New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

370 Pages·2022·15.248 MB·English

by  Asifa Tassaddiq, Muhammad Yaseen

New Advancements
in Pure and Applied
Mathematics
via Fractals and
Fractional Calculus
Edited by
Asifa Tassaddiq and Muhammad Yaseen
Printed Edition of the Special Issue Published in Fractal Fract
www.mdpi.com/journal/fractalfract
New Advancements in Pure and
Applied Mathematics via Fractals and
Fractional Calculus

New Advancements in Pure and
Applied Mathematics via Fractals and
Fractional Calculus
Editors
AsifaTassaddiq
MuhammadYaseen
MDPI•Basel•Beijing•Wuhan•Barcelona•Belgrade•Manchester•Tokyo•Cluj•Tianjin
Editors
AsifaTassaddiq MuhammadYaseen
CollegeofComputerand UniversityofSargodha
InformationSciences Pakistan
MajmaahUniversity
SaudiArabia
EditorialOffice
MDPI
St.Alban-Anlage66
4052Basel,Switzerland
This is a reprint of articles from the Special Issue published online in the open access journal
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Contents
AsifaTassaddiqandMuhammadYaseen
EditorialforSpecialIssue“NewAdvancementsinPureandAppliedMathematicsviaFractals
andFractionalCalculus”
Reprintedfrom:FractalFract.2022,6,284,doi:10.3390/fractalfract6060284 . . . . . . . . . . . . . 1
AsifaTassaddiqandRekhaSrivastava
NewResultsInvolvingRiemannZetaFunctionUsingItsDistributionalRepresentation
Reprintedfrom:FractalFract.2022,6,254,doi:10.3390/fractalfract6050254 . . . . . . . . . . . . . 5
SaimaRashid,ZakiaHammouch,HassenAydi,AbdulazizGarbaAhmadand
AbdullahM.Alsharif
NovelComputationsoftheTime-FractionalFisher’sModelviaGeneralizedFractionalIntegral
OperatorsbyMeansoftheElzakiTransform
Reprintedfrom:FractalFract.2021,5,94,doi:10.3390/fractalfract5030094 . . . . . . . . . . . . . . 21
SaimaRashid,RehanaAshraf,AhmetOcakAkdemir,ManarA.Alqudah,
ThabetAbdeljawadandMohamedS.Mohamed
AnalyticFuzzyFormulationofaTime-FractionalFornberg–WhithamModelwithPowerand
Mittag–LefflerKernels
Reprintedfrom:FractalFract.2021,5,113,doi:10.3390/fractalfract5030113 . . . . . . . . . . . . . 51
BriceydaB.DelgadoandJorgeE.Macı´as-D´ıaz
OntheGeneralSolutionsofSomeNon-HomogeneousDiv-CurlSystemswith
Riemann–LiouvilleandCaputoFractionalDerivatives
Reprintedfrom:FractalFract.2021,5,117,doi:10.3390/fractalfract5030117 . . . . . . . . . . . . . 83
MuhammadSamraiz,MuhammadUmer,ArtionKashuri,ThabetAbdeljawad,SajidIqbal
andNabilMlaiki
OnWeighted(k,s)-Riemann-LiouvilleFractionalOperatorsandSolutionofFractionalKinetic
Equation
Reprintedfrom:FractalFract.2021,5,118,doi:10.3390/fractalfract5030118 . . . . . . . . . . . . . 101
HamadjamAbboubakar,RaissaKomRegonneandKottakkaranSooppyNisar
Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination
Perspectives
Reprintedfrom:FractalFract.2021,5,149,doi:10.3390/fractalfract5040149 . . . . . . . . . . . . . 119
AsifaTassaddiq, SaniaQureshi, AmanullahSoomro, EvrenHincal, DumitruBaleanuand
AsifAliShaikh
ANewThree-StepRoot-FindingNumericalMethodandItsFractalGlobalBehavior
Reprintedfrom:FractalFract.2021,5,204,doi:10.3390/fractalfract5040204 . . . . . . . . . . . . . 151
MuhammadYaseen,SadiaMumtaz,RenyGeorgeandAzharHussain
Existence Results for the Solution of the Hybrid Caputo–Hadamard Fractional Differential
ProblemsUsingDhage’sApproach
