Fractional Calculus Operators and the Mittag-Leffler Function

260 Pages·2022·2.765 MB·English

by  Maja Andrić

Fractional Calculus
Operators and
the Mittag-Leffler
Function
Edited by
Maja Andrić
Printed Edition of the Special Issue Published in Fractal Fract
www.mdpi.com/journal/fractalfract
Fractional Calculus Operators and the
Mittag-Leffler Function

Fractional Calculus Operators and the
Mittag-Leffler Function
Editor
MajaAndric´
MDPI•Basel•Beijing•Wuhan•Barcelona•Belgrade•Manchester•Tokyo•Cluj•Tianjin
Editor
MajaAndric´
UniversityofSplit
Croatia
EditorialOffice
MDPI
St.Alban-Anlage66
4052Basel,Switzerland
This is a reprint of articles from the Special Issue published online in the open access journal
FractalandFractional(ISSN2504-3110) (availableat: https://www.mdpi.com/journal/fractalfract/
specialissues/FCOMLF).
Forcitationpurposes,citeeacharticleindependentlyasindicatedonthearticlepageonlineandas
indicatedbelow:
LastName,A.A.;LastName,B.B.;LastName,C.C.ArticleTitle. JournalNameYear,VolumeNumber,
PageRange.
ISBN978-3-0365-5367-2(Hbk)
ISBN978-3-0365-5368-9(PDF)
©2022bytheauthors. ArticlesinthisbookareOpenAccessanddistributedundertheCreative
Commons Attribution (CC BY) license, which allows users to download, copy and build upon
publishedarticles,aslongastheauthorandpublisherareproperlycredited,whichensuresmaximum
disseminationandawiderimpactofourpublications.
ThebookasawholeisdistributedbyMDPIunderthetermsandconditionsoftheCreativeCommons
licenseCCBY-NC-ND.
Contents
AbouttheEditor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
MajaAndric´
EditorialforSpecialIssue“FractionalCalculusOperatorsandtheMittag–LefflerFunction”
Reprintedfrom:FractalFract.2022,6,442,doi:10.3390/fractalfract6080442 . . . . . . . . . . . . . 1
HariM.Srivastava,Abedel-KarremN.Alomari,KhaledM.SaadandWaleedM.Hamanah
SomeDynamicalModelsInvolvingFractional-OrderDerivativeswiththeMittag-LefflerType
KernelsandTheirApplicationsBasedupontheLegendreSpectralCollocationMethod
Reprintedfrom:FractalFract.2021,5,131,doi:10.3390/fractalfract5030131 . . . . . . . . . . . . . 5
YuriKondratiev,Jose´daSilva
CesaroLimitsforFractionalDynamics
Reprintedfrom:FractalFract.2021,5,133,doi:10.3390/fractalfract5040133 . . . . . . . . . . . . . 19
PetrA.RyapolovandEugeneB.Postnikov
Mittag–Leffler Function as an Approximant to the Concentrated Ferrofluid’s Magnetization
Curve
Reprintedfrom:FractalFract.2021,5,147,doi:10.3390/fractalfract5040147 . . . . . . . . . . . . . 31
JunesangChoi,MohdIdrisQureshi,AarifHussainBhatandJavidMajid
ReductionFormulasforGeneralizedHypergeometricSeriesAssociatedwithNewSequences
andApplications
Reprintedfrom:FractalFract.2021,5,150,doi:10.3390/fractalfract5040150 . . . . . . . . . . . . . 41
BashirAhmadandSotirisK.Ntouyas
Hilfer–Hadamard Fractional Boundary Value Problems with Nonlocal Mixed Boundary
Conditions
Reprintedfrom:FractalFract.2021,5,195,doi:10.3390/fractalfract5040195 . . . . . . . . . . . . . 65
DheerandraShankerSachan,ShaileshJaloreeandJunesangChoi
