Mittag-leffler functions, related topics and applications.

548 Pages·2020·6.438 MB·English

Springer Monographs in Mathematics
Rudolf Gorenflo
Anatoly A. Kilbas
Francesco Mainardi
Sergei Rogosin
Mittag-Leffler
Functions,
Related Topics
and Applications
Second Edition
Springer Monographs in Mathematics
Editors-in-Chief
IsabelleGallagher,UFRdeMathématiques,UniversitéParis-Diderot,Paris,France
Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study,
Seoul,SouthKorea;MathematicalInstitute,UniversityofWarwick,Coventry,UK
Series Editors
Sheldon Axler, Department of Mathematics, San Francisco State University, San
Francisco, CA, USA
MarkBraverman,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,
USA
Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton,
NJ, USA
Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan
Sinan C. Güntürk, Department of Mathematics, Courant Institute of Mathematical
Science, New York, NY, USA
Claude Le Bris, Cite Descartes, Champs-sur-Marne, CERMICS-ENPC, Marne la
Vallée, France
Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay,
France
AlbertoA.Pinto,DepartmentofMathematics,UniversityofPorto,Porto,Portugal
GabriellaPinzari,DepartmentofMathematics,UniversityofPadova,Padova,Italy
Ken Ribet, Department of Mathematics, University of California, Berkeley, CA,
USA
René Schilling, Institut für Mathematische Stochastik, TU Dresden, Dresden,
Germany
Panagiotis Souganidis, Department of Mathematics, University of Chicago,
Chicago, IL, USA
Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK
ShmuelWeinberger,DepartmentofMathematics,UniversityofChicago,Chicago,
IL, USA
Boris Zilber, Department of Mathematics, Oxford University, Oxford, UK
Thisseriespublishesadvancedmonographsgivingwell-writtenpresentationsofthe
“state-of-the-art”infieldsofmathematicalresearchthathaveacquiredthematurity
neededforsuchatreatment.Theyaresufficientlyself-containedtobeaccessibleto
morethanjusttheintimatespecialistsofthesubject,andsufficientlycomprehensive
to remain valuable references for many years. Besides the current state of
knowledgeinitsfield,anSMMvolumeshouldideallydescribeitsrelevancetoand
interaction with neighbouring fields of mathematics, and give pointers to future
directions of research.
More information about this series at http://www.springer.com/series/3733
fl
Rudolf Goren o Anatoly A. Kilbas
(cid:129) (cid:129)
Francesco Mainardi Sergei Rogosin
(cid:129)
fl
Mittag-Lef er Functions,
Related Topics
and Applications
Second Edition
123
RudolfGorenflo(1930–2017) AnatolyA.Kilbas (1948–2010)
Mathematical Institute Department ofMathematics
FreeUniversity Berlin andMechanics
Berlin, Germany Belarusian State University
Minsk,Belarus
Francesco Mainardi
Department ofPhysics SergeiRogosin
andAstronomy Department ofEconomics
University of Bologna Belarusian State University
Bologna, Italy Minsk,Belarus
ISSN 1439-7382 ISSN 2196-9922 (electronic)
SpringerMonographs inMathematics
ISBN978-3-662-61549-2 ISBN978-3-662-61550-8 (eBook)
https://doi.org/10.1007/978-3-662-61550-8
MathematicsSubjectClassification: 33E12,26A33,34A08,45K05,44Axx,60G22
1stedition:©Springer-VerlagBerlinHeidelberg2014
2ndedition:©Springer-VerlagGmbHGermany,partofSpringerNature2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part of
SpringerNature.
Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany
To the memory of our colleagues and friends
Anatoly Kilbas (1948–2010) and
Rudolf Gorenflo (1930–2017)
Preface to the Second Edition
After the appearance of the first edition of our book “Mittag-Leffler Functions:
Related Topics and Applications”, we have observed a growing interest in the
subject.Manynewresearcharticlesandbookshaveappeared.Thisismainlydueto
thecentralroleoftheMittag-LefflerfunctionsinFractionalCalculusandFractional
Modeling. With this interest in mind, we decided to prepare the second edition of
ourbookonMittag-Lefflerfunctions,presentingnewideasandresultsrelatedtothe
theory and applications of this family offunctions.
New results have been added to practically all sections of the book. In Chap. 3
“TheClassicalMittag-LefflerFunction”,resultsonMittag-Lefflersummationaswell
as the notion of the Mittag-Leffler reproducing kernel Hilbert space are discussed.
We present results applying the distribution of the zeros of the Mittag-Leffler
functiontothestudyofinverseproblemsfordifferentialequationsinBanachspaces
(Chap.4“TheTwo-ParametricMittag-LefflerFunction”).Chapter5“Mittag-Leffler
FunctionsWithThreeParameters”discussesrecentresultsonLeRoytypefunctions
relatedtotheMittag-Lefflerfunctionbuthavingadifferentnature.Newapplications
related to all functions in this chapter have been added. Essentially enlarged is the
next chapter (Chap. 6) concerning Mittag-Leffler functions depending on several
parameters. Such functions have become important from both a theoretical and an
appliedpointofview.WealsodiscussthepropertiesoftheMittag-Lefflerfunctions
ofseveralvariablesandwithmatrixargument.Numericalmethodsfortheseclasses
offunctions are discussed too. We have completely rewritten the chapters dealing
with applications (Chap. 8 “Applications to Fractional Order Equations”, Chap. 9
“Applications to Deterministic Models”, and Chap. 10 “Applications to Stochastic
Models”, which are essentially enlarged versions of Chaps. 7–9 from the first edi-
tion).Thus,webrieflydiscussinChap.8themainideasoffractionalcontroltheory
and present some numerical methods applied to the study of fractional models,
including those related to the calculation of the values of the Mittag-Leffler func-
tions.Wealsoaddedanewchapter(Chap.7),whichdescribesthemainproperties
of the classical Wright function, closely related to the Mittag-Leffler function.
