Handbook of Fractional Calculus with...

Anatoly Kochubei, Yuri Luchko (Eds.)Handbook of Fractional Calculus with Applications

Handbook of Fractional Calculus with ApplicationsEdited by J. A. Tenreiro Machado

Volume 2: Fractional Differential EquationsAnatoly Kochubei, Yuri Luchko (Eds.), 2019ISBN 978-3-11-057082-3, e-ISBN (PDF) 978-3-11-057166-0,e-ISBN (EPUB) 978-3-11-057105-9

Volume 3: Numerical MethodsGeorge Em Karniadakis (Ed.), 2019ISBN 978-3-11-057083-0, e-ISBN (PDF) 978-3-11-057168-4,e-ISBN (EPUB) 978-3-11-057106-6

Volume 4: Applications in Physics, Part AVasily E. Tarasov (Ed.), 2019ISBN 978-3-11-057088-5, e-ISBN (PDF) 978-3-11-057170-7,e-ISBN (EPUB) 978-3-11-057100-4

Volume 5: Applications in Physics, Part BVasily E. Tarasov (Ed.), 2019ISBN 978-3-11-057089-2, e-ISBN (PDF) 978-3-11-057172-1,e-ISBN (EPUB) 978-3-11-057099-1

Volume 6: Applications in ControlIvo Petráš (Ed.), 2019ISBN 978-3-11-057090-8, e-ISBN (PDF) 978-3-11-057174-5,e-ISBN (EPUB) 978-3-11-057093-9

Volume 7: Applications in Engineering, Life and Social Sciences, Part ADumitru Bǎleanu, António Mendes Lopes (Eds.), 2019ISBN 978-3-11-057091-5, e-ISBN (PDF) 978-3-11-057190-5,e-ISBN (EPUB) 978-3-11-057096-0

Volume 8: Applications in Engineering, Life and Social Sciences, Part BDumitru Bǎleanu, António Mendes Lopes (Eds.), 2019ISBN 978-3-11-057092-2, e-ISBN (PDF) 978-3-11-057192-9,e-ISBN (EPUB) 978-3-11-057107-3

Anatoly Kochubei, Yuri Luchko (Eds.)

Handbook ofFractional Calculuswith Applications|Volume 1: Basic Theory

Series edited by Jose Antonio Tenreiro Machado

EditorsProf. Dr. Anatoly KochubeiNational Academy of Sciences of UkraineInstitute of MathematicsTereschenkivska str., 3Kiev [email protected]

Prof. Dr. Yuri LuchkoBeuth Hochschule für Technik BerlinFB II Mathematik-Physik-ChemieLuxemburger Str. 1013353 [email protected]

Series EditorProf. Dr. Jose Antonio Tenreiro MachadoDepartment of Electrical EngineeringInstituto Superior de Engenharia do PortoInstituto Politécnico do Porto4200-072 [email protected]

ISBN 978-3-11-057081-6e-ISBN (PDF) 978-3-11-057162-2e-ISBN (EPUB) 978-3-11-057063-2

Library of Congress Control Number: 2018967839

Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

© 2019 Walter de Gruyter GmbH, Berlin/BostonCover image: djmilic / iStock / Getty Images PlusTypesetting: VTeX UAB, LithuaniaPrinting and binding: CPI books GmbH, Leck

www.degruyter.com

Contents

Preface| V

J. A. Tenreiro Machado and Virginia KiryakovaRecent history of the fractional calculus: data and statistics| 1

Anatoly N. Kochubei and Yuri LuchkoBasic FC operators and their properties| 23

Rudolf HilferMathematical and physical interpretations of fractional derivatives andintegrals| 47

Virginia KiryakovaGeneralized fractional calculus operators with special functions| 87

Anatoly N. KochubeiGeneral fractional calculus| 111

Virginia Kiryakova and Yuri LuchkoMultiple Erdélyi–Kober integrals and derivatives as operators of generalizedfractional calculus| 127

Mateusz KwaśnickiFractional Laplace operator and its properties| 159

Yuri Luchko and Virginia KiryakovaApplications of the Mellin integral transform technique in fractional calculus| 195