Reprintedfrom:FractalFract.2022,6,17,doi:10.3390/fractalfract6010017 . . . . . . . . . . . . . . 177
AsifaTassaddiq,MuhammadSajjadShabbir,QamarDinandHumeraNaaz
Discretization,Bifurcation,andControlforaClassofPredator-PreyInteractions
Reprintedfrom:FractalFract.2022,6,31,doi:10.3390/fractalfract6010031 . . . . . . . . . . . . . . 195
v
MuhammadYaseen,QamarUnNisaArif,RenyGeorgeandSanaKhan
Comparative Numerical Study of Spline-Based Numerical Techniques for Time Fractional
CattaneoEquationintheSenseofCaputo–Fabrizio
Reprintedfrom:FractalFract.2022,6,50,doi:10.3390/fractalfract6020050 . . . . . . . . . . . . . . 217
MuhammadSamraiz,ZahidaPerveen,GauharRahman,MuhammadAdilKhanand
KottakkaranSooppyNisar
Hermite-HadamardFractionalInequalitiesforDifferentiableFunctions
Reprintedfrom:FractalFract.2022,6,60,doi:10.3390/fractalfract6020060 . . . . . . . . . . . . . . 239
ZulfiqarAhmadNoor,ImranTalib,ThabetAbdeljawadandManarA.Alqudah
Numerical Study of Caputo Fractional-Order Differential Equations by Developing New
OperationalMatricesofVieta–LucasPolynomials
Reprintedfrom:FractalFract.2022,6,79,doi:10.3390/fractalfract6020079 . . . . . . . . . . . . . . 257
HumairaYasmin
NumericalAnalysisofTime-FractionalWhitham-Broer-KaupEquationswith
Exponential-DecayKernel
Reprintedfrom:FractalFract.2022,6,142,doi:10.3390/fractalfract6030142 . . . . . . . . . . . . . 277
ChoukriDerbazi,ZidaneBaitiche,MohammedS.Abdo,KamalShah,BahaaeldinAbdalla
andThabetAbdeljawad
Extremal Solutions of Generalized Caputo-Type Fractional-OrderBoundary Value Problems
UsingMonotoneIterativeMethod
Reprintedfrom:FractalFract.2022,6,146,doi:10.3390/fractalfract6030146 . . . . . . . . . . . . . 295
KamsingNonlaopon,MuhammadUzairAwan,MuhammadZakriaJaved,Hu¨seyinBudak
andMuhammadAslamNoor
Someq-FractionalEstimatesofTrapezoidlikeInequalitiesInvolvingRaina’sFunction
Reprintedfrom:FractalFract.2022,6,185,doi:10.3390/fractalfract6040185 . . . . . . . . . . . . . 309
TabindaNahidandJunesangChoi
CertainHybridMatrixPolynomialsRelatedtotheLaguerre-ShefferFamily
Reprintedfrom:FractalFract.2022,6,211,doi:10.3390/fractalfract6040211 . . . . . . . . . . . . . 329
RanaSafdarAli,AimanMukheimer,ThabetAbdeljawad,ShahidMubeen,SabilaAli,
GauharRahmanandKottakkaranSooppyNisar
SomeNewHarmonicallyConvexFunctionTypeGeneralizedFractionalIntegralInequalities
Reprintedfrom:FractalFract.2021,5,54,doi:10.3390/fractalfract5020054 . . . . . . . . . . . . . . 349
vi
fractal and fractional
Editorial
Editorial for Special Issue “New Advancements in Pure and
Applied Mathematics via Fractals and Fractional Calculus”
AsifaTassaddiq1,*andMuhammadYaseen2
1 DepartmentofBasicSciencesandHumanities,CollegeofComputerandInformationSciences,
MajmaahUniversity,AlMajmaah11952,SaudiArabia
2 DepartmentofMathematics,UniversityofSargodha,Sargodha40100,Pakistan;[email protected]
* Correspondence:[email protected]
Fractionalcalculushasreshapedscienceandtechnologysinceitsfirstappearancein
aletterreceivedtoGottfriedWilhelmLeibnizfromGuil-laumedel’Hôpitalintheyear
1695.Theexistenceoffractionalbehaviorinnaturecannotbedenied.Anyphenomenon
withapulse,rhythm,orpatternhasthepotentialtobeafractal.ThegoalofthisSpecial
Issueistoexplorenewdevelopmentsinbothpureandappliedmathematicsasaresult
offractionalbehavior. ThisassertionissupportedbythepapersinthisSpecialIssue.