Consequently, thestructureofAppendix F“HigherTranscendentalFunctions”has
beenchanged, since nowinApp.F.2wedeal mainly with thegeneralized Wright
vii
viii PrefacetotheSecondEdition
function,focussingnotonlyonthegeneralfunctionofthistype(theso-calledFox–
Wright function W ) but also on the most applicable special cases W and W .
p q 1 1 0 2
EssentialchangeswerealsomadetoAppendixE“ElementsofFractionalCalculus”
inordertooutlinetheroleoflesspopularfractionalconstructionsandtoshowwhich
specific properties of these constructions give potential further applications of
Grünwald–Letnikov, Marchaud, Hadamard, Erdélyi–Kober, and Riesz fractional
derivatives.
Our book project could not have been realized without the constant support of
ourcolleaguesandfriends.WearegratefultoRobertoGarrappaforpreparingshort
reviews of his results and allowing us to include them in the book. Additional
thanks are due to Alexander Apelblat, Roberto Garra, Andrea Giusti, George
Karneadakis,VirginiaKiryakova,YuriLuchko,ArakMathai,EdmundoCapelasde
Oliveira, Gianni Pagnini, Enrico Scalas, José Tenreiro Machado, and Vladimir
Uchaikin.Ourwives,GiovannaandMaryna,weresopolitetoallowustospendso
much time on the book. The 2nd edition was discussed in Bologna and Berlin by
threeofus,butasofOctober20,2017,ProfessorRudolfGorenfloisnolongerwith
us.Wetookthelibertytodedicatethissecondeditiontoourmissedcolleaguesand
friends, Anatoly Kilbas and Rudolf Gorenflo, keeping them as co-authors because
of their essential role in realizing this project.
Bologna, Italy Francesco Mainardi
Minsk, Belarus Sergei Rogosin
March 2020
Preface to the First Edition
ThestudyoftheMittag-Lefflerfunctionanditsvariousgeneralizationshasbecome
a very popular topic in Mathematics and its Applications. However, during the
twentieth century, this function was practically unknown to the majority of scien-
tists, since it was ignored in most common books on special functions. As a
noteworthyexceptionthehandbook“HigherTranscendentalFunctions”,vol.3,by
A. Erdelyi et al. deserves to be mentioned.
NowtheMittag-Lefflerfunctionisleavingitsisolatedroleas Cinderella(using
the term coined by F.G. Tricomi for the incomplete gamma function).
Therecentgrowinginterestinthisfunctionismainlyduetoitscloserelationto
the Fractional Calculus and especially to fractional problems which come from
applications.
Our decision to write this book was motivated by the need to fill the gap in the
literature concerning this function, to explain its role in modern pure and applied
mathematics, andto give thereader an idea of how one can use such a functionin
the investigation of modern problems from different scientific disciplines.
ThisbookisafruitofcollaborationbetweenresearchersinBerlin,Bolognaand
Minsk.Ithashighlyprofitedfromvisits ofSR totheDepartment ofPhysicsatthe
University of Bologna and from several visits of RG to Bologna and FM to the
DepartmentofMathematicsandComputerScienceatBerlinFreeUniversityunder
the European ERASMUS exchange. RG and SR appreciate the deep scientific
atmosphere at the University of Bologna and the perfect conditions they met there
for intensive research.
We are saddened that our esteemed and always enthusiastic co-author
Anatoly A. Kilbas is no longer with us, having lost his life in a tragic accident
on28June2010intheSouth ofRussia.Wewillkeephim,andourinspiringjoint
work with him, in living memory.
Berlin, Germany Rudolf Gorenflo
Bologna, Italy Francesco Mainardi
Minsk, Belarus Sergei Rogosin
March 2014
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Historical Overview of the Mittag-Leffler Functions . . . . . . . . . . . . 7
2.1 A Few Biographical Notes On Gösta Magnus
Mittag-Leffler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Contents of the Five Papers by Mittag-Leffler
on New Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Further History of Mittag-Leffler Functions . . . . . . . . . . . . . . . 12
3 The Classical Mittag-Leffler Function. . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Relations to Elementary and Special Functions. . . . . . . . . . . . . 21
3.3 Recurrence and Differential Relations. . . . . . . . . . . . . . . . . . . . 23
3.4 Integral Representations and Asymptotics. . . . . . . . . . . . . . . . . 24
3.5 Distribution of Zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Further Analytic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6.1 Additional Integral Properties . . . . . . . . . . . . . . . . . . . 37
3.6.2 Mittag-Leffler Summation of Power Series. . . . . . . . . . 41
3.6.3 Mittag-Leffler Reproducing Kernel Hilbert Spaces . . . . 44
3.7 The Mittag-Leffler Function of a Real Variable. . . . . . . . . . . . . 46
3.7.1 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.7.2 The Complete Monotonicity Property . . . . . . . . . . . . . 53
3.7.3 Relation to Fractional Calculus . . . . . . . . . . . . . . . . . . 55
3.8 Historical and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . 57
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 The Two-Parametric Mittag-Leffler Function . . . . . . . . . . . . . . . . . 63
4.1 Series Representation and Properties of Coefficients . . . . . . . . . 64
4.2 Explicit Formulas. Relations to Elementary and Special
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Differential and Recurrence Relations. . . . . . . . . . . . . . . . . . . . 66
xi