Yuri LuchkoFractional Fourier transform| 225

Yuri LuchkoThe Wright function and its applications| 241

Rudolf Gorenflo, Francesco Mainardi, and Sergei RogosinMittag-Leffler function: properties and applications| 269

Richard B. ParisAsymptotics of the special functions of fractional calculus| 297

VIII | Contents

Moulay Rchid Sidi Ammi and Delfim F.M. TorresAnalysis of fractional integro-differential equations of thermistor type| 327

Ricardo Almeida and Delfim F.M. TorresA survey on fractional variational calculus| 347

Teodor M. Atanacković, Sanja Konjik, and Stevan PilipovićVariational principles with fractional derivatives| 361

MarkMeerschaert and Hans Peter SchefflerContinuous time random walks and space-time fractional differentialequations| 385

MarkM. Meerschaert, Erkan Nane, and P. VellaisamyInverse subordinators and time fractional equations| 407

MarkM. MalamudSpectral theory of fractional order integration operators, their direct sums, andsimilarity problem to these operators of their weak perturbations| 427

Anatoly N. KochubeiFractional differentiation in p-adic analysis| 461

Index| 473

Preface

Fractional calculus (FC) originated in 1695, nearly at the same time as conventionalcalculus. However, FC attracted limited attention and remained a pure mathematicalexercise, in spite of the contributions of important mathematicians, physicists andengineers. FC had a rapid development during the last few decades, both in mathe-matics and in the applied sciences, it being nowadays recognized as an excellent toolfor describing complex systems, phenomena involving long rangememory effects andnon-locality. A hugenumber of research papers andbooks devoted to this subject havebeen published, and presently several specialized conferences andworkshops are be-ing organized each year. The FC popularity in all fields of science is due to its suc-cessful application in mathematical models, namely in the form of FC operators andfractional integral and differential equations. Presently, we are witnessing consider-able progress both as regards theoretical aspects and applications of FC in areas suchas physics, engineering, biology, medicine, economy, or finance.

The popularity of FC has attracted many researchers from all over the world andthere is a demand for works covering all areas of science in a systematic and rigorousform. In fact, the literature devoted to FC and its applications is huge, but readers areconfronted with a high heterogeneity and, in some cases, with misleading and inac-curate information. The Handbook of Fractional Calculus with Applications (HFCA)intends to fill that gap and provides the readers with a solid and systematic treatmentof the main aspects and applications of FC. Motivated by these ideas, the editors ofthe volumes involved a team of internationally recognized experts for a joint publish-ing project offering a survey of their own and other important results in their fields ofresearch. As a result of these joint efforts, a modern survey of FC and its applications,reflecting present day scientific knowledge, is now availablewith theHFCA. This workis distributed in the form of several distinct volumes, each one developed under thesupervision of its editors.

The first volume of HFCA is devoted to the basic theory of FC and presents both aselection of the well-known fundamental results and some modern new trends in FC.It starts with a survey of the recent history of FC, including lists of the FC books pub-lished until now, FC conferences, and journals focusing on FC and its applications.In the second part of the volume, the basic FC operators and their properties as wellas their mathematical and physical interpretations are discussed. Moreover, some se-lected generalizations of the basic FC operators like the multiple Erdélyi–Kober oper-ators, the fractional Laplace operator, and general FC are presented. The next part ofthe volume provides an overview of the FC special functions and the integral trans-forms that play a prominent role in FC. In particular, the fractional Fourier transformand the Mellin integral transform as well as their numerous applications in FC arediscussed. The most important functions of FC—the Wright function and the Mittag-Leffler function—are presented along with their applications in FC and useful proper-

https://doi.org/10.1515/9783110571622-201

VI | Preface

ties including their asymptotical behavior. The fourth part of the volume provides ashort survey of basic results in one of the modern branches of FC: fractional calculusof variations. In addition, a special class of fractional integral equations of thermis-tor type is discussed here. The very important link between FC and probability theoryis subject of the fifth part of the volume. The chapters’ authors explain in detail howboth the time- and the space-fractional partial differential equations appear in a nat-ural way on the macro-level starting from special stochastic processes on the micro-level. Finally, the last part of the volume presents some FC results not very well knowneven to the FC experts, including the surveys of the spectral theory of direct sums ofmultiples of the Riemann–Liouville fractional integrals and fractional differentiationin p-adic analysis.