Thevarietyoftopicscoveredheredemonstratestheimportanceoffractionalcalculusin
variousfieldsandprovidesadequatecoveragetoappealtotheinterestsofeachreader.This
SpecialIssueofFractalandFractionalwaspostedinearly2021withthegoalofexploringthe
variousconnectionsbetweenfractionalcalculusanditsapplicationsinpureandapplied
mathematics. Initially, a deadline was set and has been extended to 5 April 2022, in
considerationoftheauthor’sinterest.Intotal,wereceived74submissions.Followinga
thoroughpeer-reviewprocess,seventeenofthemwereeventuallypublishedand,keeping
withtheoriginalconceptofthisSpecialIssue,havenowbeencompiledintothisbook.The
followingaredetailsofthepaperspublishedinourSpecialIssue:
Citation:Tassaddiq,A.;Yaseen,M. Alietal.[1]developedanewversionofgeneralizedfractionalHadamardandFejér–
EditorialforSpecialIssue“New Hadamard-typeintegralinequalitiesthatcanbeusedtoinvestigatethestabilityandcontrol
AdvancementsinPureandApplied ofcorrespondingfractionaldynamicequations.
MathematicsviaFractalsand
Fisher’sequationisaprecisemathematicalresultderivedfrompopulationdynamics
FractionalCalculus”.FractalFract.
andgenetics,specificallychemistry.Rashidetal.[2]usedahybridtechniqueinconjunction
2022,6,284. https://doi.org/
withanewiterativetransformmethodtosolvethenonlinearfractionalFishermodel.
10.3390/fractalfract6060284
Furthermore,whiletheproposedprocedureishighlyrobust,explicit,andviablefornon-
Received:19May2022 linearfractionalPDEs,ithasthepotentialtobeconsistentlyappliedtoothermultifaceted
Accepted:24May2022 physicalprocesses.
Published:25May2022 Itisworthnotingthattheproposedfuzzinessapproachistovalidatethesuperiority
anddependabilityofconfiguringnumericalsolutionstononlinearfuzzyfractionalpartial
Publisher’sNote:MDPIstaysneutral
withregardtojurisdictionalclaimsin differentialequationsarisinginphysicalandcomplexstructures. Asaresult,in[3],the
publishedmapsandinstitutionalaffil- authorsevaluateasemi-analyticalmethodinconjunctionwithanewhybridfuzzyintegral
iations. transformandtheAdomiandecompositionmethodusingthefuzzinessconceptknownas
theElzakiAdomiandecompositionmethod(EADM).
In[4],theauthorsanalyzedthesolutionsofanonlineardiv-curlsystemwithfractional
derivativesoftheRiemann–LiouvilleorCaputotypes. Tothatend,thefractional-order
Copyright: © 2022 by the authors. vectoroperatorsofdivergence,curl,andgradientwereidentifiedascomponentsofthe
Licensee MDPI, Basel, Switzerland. quaternionicfractionalDiracoperator.Generalsolutionstosomenon-homogeneousdiv-
Thisarticleisanopenaccessarticle curlsystemswerederivedthatconsiderthepresenceoffractional-orderderivativesofthe
distributed under the terms and Riemann–LiouvilleorCaputotypesasoneofthemostimportantresultsofthismanuscript.
conditionsoftheCreativeCommons
Anintegro-differentialkineticequationwasderivedin[5]byusingnovelfractional
Attribution(CCBY)license(https://
operatorsanditssolutionusingweightedgeneralizedLaplacetransforms.Theweighted
creativecommons.org/licenses/by/
(k,s)-Riemann–Liouvillefractionalintegralanddifferentialoperatorsaredefinedbythe
4.0/).
FractalFract.2022,6,284.https://doi.org/10.3390/fractalfract6060284 1 https://www.mdpi.com/journal/fractalfract
FractalFract.2022,6,284
authors. The paper includes some specific properties of the operators as well as the
weightedgeneralizedLaplacetransformofthenewoperators.
Themodelsthatincludevaccinationasacontrolmeasureareveryimportant. In
lightofthis,theauthorsdevelopedandmathematicallyinvestigatedintegerandfractional
modelsoftyphoidfevertransmissiondynamicsin[6].Severalnumericalsimulationswere
run,allowingustoconcludethatsuchdiseasesmaybecombatedthroughvaccination
combinedwithenvironmentalsanitation.
Chemical,electrical,biochemical,geometrical,andmeteorologicalmodelsareexam-
plesofnonlinearmodelsusedinscienceandengineering.Theauthorsof[7]investigated
theglobalfractalbehaviorofanewnonlinearthree-stepmethodwithtenth-orderconver-
gence.Basinsofattractionconsidervarioustypesofcomplexfunctions.Whencompared
tootherwell-knownmethods,theproposedmethodachievesthespecifiedtoleranceinthe
smallestnumberofiterationswhileassumingdifferentinitialguesses.