CertainRecurrenceRelationsofTwoParametricMittag-LefflerFunctionandTheirApplication
inFractionalCalculus
Reprintedfrom:FractalFract.2021,5,215,doi:10.3390/fractalfract5040215 . . . . . . . . . . . . . 91
GhulamFarid,MuhammadYussoufandKamsingNonlaopon
Feje´r–HadamardTypeInequalitiesfor(α,h-m)-p-ConvexFunctionsviaExtendedGeneralized
FractionalIntegrals
Reprintedfrom:FractalFract.2021,5,253,doi:10.3390/fractalfract5040253 . . . . . . . . . . . . . 109
AbdelhalimEbaidandHindK.Al-Jeaid
TheMittag–LefflerFunctionsforaClassofFirst-OrderFractionalInitialValueProblems:Dual
SolutionviaRiemann–LiouvilleFractionalDerivative
Reprintedfrom:FractalFract.2022,6,85,doi:10.3390/fractalfract6020085 . . . . . . . . . . . . . . 125
ZhiqiangZhang,GhulamFarid,SajidMehmood,KamsingNonlaoponandTaoYan
Generalizedk-FractionalIntegralOperatorsAssociatedwithPo´lya-Szego¨andChebyshevTypes
Inequalities
Reprintedfrom:FractalFract.2022,6,90,doi:10.3390/fractalfract6020090 . . . . . . . . . . . . . . 139
v
Saima Naheed, Shahid Mubeen, Gauhar Rahman, Zareen A. Khan, Kottakkaran Sooppy
Nisar
CertainIntegralandDifferentialFormulasInvolvingtheProductofSrivastava’sPolynomials
andExtendedWrightFunction
Reprintedfrom:FractalFract.2022,6,93,doi:10.3390/fractalfract6020093 . . . . . . . . . . . . . . 151
AbdulrahmanF.Aljohani, AbdelhalimEbaid, EbrahemA.Algehyne, YussriM.Mahrous,
CarloCattani,HindK.Al-Jeaid
The Mittag-Leffler Function for Re-Evaluating the Chlorine Transport Model: Comparative
Analysis
Reprintedfrom:FractalFract.2022,6,125,doi:10.3390/fractalfract6030125 . . . . . . . . . . . . . 171
Yi-XiaLi,MuhammadSamraiz,AyeshaGul,MiguelVivas-Cortez,GauharRahman
Hermite-HadamardFractionalIntegralInequalitiesviaAbel-GontscharoffGreen’sFunction
Reprintedfrom:FractalFract.2022,6,126,doi:10.3390/fractalfract6030126 . . . . . . . . . . . . . 185
DineshKumar,JetaRamandJunesangChoi
DirichletAveragesofGeneralizedMittag-LefflerTypeFunction
Reprintedfrom:FractalFract.2022,6,297,doi:10.3390/fractalfract6060297 . . . . . . . . . . . . . 205
MajaAndric´
FractionalIntegralInequalitiesofHermite–HadamardTypefor(h,g;m)-ConvexFunctionswith
ExtendedMittag-LefflerFunction
Reprintedfrom:FractalFract.2022,6,301,doi:10.3390/fractalfract6060301 . . . . . . . . . . . . . 217
ZhehaoZhang
ANewFractionalPoissonProcessGovernedbyaRecursiveFractionalDifferentialEquation
Reprintedfrom:FractalFract.2022,6,418,doi:10.3390/fractalfract6080418 . . . . . . . . . . . . . 233
vi
About the Editor
MajaAndric´
Maja Andric´ is an Associate Professor in the Faculty of Civil Engineering, Architecture and
GeodesyUniversityofSplit,Croatia.In2011shereceivedherPh.D.inMathematicsfromtheFaculty
of Natural and Mathematical Sciences, University of Zagreb. The main field of her research is
MathematicalAnalysiswithaspecialfocusontheTheoryofInequalitiesandFractionalCalculus.
Shepublished2scientificbooks(monographs)and26scientificpapers. Since2014,shehasbeena
memberoftheEditorialBoardoftheinternationaljournal,FractionalDifferentialCalculus,asAssociate
Editor.
vii