Our special thanks go to the authors of the individual chapters, which are excel-lent surveys of selected classical and new results in several important fields of FC. Theeditors believe that the HFCA will represent a valuable and reliable reference workfor all scholars and professionals willing to perform research in this challenging andtimely scientific area.

Anatoly Kochubei and Yuri Luchko

J. A. Tenreiro Machado and Virginia KiryakovaRecent history of the fractional calculus:data and statisticsAbstract: Fractional Calculus (FC) was a bright idea of Gottfried Leibniz originating inthe end of the seventeenth century. The topic was developed mainly in a mathemati-cal framework, but during the last decades FC was recognized to represent an usefultool for understanding and modeling many natural and artificial phenomena. Scien-tific areas including not only mathematics and physics, but also engineering, biology,finance, economy, chemistry and human sciences successfully applied FC concepts.The huge progress can be measured by the increasing number of papers, books, andconferences. This chapter presents a brief historical sketch and some bibliographicmetrics of the evolution that occurred during the previous five decades.

Keywords: fractional calculus, fractional order differential equations, fractional ordermathematical models

MSC 2010: 26A33, 01A60, 01A61, 01A67, 34A08, 35R11, 60G22

1 Introduction

In the year 1695 in a letter from Guillaume l’Hôpital to Gottfried Leibniz the questionwas raised if the derivative of integer order could be extended to fractional values.Leibniz studied the problem for the power function and replied “It leads to a paradox,from which one day useful consequences will be drawn.” During the succeeding cen-turies the topic was further developed, but it remained in the limited scope of puremathematics.

The first conference dedicated to FC was due to Bertram Ross who organized theInternational Conference on Fractional Calculus and its Applications at the Univer-sity of New Haven, during June 1974, and who edited the proceedings [192]. The firstmonograph is due to Keith B. Oldham and Jerome Spanier [162].

Acknowledgement: The authors thank the colleagues that helped by providing various informationdata andsources tomake the surveyas complete aspossible.Virginia Kiryakova’sworkon this chapteris on the program of the projects (2017–2019) under bilateral agreements of Bulgarian Academy ofSciences with Serbian andMacedonian Academies of Sciences and Arts, and COST Action program CA15225.

J. A. Tenreiro Machado, Institute of Engineering, Polytechnic of Porto, Department of ElectricalEngineering, Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, PortugalVirginia Kiryakova, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad.G. Bontchev Str., Block 8, Sofia 1113, Bulgaria

https://doi.org/10.1515/9783110571622-001

2 | J.A. T. Machado and V. Kiryakova

We can list (in alphabetical order) some important FC researchers since 1695, ex-cluding alive ones. Readers can find further details in [54, 122, 116, 115, 120, 121, 114],and posters at http://www.math.bas.bg/~fcaa.– Abel, Niels Henrik (5 August 1802–6 April 1829), Norwegian mathematician– Al-Bassam, Mohamad Ali (7 December 1923–d. 2004), mathematician of Iraqi ori-

gin– Bagley, Ronald (31 May 1947–4 May 2017), American mechanical engineer– Bhrawy, Ali Hassan (11 February 1975–3 February 2017), Egyptian mathematician– Cole, Kenneth Stewart (10 July 1900–18 April 18 1984), American physicist– Cole, Robert H. (26 October 1914–17 November 1990), American physicist– Cossar, James (d. 24 July 1998), British mathematician– Davis, Harold Thayer (5 October 1892–14 November 1974), American mathemati-

cian– Djrbashjan,MkhtarM. (11 September 1918–6May 1994), Armenianmathematician– Erdélyi, Arthur (2 October 1908–12 December 1977), Hungarian-born Britishmath-