TheauthorsinvestigatetheexistenceresultsforthehybridCaputo–Hadamardfrac-
tionalboundaryvalueproblemin[8].TheproposedBVP’sinclusionversionwiththree-
pointhybridCaputo–Hadamardterminalconditionsisalsoconsidered,andtherelated
existenceresultsareprovided.Toaccomplishtheseobjectives,Dhage’swell-knownfixed-
pointtheoremsforbothBVPsareapplied. Furthermore, twonumericalexamplesare
presentedtovalidatetheanalyticalfindings.
Theauthorsof[9]developedafeedback-controlstrategytocontrolthechaoscaused
bybifurcation. Theproposedmodel’sfractaldimensionswerecomputed. Tofurther
confirmthecomplexityandchaoticbehavior,themaximumLyapunovexponentsand
phaseportraitsweredepicted.Finally,numericalsimulationswerepresentedtovalidate
thetheoreticalandanalyticalresults.
Numerical analysis is always necessary to demonstrate the efficacy of proposed
schemes.Keepingthisinmind,theauthorsin[10]concentratedonnumericallyaddress-
ingthetimefractionalCattaneoequationinvolvingtheCaputo–Fabrizioderivativeusing
spline-basednumericaltechniques.Themainadvantageoftheschemesisthattheapproxi-
mationsolutionisgeneratedasasmoothpiecewisecontinuousfunction,whichallowsto
approximateasolutionatanypointinthedomainofinterest.
Certainconvexands-convexfunctionshaveapplicationsinoptimizationtheory.Asa
result,in[11],theauthorsinvestigatedavarietyofmean-typeintegralinequalitiesfora
well-knownHilferfractionalderivative.Someidentitieswerealsoestablishedinorderto
infermoreinterestingmeaninequalities.TheCaputofractionalderivativeconsequences
werepresentedasspecialcasestotheirgeneralconclusions.
The authors of [12] proposed a numerical method for solving Caputo fractional-
order differential equations based on the operational matrices of shifted Vieta–Lucas
polynomials(VLPs)(FDEs).Anewoperationalmatrixoffractional-orderderivativesin
theCaputosensewasderived,whichwasthenusedinconjunctionwiththespectraltau
andspectralcollocationmethodstoreducetheFDEstoasystemofalgebraicequations.
Numericalexampleswereprovidedtodemonstratetheaccuracyofthismethod,which
demonstratedthattheobtainedresultsagreewellwiththeanalyticalsolutionsforboth
linearandnonlinearFDEs.
Asemi-analyticalanalysisofthefractional-ordernon-linearcoupledsystemofWhitham–
Broer–Kaupequationswaspresentedin[13]. Thefractionalderivativewasconsidered
intheCaputo–Fabriziosense. Whentheanalyticalandactualsolutionsarecompared,
itisclearthattheproposedapproacheseffectivelysolvecomplexnonlinearproblems.
Furthermore,theproposedmethodologiescontrolandmanipulatetheobtainednumerical
solutionsinanextrememannerinalargeacceptableregion.
Theauthorsof[14]derivedsomesuitableresultsforextremalsolutionstoaclassof
generalizedCaputo-typenonlinearfractionaldifferentialequations(FDEs)withnonlinear
boundaryconditions(NBCs).Theaforementionedoutcomeswereobtainedbyemploying
themonotoneiterativemethod,whichemploystheprocedureofupperandlowersolutions.
Therearetwosequencesofextremalsolutionsgenerated,oneofwhichconvergestothe
2
FractalFract.2022,6,284
uppersolutionandtheothertothecorrespondinglowersolution.Themethoddoesnot
requireanypriordiscretizationorcollocationtogeneratetheaforementionedupperand
lowersolutionsequences.
Q-calculusisanon-trivialandusefulgeneralizationofcalculus.Theauthorsof[15]
presentedtwonewidentitiesinvolvingq-Riemann–Liouvillefractionalintegrals. New
q-fractionalestimatesoftrapezoidal-likeinequalitieswerederivedusingtheseidentitiesas
auxiliaryresults,inessenceoftheclassofgeneralizedexponentialconvexfunctions.
Thedefinitionandapplicabilityofnewfamiliesofpolynomialsgeneratingfunction
andoperationalrepresentationsarealwaysofgreatinterest.Theauthorsof[16]usedoper-
ationaltechniquestoinvestigateanewtypeofpolynomial,specificallytheGould–Hopper–
Laguerre–Sheffermatrixpolynomials.Furthermore,theseparticularmatrixpolynomials
wereinterpretedintermsofquasi-monomiality.Theintegraltransformwasusedtoinvesti-
gatethepropertiesoftheextendedversionsoftheGould–Hopper–Laguerre–Sheffermatrix
polynomials.Therewerealsoexamplesofhowtheseresultsapplytospecificmembersof
thematrixpolynomialfamily.
