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Developments in Mathematics Trifce Sandev Živorad Tomovski Fractional Equations and Models Theory and Applications Developments in Mathematics Volume 61 SeriesEditors Krishnaswami Alladi, Department of Mathematics, University of Florida, Gainesville,FL,USA Pham HuuTiep, Departmentof Mathematics,RutgersUniversity,Piscataway, NJ, USA LoringW.Tu,DepartmentofMathematics,TuftsUniversity,Medford,MA,USA TheDevelopmentsinMathematics(DEVM)bookseriesisdevotedtopublishing well-written monographs within the broad spectrum of pure and applied mathematics. Ideally,eachbookshouldbeself-containedandfairlycomprehensiveintreating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High quality edited volumes conveyingcurrent state-of-the-art research will occasionally also be considered for publication. The DEVMseriesappealstoavarietyofaudiencesincludingresearchers,postdocs,and advancedgraduatestudents. Moreinformationaboutthisseriesathttp://www.springer.com/series/5834 Trifce Sandev • Živorad Tomovski Fractional Equations and Models Theory and Applications 123 TrifceSandev ŽivoradTomovski ResearchCenterforComputerScience FacultyofNaturalSciences andInformationTechnologies andMathematics,InstituteofMathematics MacedonianAcademyofSciencesandArts Ss.CyrilandMethodiusUniversity Skopje inSkopje Skopje FacultyofScience DepartmentofMathematics UniversityofOstrava Ostrava,TheCzechRepublic ISSN1389-2177 ISSN2197-795X (electronic) DevelopmentsinMathematics ISBN978-3-030-29613-1 ISBN978-3-030-29614-8 (eBook) https://doi.org/10.1007/978-3-030-29614-8 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To thememoryof mymother,StajkaSandeva TS Dedicatedtomyson, daughter,and wife ŽT Preface This book is a result of more than 10 years of research in the field of fractional calculus and its application in stochastic processes and anomalousdynamics.The aimistoprovideanintroductiontothetheoryoffractionalcalculusandfractional differential equations, fractional stochastic and kinetic models, related special functions,andtheirapplications. ThebookcoversthemathematicalfoundationoftheMittag-LefflerandFoxH- functions,fractionalintegralsandderivatives,andmanyrecentnoveldefinitionsof generalizedoperatorswhich appear to have many applicationsnowadays.We pay specialattentiontotheanalysisofthecompletemonotonicityofthethree-parameter Mittag-Lefflerfunction,whichisaveryimportantprerequisiteforitsapplicationin modelingdifferentanomalousdynamicsprocesses.Wegiveanumberofdefinitions andusefulpropertiesofdifferentfractionaloperators,startingwiththosenamedas theRiemann–LiouvillefractionalderivativeandintegralandtheCaputofractional derivative, and then proceeding to more complicated and recently introduced composite (or so-called Hilfer) derivatives, Prabhakar integral and derivatives, Hilfer-Prabhakar derivatives, tempered derivatives, generalized distributed order derivatives,andgeneralizedintegraloperatorswith Mittag-Lefflerfunctionsin the kernel.Manyusefulpropertiesandrelationsinfractionalcalculuswhichhavebeen usedinmodelinganomalousdiffusionandnon-exponentialrelaxationarepresented. The Cauchy-type problems of fractional differential equations and their solu- tions, existence and uniquenesstheorems, and differentmethods for solving frac- tional differential equations are presented in this book. Volterra type integral equations are analyzed and equivalence with the Cauchy-type problems has been shown. We also give an exhaustivepresentation of applicationsof the operational method for solving fractional differential equations where, as solutions, the so- calledmultinomialMittag-Lefflerfunctionsareobtained. The book pays special attention to derivation of the fractional diffusion and Fokker-Planck equations within the continuous time random walk theory, and to theirsolutionsandapplications.Theelegantsubordinationapproachispresentedto connectthesolutionsofthefractionaldiffusionequationswiththeclassicalonefor Brownianmotion. We show that all the well-knownfractionaldiffusionequations vii viii Preface (mono-fractional, bi-fractional, distributed order, tempered, etc.) in normal (or Caputo) form and in modified (or Riemann–Liouville) form are special cases of the generalized diffusion equations in normal and modified form with memory kernels. The non-negativity of the corresponding solutions is shown by applying thedefinitionsandpropertiesofthecompletelymonotoneandBernsteinfunctions. Methodsofsolvingfractionaldiffusion-waveequationsinafiniteandintheinfinite domainarealso presentedinthisbook.Furthermore,detailedanalysisofthenon- negativityofthegeneralizedfractionalwaveequationwithmemorykernelbyhelp