ematician– Euler, Leonhard (15 April 1707–18 September 1783), Swiss mathematician and

physicist– Feller, William (Vilim) (7 July 1906–14 January 1970), Croatian–American mathe-

matician– Fourier, Jean Baptiste Joseph (21March 1768–16May 1830), Frenchmathematician

and physicist– Gelfand, Israel (Israïl) Moiseevich (2 September 1913–5 October 2009), Russian

mathematician– Gemant, Andrew (27 July 1895–1 February 1983), American physicist– Gerasimov, Alexey N. (24 March 1897–14 March 1968) Russian (Soviet) physicist

working in the field of mechanics– Gorenflo, Rudolf (31 July 1930–20 October 2017), German mathematician– Grünwald, AntonKarl (23November 1838–2 September 1920), Germanmathemati-

cian– Hadamard, Jacques Salomon (8 December 1865–17 October 1963), French mathe-

matician– Hardy,GodfreyHarold (7 February 1877–1December 1947), Englishmathematician– Heaviside, Oliver (18 May 1850–3 February 1925), English electrical engineer,

mathematician, and physicist– Holmgren,Hjalmar J. (22December 1822–29August 1885), Swedishmathematician– de l’Hôpital, Guillaume François Antoine (1661–2 February 1704), French mathe-

matician– Kilbas, Anatoly A. (20 July 1948–28 June 2010), Belarusian mathematician– Kober, Hermann (1888–4 October 1973), mathematician born in Poland, studied

and worked in Germany, later in Great Britain– Krug, Anton (19th–20th century), German mathematician

Recent history of the fractional calculus: data and statistics | 3

– Lacroix, Sylvestre François (28 April 1765–24 May 1843), French mathematician– Lagrange, Joseph-Louis (25 January 1736–10 April 1813), Italian-French mathe-

matician and astronomer– Laplace, Pierre-Simon (23 March 1749–5 March 1827), French mathematician and

astronomer– Laurent, Paul Matthieu Hermann (2 September 1841–19 February 1908), French

mathematician– Leibniz, Gottfried Wilhelm (1 July 1646–14 November 1716), German mathemati-

cian and philosopher– Letnikov, Aleksey V. (1 January 1837–27 February 1888), Russian mathematician– Lévy, Paul Pierre (15 September 1886–15 December 1971), French mathematician– Liouville, Joseph (24 March 1809–8 September 1882), French mathematician– Littlewood, John Edensor (9 June 1885–6 September 1977), British mathematician– Love, Eric Russel (31 March 1912–7 August 2001), British-Australian mathemati-

cian– Marchaud, André (1887–1973), French mathematician– Mikolás, Miklós (5 April 1923–2 February 2001), Hungarian mathematician– Mittag-Leffler, Magnus Gustaf (Gösta) (16 March 1846–7 July 1927), Swedish math-

ematician– Montel, Paul Antoine Aristide (29 April 1876–22 January 1975), Frenchmathemati-

cian– Nagy, Béla Szőkefalvi (29 July 1913–22 December 1998), Hungarianmathematician– Nayfeh, Ali Hasan (21 December 1933–27 March 2017), Palestinian–American me-

chanical engineer– Nekrasov, Pavel Alekseevich (13 February 1853–20December 1924), Russianmath-

ematician– Newton, Isaac (4 January 1643–31 March 1727), English physicist, mathematician,

astronomer, natural philosopher, alchemist, and theologian– Nigmatullin, Rashid Shakirovich (5 January 1923–7 July 1991), Tatarstan, Russian

Federation (Soviet Union) scientist in the field of radio-engineering, radioelec-tronics, electrochemistry, etc.