LaplacetransformoftheRiemannzetafunctionusingitsdistributionalrepresentation
wascomputed,whichplayedacriticalroleinapplyingtheoperatorsofgeneralizedfrac-
tionalcalculustothiswell-studiedfunction[17].Asaresult,asspecialcases,similarnew
imagescanbeobtainedusingvariousotherpopularfractionaltransforms.TheRiemann
zetafunctionwasusedtoformulateandsolveanewfractionalkineticequation.Following
that,anewrelationshipinvolvingtheLaplacetransformoftheRiemannzetafunctionand
theFox–Wrightfunctionwasinvestigated,whichsignificantlysimplifiedtheresults.
Tosummarize, thisspecialselectioncoversthescopeofongoingactivitiesinthe
contextoffractionalcalculusbypresentingalternativeperspectives,viablemethods,new
derivatives,andstrategiestosolvepracticalissues.Aseditors,wepresumethatthiswillbe
followedbyasetofSpecialIssuesandtextstofurtherinvestigatethistheme.
AstheguesteditorsofthisSpecialIssue, wewouldliketotakethisopportunity
tothankallofthereviewers,editorialboardmembers,andeditorswhoassistedusin
perfectingthecontentofthisvolume.WewouldalsoliketothankMs.CecileZhengfrom
thejournalofficeforherpromptassistanceacrosstheSpecialIssuemanagementprocess.
AllauthorcontributionstothisSpecialIssuearegreatlyacknowledgedwiththanks.
Funding:Thisresearchreceivednoexternalfunding.
ConflictsofInterest:Theauthorsdeclarenoconflictofinterest.
References
1. Ali,R.;Mukheimer,A.;Abdeljawad,T.;Mubeen,S.;Ali,S.;Rahman,G.;Nisar,K.SomeNewHarmonicallyConvexFunction
TypeGeneralizedFractionalIntegralIntegralInequalities.FractalFract.2021,5,54.[CrossRef]
2. Rashid,S.;Hammouch,Z.;Aydi,H.;Ahmad,A.;Alsharif,A.NovelComputationsoftheTime-FractionalFisher’sModelvia
GeneralizedFractionalIntegralOperatorsbyMeansoftheElzakiTransform.FractalFract.2021,5,94.[CrossRef]
3. Rashid,S.;Ashraf,R.;Akdemir,A.;Alqudah,M.;Abdeljawad,T.;Mohamed,M.AnalyticFuzzyFormulationofaTime-Fractional
Fornberg–WhithamModelwithPowerandMittag–LefflerKernels.FractalFract.2021,5,113.[CrossRef]
4. Delgado,B.;Macias-Diaz,J.OntheGeneralSolutionsofSomeNon-HomogeneousDiv-CurlSystemswithRiemann–Liouville
andCaputoFractionalDerivatives.FractalFract.2021,5,117.[CrossRef]
5. Samraiz,M.;Umer,M.;Kashuri,A.;Abdeljawad,T.;Iqbal,S.;Mlaiki,N.OnWeighted(k,s)-Riemann-LiouvilleFractional
OperatorsandSolutionofFractionalKineticEquation.FractalFract.2021,5,118.[CrossRef]
6. Abboubakar,H.;KomRegonne,R.;SooppyNisar,K.FractionalDynamicsofTyphoidFeverTransmissionModelswithMass
VaccinationPerspectives.FractalFract.2021,5,149.[CrossRef]
7. Tassaddiq,A.;Qureshi,S.;Soomro,A.;Hincal,E.;Baleanu,D.;Shaikh,A.ANewThree-StepRoot-FindingNumericalMethod
andItsFractalGlobalBehavior.FractalFract.2021,5,204.[CrossRef]
8. Yaseen,M.;Mumtaz,S.;George,R.;Hussain,A.ExistenceResultsfortheSolutionoftheHybridCaputo-HadamardFractional
DifferentialProblemsUsingDhage’sApproach.FractalFract.2022,6,17.[CrossRef]
9. Tassaddiq,A.;Shabbir,M.;Din,Q.;Naaz,H.Discretization,Bifurcation,andControlforaClassofPredator-PreyInteractions.
FractalFract.2022,6,31.[CrossRef]
10. Yaseen,M.;Arif,Q.U.N.;George,R.;Khan,S.ComparativeNumericalStudyofSpline-BasedNumericalTechniquesforTime
FractionalCattaneoEquationintheSenseofCaputo-Fabrizio.FractalFract.2022,6,50.[CrossRef]
3