fractal and fractional
Editorial
Editorial for Special Issue “Fractional Calculus Operators and
the Mittag–Leffler Function”
MajaAndric´
FacultyofCivilEngineering,ArchitectureandGeodesy,UniversityofSplit,MaticeHrvatske15,
21000Split,Croatia;[email protected]
Amongthenumerousapplicationsofthetheoryoffractionalcalculusinalmostall
appliedsciences, applicationsinnumericalanalysisandvariousfieldsofphysicsand
engineeringstandout.Applicationsofinequalitiesinvolvingfunctionintegralsandtheir
derivatives,aswellasapplicationsoffractionaldifferentiationinequalities,havemotivated
manyresearcherstoinvestigateextensionsandgeneralizationsusingvariousfractional
differentialandintegraloperators.OfparticularimportanceistheMittag–Lefflerfunction
which,withitsgeneralizations,appearsasasolutiontodifferentialorintegralequations
offractionalorder. Thisproducednewresultsformoregeneralizedfractionalintegral
operatorscontainingtheMittag–Lefflerfunctionintheirkernels.
ThisSpecialIssuecontains15publishedpapers. In[1],theauthorsusederivatives
offractionalorder,whicharebasedongeneralizedkernelsoftheMittag–Lefflertype,to
presentthreemodelsoffractionalorder. Thesemodelsinvolvefractionalordinaryand
partialdifferentialequations.Theaccuracyofthesolutionwascheckedincomparisonwith
theexactsolutionandbycalculatingtheresidualerrorfunction.
Theasymptoticbehaviorofrandomtimechangesindynamicalsystemsisstudied
in[2].Astimevariesrandomly,threeclassesareproposedthatshowdifferentpatternsof
asymptoticdecay.
Citation:Andric´,M.Editorialfor
In[3],theauthorsshowedthatthestaticmagnetizationcurveofhighlyconcentrated
SpecialIssue“FractionalCalculus
ferrofluidscanbeaccuratelyapproximatedbytheMittag–Lefflerfunctionoftheinverse
OperatorsandtheMittag-Leffler
externalmagneticfield.
Function”.FractalFract.2022,6,442.
Byintroducingtwoseriesofnewnumbersandtheirderivatives[4],andchoosingto
https://doi.org/10.3390/
usesixwell-knowngeneralizedKummeradditionformulas,theauthorsestablishsixclasses
fractalfract6080442
ofgeneralizedadditionformulasanddemonstratesixidentitiesregardingfinitesums.
Received:2August2022 Thepaper[5]dealswiththeexistenceanduniquenessofthesolutionfortheHilfer–
Accepted:12August2022 Hadamardfractionaldifferentialequation,supplementedwithmixednon-localbound-
Published:14August2022 aryconditions.
Publisher’sNote:MDPIstaysneutral In [6], the authors developed some new recurrence relations for two parametric
withregardtojurisdictionalclaimsin Mittag–Lefflerfunctionsanddiscussedsomeapplicationsoftheserecurrencerelations.
publishedmapsandinstitutionalaffil- ByapplyingRiemann–Liouvillefractionalintegralsanddifferentialoperators,fournew
iations. relationsbetweenFox–Wrightfunctionswithcertainspecialcasesareestablished.
Thegoalofthepaper[7]wastofindnewversionsofFejér–Hadamard-typeinequalities
for(α,h−m)−p-convexfunctionsbymeansofexpandedfractionaloperatorswithMittag–
Lefflerfunctions.Theseinequalitiesholdatthesametimefordifferenttypesofconvexities
Copyright: © 2022 by the author. anddifferenttypesoffractionalintegrals.
Licensee MDPI, Basel, Switzerland. In[8],anewapproachisgivenforsolvingaclassoffirst-orderfractionalinitialvalue
Thisarticleisanopenaccessarticle
problems.TheresultsarebasedontheRiemann–Liouvillefractionalderivativeandthe
distributed under the terms and
solutionsareexpressedintermsofMittag–Leffler,exponential,andtrigonometricfunctions.
conditionsoftheCreativeCommons
Thepaper[9]aimstodefineanoperatorthatcontainstheMittag–Lefflerfunctionin
Attribution(CCBY)license(https://
itskernel,whichleadstothedeductionofmanyalreadyexistingoperators.Theresultsof
creativecommons.org/licenses/by/
4.0/).
FractalFract.2022,6,442.https://doi.org/10.3390/fractalfract6080442 1 https://www.mdpi.com/journal/fractalfract