– Nonnenmacher, Theo F. (7 March 1933–1 March 2016), German physicist– Pincherle, Salvatore (11 March 1853–10 July 1936), Italian mathematician– Post, Emil Leon (11 February 1897–21April 1954), Polish–Americanmathematician– Rabotnov, Yury Nikolaevich (24 February 1914–15 May 1985), Russian (Soviet) sci-

entist in the field of mechanics– Riemann, Georg Friedrich Bernhard (17 September 1826–20 July 1866), German

mathematician– Riesz, Marcel (16 November 1886–4 September 1969), Hungarian mathematician– Ross, Bertram (1917–27 October 1993), American mathematician– Rossikhin, Yury (14 March 1944–29 March 2017), Russian scientist in Mechanics


– Rozovskiy, Moses Isaakovich (12 October 1906–1994), Soviet scientist in the fieldof mathematics and mechanics

– Scott Blair, George William (23 July 1902–1987), British chemist– Shilov, Georgiy Evgen’evich (3 February 1917–17 January 1975), Soviet mathemati-

cian– Sneddon, Ian Naismith (8 December 1919–4 November 2000), Scottish mathe-

matician– Sonine, Nikolay Ya. (22 February 1849–27 February 1915), Russian mathematician– Stankovic, Bogoljub (1 September 1924–16 May 2018), Serbian mathematician– Tardy, Placido (23 October 1816–2 November 1914), Italian mathematician– Wallis, John (23 November 1616–28 October 1703), English mathematician– Weierstrass, Karl Theodor Wilhelm (31 October 1815–19 February 1897), German

mathematician– Weyl, Hermann Klaus Hugo (9 November 1885–8 December 1955), German math-

ematician– Widder, David Vernon (25 March 1898–8 July 1990), American mathematician– Zygmund, Antoni (25 December 1900–30 May 1992), Polish-born American math-

ematician

During the last decades applied sciences paid attention to FC and verified that it con-stitutes a formidable tool to describe many natural and artificial phenomena embed-ding long range memory effects, which classical integer order calculus neglects. Thisstate of affairs motivated a huge development, in theoretical, numerical and appliedaspects, and the emergence of a plethora of models and new proposals. This chap-ter presents a brief list of the contributors for the FC progress during four centuriesand assembles the main bibliographic information in books available in the time ofwriting.

Evidence for thewide spread of FC as a tool inmathematical disciplines and otherrelated areas is the following. Since 2010, theMathematics Subject Classification (MSC2010) includes not only the basic index 26A33, but many other positions related to FC,like 28A80, 33C60, 33E12, 33E30, 34A08, 34K37, 35R11, 44A20, 44A40, 45E10, 60G22,93B99, 93D05, etc. In MSC 2020, we expect yet new FC related positions to appear.

Many of the open problems and hypotheses from the early years (20th century) ofFC contemporary development (posed, for example, at 1st andnext conferences onFC)have been solved or already out of question. However, new ones, questions about thepresent state of the art, and challenges for the futurewerediscussedat recentmeetingsas ICFDA 2014, ICFDA 2016 and ICFDA 2018 (Section 4). Considering the successfuldevelopment and ever wider popularity of FC, the last years’ flood of publications andtopic exploitation could now threaten the FC prestige achieved by the serious effortsand contributions by predecessors. The Round Table sessions [112, 119, 117] aimed atputting forward new problems and challenges and they addressed the needs of the FCcommunity.


The chapter is organizedas follows. Section 2 lists thebooks editedandbookswithauthor, addressing FC. Section 3 reports a list of periodic conferences that are devoted,entirely or partly, to FC. Section 4 provides information on journals specialized on thetopics of FC and its applications.

2 Books published

In this section we collect the books with author(s) and books with editor(s) publishedsince the middle of the twentieth century up to the year 2018. The list of books withauthor and with editor(s) is presented in Table 1.

The evolution of FC in the light of indices such as the number of published bookswith author(s) andbooks edited (more than 195 and46, respectively)maybenot totallyrepresentative, as any other measure of scientific progress. Nevertheless, the data col-lected and represented in Figure 1 is unequivocal. The trends of the type lnN = a+bx,a, b ∈ ℝ, with N for the number of books with author and x for the year reveals a clearpattern of continuous growth (see Figure 2).

