such as forward and backward jump operators, graininnes function, right and left
scattered,denseandisolatedpoints,aswellaswiththebasiccalculusconceptson
timescalessuchasthedeltadifferentiationandintegrationandtheirproperties,el-
ementaryfunctionsontimescales,Taylorformula.Forthereaderswhoarestudying
thetimescalesforthefirsttime,wesuggestlearningthesebasicnotionsandconcepts
fromthenumerousreferencesgiveninthisbookandelsewhere.
Thetextmaterialofthisbookispresentedinahighlyreadable,mathematically
solidformat.Manypracticalproblemsareillustrated,displayingawidevarietyofso-
lutiontechniques.Theauthorswelcomeanysuggestionsfortheimprovementofthe
text.
Paris/Ankara,July2022 SvetlinGeorgiev
İnciErhan
Contents
Preface|V
1 Polynomialinterpolation|1
1.1 Lagrangeinterpolation|1
1.2 σ-Lagrangeinterpolation|9
1.3 Hermiteinterpolation|21
1.4 σ-Hermiteinterpolation|31
1.5 Deltadifferentiation|39
1.6 Advancedpracticalproblems|41
2 Numericalintegration|44
2.1 Newton–Cotesformulae|44
2.2 σ-Newton–Cotesformulae|51
2.3 Errorestimates|60
2.4 σ-Errorestimates|61
2.5 Compositequadraturerules|62
2.6 σ-Compositequadraturerules|67
2.7 TheEuler–Maclaurenexpansion|75
2.8 Theσ-Euler–MaclaurenExpansion|76
2.9 ConstructionofGaussquadraturerules|79
2.10 ErrorestimationforGaussquadraturerules|83
2.11 σ-Gaussquadraturerules|83
2.12 Errorestimationforσ-Gaussquadraturerules|85
2.13 Advancedpracticalproblems|86
3 Piecewisepolynomialapproximation|89
3.1 Linearinterpolatingsplines|89
3.2 Linearinterpolatingσ-splines|93
3.3 Cubicsplines|100
3.4 Hermitecubicsplines|114
3.5 Advancedpracticalproblems|119
4 TheEulermethod|122
4.1 Analyzingthemethod|122
4.2 Localtruncationerror|123
4.3 Globaltruncationerror|126
4.4 Numericalexamples|128
4.5 Advancedpracticalproblems|146
5 Theorder2Taylorseriesmethod–TS(2)|149
VIII | Contents
5.1 Analyzingthemethod|149
5.2 ConvergenceoftheTS(2)method|151
5.3 Thetrapezoidrule|155
5.4 Numericalexamples|157
5.5 Advancedpracticalproblems|166
6 TheorderpTaylorseriesmethod–TS(p)|168
6.1 AnalyzingtheorderpTaylorseriesmethod|168
6.2 ConvergenceanderroranalysisoftheTS(p)method|170
6.3 The2-stepAdams–Bashforthmethod–AB(2)|176
6.4 Numericalexamples|177
6.5 Advancedpracticalproblems|186
7 Linearmultistepmethods–LMMs|188
7.1 Two-stepmethods|188
7.2 Consistencyoftwo-stepmethods|189
7.3 Constructionoftwo-stepmethods|191
7.4 k-Stepmethods|195
7.5 Consistencyofk-stepmethods|196
7.6 Numericalexamples|198
7.7 Advancedpracticalproblems|203
8 Runge–Kuttamethods–RKMs|204
8.1 One-stagemethods|204
8.2 Two-stagemethods|205
8.3 Three-stagemethods|207
8.4 s-Stagemethods|209
8.5 Numericalexamples|210
8.6 Advancedpracticalproblems|226
9 Theseriessolutionmethod–SSM|227
9.1 Preliminariesonseriesrepresentations|227
9.2 TheSSMfordynamicequations|230
9.3 TheSSMforCaputofractionaldynamicequation|232
9.4 Numericalexamples|236
9.5 Advancedpracticalproblems|248
10 TheAdomianpolynomialsmethod|249
10.1 Analyzingthemethod|249
10.2 First-ordernonlineardynamicequations|255
10.3 Numericalexamples|257
10.4 Advancedpracticalproblems|265
Contents | IX
11 Weaksolutionsandvariationalmethodsforsomeclassesoflinear
first-orderdynamicsystems|266
11.1 Variationalmethodsforfirst-orderlineardynamicsystems–I|266
11.2 Variationalmethodsforfirst-orderlineardynamicsystems–II|273
11.3 Advancedpracticalproblems|279
12 Variationalmethodsfornonlineardynamicequations|281
12.1 Existenceofsolutions|281
12.2 Necessaryconditionsfortheexistenceofsolutions|288
12.3 Advancedpracticalproblems|299
A Rolle’stheorem|303
B FréchetandGâteauxderivatives|309
B.1 Remainders|309
B.2 DefinitionanduniquenessoftheFréchetderivative|311
B.3 TheGâteauxderivative|317
C Pötzsche’schainrules|319
C.1 Measurechains|319
C.2 Pötzsche’schainrule|321
C.3 AgeneralizationofPötzsche’schainrule|323
D Lebesgueintegration.Lp-spaces.Sobolevspaces|329
D.1 TheLebesguedeltaintegral|329
D.2 Thespaces𝕃p(𝕋)|343
D.3 Sobolev-typespacesandgeneralizedderivatives|346
D.4 Weaksolutionsofdynamicalsystems|360
D.5 AGronwall-typeinequality|372
E Mazur’stheorem|377
Bibliography|379
Index|381