New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus PDF
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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 FractalandFractional(ISSN2504-3110) (availableat: https://www.mdpi.com/journal/fractalfract/ specialissues/pureandappliedmath). Forcitationpurposes,citeeacharticleindependentlyasindicatedonthearticlepageonlineandas indicatedbelow: LastName,A.A.;LastName,B.B.;LastName,C.C.ArticleTitle. JournalNameYear,VolumeNumber, PageRange. ISBN978-3-0365-4905-7(Hbk) ISBN978-3-0365-4906-4(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 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
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