3 Conferences devoted to fractional calculus

In this section we list some conferences dedicated, entirely or partly (with special ses-sions), to FC during the last decades. FC conferences without periodicity are not in-cluded. Formore details, such as organizers, places, timetable, and other information,see the survey [114].– Action thématique “Les systèmes à dérivée non entière” (SDNE 2001, 2002, 2003,

2004, 2005, 2006)– Conference on Non-integer Order Calculus and its Applications (RαRNR 2009,

2010, 2011, 2012, . . . , 2016, 2017)– Fractional Calculus Day (FCDay 2009, 2013, 2015, 2016, 2017)– International Carpathian Control Conference (ICCC 2000, 2001, 2002, 2003, . . . ,

2016, 2017, 2018)– International Conference on Analytic Methods of Analysis and Differential Equa-

tions (AMADE 1999, 2001, 2003, 2006, 2009, 2011, 2012, 2015, 2018)– International Conference (Workshop) on Fractional Differentiation and its Appli-

cations (FDA 2004, 2006, 2008, 2010, 2012, 2013; ICFDA 2014, 2016, 2018)– International Conference Modern Methods, Problems and Applications of Opera-

tor (OTHA 2011, 2012, . . . , 2017, 2018)– International Conference Transform Methods and Special Functions (TMSF 1994,

1996, 1999, 2003, 2010, 2011, 2014, 2017)


Table 1: Published books with author and with editor(s) since the 60s.

Year Books with author Books with editor

1960–1979 [55, 205, 38, 135, 162, 186, 136, 206, 139] [192]1980–1989 [209, 210, 166, 155, 183, 202, 156, 157] [140]1990 [212] [78]1991 [159, 158, 167, 71]1992 [208, 39]1993 [203, 149] [90]1994 [242, 96, 54] [229]1995 [168] [196]1996 [160, 193]1997 [40, 132, 91]1998 [143] [197, 138]1999 [179, 170, 224, 234]2000 [77]2001 [12, 19] [22]2002 [237, 177, 63, 83, 82]2003 [233, 174, 153]2004 [94]2005 [84, 184, 79] [142, 57]2006 [124, 95]2007 [216, 198]2008 [246, 220, 128, 134, 241, 240, 228, 20, 30, 49] [150, 92, 99]2009 [8, 102, 137, 33, 100, 87, 175] [123, 23]2010 [125, 37, 152, 187, 52] [110, 118, 73]2011 [176, 163, 98, 75, 141, 105, 5, 213, 243, 88, 50, 106, 195,

101][97, 24]

2012 [21, 11, 161, 2, 111, 85, 204, 127, 250, 51, 227, 244, 221] [93, 191]2013 [43, 65, 172, 81, 222, 80, 219, 42, 27, 238] [46, 151, 32, 107, 6]2014 [76, 247, 129, 13, 14, 86, 89, 144, 169, 3, 70, 28, 64, 145] [47, 48, 45, 62]2015 [181, 180, 164, 9, 69, 7, 225, 60, 72, 18, 131, 199, 171, 26,

108, 235, 188, 194, 25, 245, 126, 35, 146][104]

2016 [34, 248, 44, 59, 74, 154, 182, 109, 113, 173, 249, 147, 185] [190, 53, 207, 41]2017 [36, 239, 61, 231, 31, 68, 66, 236, 15, 217, 218, 133, 4,

201, 200, 178, 17]2018 [226, 230, 10, 148, 67, 1, 211, 58, 29, 223, 232, 189, 103,

130][215, 214, 56, 165,16]

– International Symposium on Fractional Signals and Systems (FSS 2009, 2011,2013, 2015, 2017)

– Mini-Symposium: Fractional Derivatives and Their Applications FDTA at EU-ROMECH Nonlinear Dynamics Conference (FDTA-ENOC 2005, 2008, 2011, 2014,2017)

– SymposiumonFractionalDerivatives andTheirApplication (FDTA2003, 2005, . . . ,2015, 2016, 2017).


Figure 1: Timeline of FC during 1650–1950.


Figure 2: Number N of books with author and with editor versus year x, and trends of the type lnN =a + bx, a,b ∈ ℝ during the period 1965–2018.

4 Journals specialized in fractional calculus

Finally, we present a list of journals specialized in FC, in order of their appearance.Since their names are somewhat similar, it is important to distinguish these journalsby exact wording and abbreviations.