ofthecompletelymonotone,completeBernsteinandStieltjesfunctionsispresented indetail. At the endof the book,an importantclass of stochastic processesgovernedby the generalized Langevin equations is covered. The role, especially, of the three- parameter and multinomial Mittag-Leffler functions and Tauberian theorems in findingsolutionsoftheseequationsispresented.Variousdiffusivebehaviors,such assubdiffusion,normaldiffusion,superdiffusion,ultraslowdiffusion,etc.,andthe crossover from one to another diffusive regime are described by these equations. Manyrelevantreferencesregardingapplicationsofthesemodelsaregiven. This book is intended for diverse scientific communities and scholars working in the field of application of fractional calculus and fractional differential and integralequationsindescribinganomalousdynamicsincomplexsystems.Students andresearchersin mathematics,physics,chemistry,biology,andengineeringmay benefitfromreadingthebookowingtothesystematicpresentationofmanyuseful toolsinthefractionalcalculustheory. Skopje,Macedonia TrifceSandev Skopje,Macedonia ŽivoradTomovski January2018 Acknowledgements TS sincerely thanks Prof. Dr. Živorad Tomovski for the given trust and support as a PhD supervisor as well as for the continuous fruitful (more than 10 years) cooperation,whichresultedinpreparationofthisbook.TSparticularlythanksProf. Dr. Ralf Metzler (University of Potsdam) for the numerous helpful discussions, suggestionsandguidance,andfortheexcellentcooperationwhichlastsfrom2010 until now. TS acknowledges the hospitality and financial support from the Max Planck Institute for the Physics of Complex Systems (MPIPKS) in Dresden— Germany,in2014,2015,and2016,whilehewasapostdoctoralresearcherandguest scientistattheInstitute,andwhereapartofthisbookhasbeenwritten.TSespecially thanks Prof. Dr. Holger Kantz (MPIPKS, Dresden), Prof. Dr. Aleksei Chechkin (AkhiezerInstituteforTheoreticalPhysics,Kharkov),Dr.AlexanderIomin(Tech- nion, Haifa), Prof. Dr. Johan L.A. Dubbeldam (Delft University of Technology), Prof. Dr. Irina Petreska (Ss. Cyril and MethodiusUniversityin Skopje),and Prof. Dr.ErvinKaminskiLenzi(UniversidadeEstadualdePontaGrossa)fortheexcellent collaboration and support all these years. TS also acknowledges DFG (Deutsche Forschungsgemeinschaft)research grant for the project“Zufaellige Suchprozesse, LevyfluegeundRandomwalksaufkomplexenNetzwerken”betweenGermanyand Macedonia. TS is especially grateful to Academician Ljupco Kocarev from the MacedonianAcademyofSciencesandArtsforthecooperationandgreatsupportin hisresearch. ŽTacknowledgesfundingfromDAADGermanfellowshipprogram,duringhis visitsatICP—UniversityofStuttgartfor3monthsin2008,andtheDepartmentof Physics,TUMunichfor3monthsin2011.ŽTalsoacknowledgessupportfromthe Netherlands Research organization (NWO), with NWO visitor grants, during his visits atthe Departmentof AppliedMathematics,TU Delftfor3 monthsin 2011, 1 month in 2013, 3 months during the period 2015–2016, and 3 months during the period 2017–2018; the Berlin Einstein Foundation, through a visitor grant, during his visit at the Weierstrass Institute for Applied Analysis and Stochastics in Berlin for 3 months in 2013. Finally, ŽT acknowledges funding from the European Commission and Croatian Ministry of Science, Education and Sports ix x Acknowledgements Co-FinancingAgreementNo.291823,projectfinancingfromtheMarieCurieFP7- PEOPLE-2011-COFUND program NEWFELPRO, during his 2-years position at theDepartmentofMathematics,UniversityofRijeka,Croatia.ŽTespeciallythanks Prof. Dr. Tibor Pogány (University of Rijeka, Croatia), Prof. Dr. Roberto Garra, and Prof.Dr. FedericoPolito (SapienzaUniversityof Rome, Italy) for the fruitful collaboration. The authors would also like to acknowledge the support from the Austrian Agency for InternationalCooperation in Education and Research (OeAD-GmbH) and the Ministry of Education and Science of the Republic of Macedonia, within thebilateralprojectprogrambetweenMacedoniaandAustria. A great impact in preparation of this book was also made by: W. Deng and P. Xu (Lanzou), G. Farid (Faisalabad), R. Gorenflo (Berlin), R. Hilfer (Stuttgart), N. Korabel (Manchester), A. Liemert (Ulm), Y. Luchko (Berlin), V. Méndez (Barcelona), G. Pagnini (Bilbao), J. Pecaric (Zagreb), R.K. Saxena (Jodhpur), R. Sibatov(Ulyanovsk),I.M.Sokolov(Berlin),H.M.Srivastava(Victoria),V.Urumov (Skopje).Theauthorssincerelythankallofthem. AsinceregratitudemustalsobeexpressedtoDr.FrancescaBonadei—Executive Editor,andMs.FrancescaFerrari—AssistantEditorforthediligentmanagementof thewholeprocessfromsubmissiontopublicationofthisbook. Lastbutnotleast,weareenormouslygratefultoourfamiliesforallthehelpand supportinourresearch.

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