Journal of Fractional Calculus (JFC); ISSN 0918-5402;Publisher: Descartes Press, Japan;Ed.-in-Chief: Katsuyuki Nishimoto (Japan);Starting year: Vol. 1 (1992), info available for Vol. 21–24 (2002),next information n/a at Internet.

Fractional Calculus and Applied Analysis (FCAA; Fract. Calc. Appl. Anal.);ISSN: 1311-0454 (print), ISSN: 1314-2224 (online);Publishers: Institute of Math. and Inform. – Bulg. Acad. Sci. (1998–2010),

Versita and Springer (2011–2014), De Gruyter (since 2015);Website (current): http://www.degruyter.com/view/j/fca;Ed.-in-Chief: Virginia Kiryakova (Bulg. Acad. Sci., Bulgaria);


Starting year: Vol. 1 (1998), current Vol. 21 (2018); 6 issues / year;Impact Factor (SCI) IF (2017) = 2.865, 5-year IF = 3.323, CiteScore = 3.06.

Communications in Fractional Calculus (CFC); ISSN: 2218-3892;Publisher: Asian Academic Publisher Ltd, China;Website was: http://www.nonlinearscience.com/journal_2218-3892.php

(now n/a), some currently available traces:http://en.journals.sid.ir/JournalList.aspx?ID=24970http://blog.sciencenet.cn/blog-298018-343352.html;Editor: Lan Xu;Starting year: Vol. 1 (2010), next information n/a.

Fractional Differential Calculus (FDC; earlier: Fractional Differential Equations);ISSN: 1847-9677;Publisher: Ele-Math (Element d.o.o.), Croatia;Website: http://fdc.ele-math.com/;Current Eds.-in-Chief: Mokhtar Kirane (Univ. de La Rochelle, France),

Josip Pečarić (Univ. of Zagreb, Croatia),Sabir Umarov (Univ. of New Haven, USA);

Starting year: Vol. 1 (2011), current Vol. 8 (2018); 2 issues / year.

Journal of Fractional Calculus and Applications (JFCA);ISSN: 2090-584X (print), ISSN; 2090-5858 (print);Publisher: South Valley University, Egypt;Website: http://fcag-egypt.com/journals/jfca/;Managing Eds.: A.M. A. El-Sayed (Alexandria Univ., Egypt),

S. Z. Rida (South Valley Univ., Egypt);Starting year: Vol. 1 (2011), current Vol. 9 (2018); 2 issues / year.

Progress in Fractional Differentiation and Applications(PFDA; Progr. Fract. Differ. Appl.);

ISSN 2356-9336 (print), ISSN 2356-9344 (online);Publisher: Natural Sciences Publ., USA;Website: http://naturalspublishing.com/show.asp?JorID=48&pgid=0;Ed.-in-Chief: Dumitru Baleanu (Çankaya University, Turkey);Starting year: Vol. 1 (2015), current Vol. 4 (2018); 4 issues / year.

Fractal and Fractional (FF; Fractal Fract.);ISSN 2504-3110, Open Online Access;Publisher: MDPI AG, Switzerland;Website: http://www.mdpi.com/journal/fractalfract;Ed.-in-Chief: Carlo Cattani (University of Tuscia, Italy);Starting year: Vol. 1 (2017), No 1 (Dec. 2017), current Vol. 2 (2018).


Electronic Newsletter “FDA Express”(Fractional Derivative and Applications Express)distributed monthly by e-mail to subscribers and available online;

Publisher: Institute of Soft Matter Mechanics, Hohai University, China;Website: http://em.hhu.edu.cn/fda/;Editor: Wen Chen, Team: Hongguang Sun, Guofei Pang, Xindong Hei,

Yingjie Liang and Lin Chen (China);Starting year: Vol. 1 (2011), current Vol. 26 (2018); 3 vols. (12 issues) / year.

Several special and topical issues ofmanyother prestigious journals arededicatedto the topic of FC and its applications. Also, there are many websites and blogs cre-ated by colleagues, where useful information, files, links, etc. related to FC are avail-able. More details as regards special issues of journals, courses, tutorials, resourcesfrom Matlab, Wolfram MathWorld, other packages on computational aspects for FC,patents, and others were provided in our previous survey of 2011, [122].

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Anatoly N. Kochubei and Yuri LuchkoBasic FC operators and their propertiesAbstract: In this chapter, a brief introduction to themain notions and constructions offractional calculus operators is presented. We start with the Riemann–Liouville inte-grals and derivatives and continue with other kinds of fractional integrals and deriva-tives, including the Caputo derivative, the Grünwald–Letnikov derivative, the Mar-chaudderivative, theWeyl integrals andderivatives of periodic functions, the Erdelyi–Kober integrals andderivatives, theHadamard integrals andderivatives, and the Rieszand the Feller potentials and fractional derivatives.We alsomention the fractional cal-culus operators introduced by Hilfer, Nakhushev, and Pskhu, as well as the fractionalderivatives of distributed order and the general fractional integrals and derivatives.

Keywords: fractional integral, fractional derivative, fractional Laplacian

MSC 2010: 26A33, 35S05

1 IntroductionThe theory of differentiation and integration of an arbitrary (not necessarily integer)order, nowadays referred to as the Fractional Calculus (FC), is nearly as old as conven-tional calculus. Leibniz, L’Hôspital, Jacob Bernoulli, Euler, Laplace, Fourier and otherprominent mathematicians who essentially contributed to the development of analy-sis, tried to interpret some of their results for integrals and derivatives of non-integerorder and suggested first ideas regarding possible definitions of fractional derivatives.In particular, Euler introduced his Gamma-function as an extension of the factorialfunction and noticed that the formula for the nth derivative of a power function canbe meaningfully interpreted also in the case of a non-integer order derivative.

In Abel’s paper [1], devoted to the generalization of the tautochrone problemand published in 1823, he introduced both the fractional integrals called nowadaysRiemann–Liouville fractional integrals and the fractional derivatives called nowa-days Caputo fractional derivatives, and he employed their relationship to derive aclosed form solution to the generalized tautochrone problem. Unfortunately, in hismore famous paper [2], devoted to the same problem, he did not use notations for the

Anatoly N. Kochubei, National Academy of Sciences of Ukraine, Institute of Mathematics,Tereshchenkivska 3, 01004 Kyiv, Ukraine, e-mail: [email protected] Luchko, Beuth University of Applied Sciences Berlin, Department of Mathematics, Physics, andChemistry, Luxemburger Str. 10, 13353 Berlin, Germany, e-mail: [email protected]

https://doi.org/10.1515/9783110571622-002

24 | A.N. Kochubei and Yu. Luchko

fractional integrals and derivatives anymore and thus his pioneering contribution toFC remained mostly unknown (see the recent paper [41] for more details).

In the series of publications that started with [29], dated 1832, Liouville system-atically worked out a rather complete theory of fractional integro-differentiation. Hisfirst ideawas to introduce a formula for fractional differentiation of series with respectto the exponential functions. Later on, he introduced the “Liouville” form of the frac-tional integral and considered a series of its applications in physics.

Another approach to defining the fractional derivatives was suggested in theworkby Grünwald [12] and Letnikov [28]. For a given function f , they introduced the differ-ences of an arbitrary order α > 0 with a step h ∈ ℝ and then defined the fractionalderivatives as the limits of the quotients of the corresponding differences like in thecase of the integer order derivatives.

Later on, several other important definitions of the fractional integro-differentialoperators were suggested (see [46] for a very detailed and excellent written survey ofthe FC history). They were all motivated either by mathematical considerations (say,by extension of the existing FC operators to the new classes of functions) or by ap-plications. In this short survey, we present the most important notions of fractionalderivatives and integrals and their basic properties.

The rest of the chapter is organized as follows. In Section 2, theRiemann–Liouvillefractional integral defined on the finite and infinite intervals and its basic propertiesare discussed. Section 3 is devoted to the Caputo fractional derivative. Also the Hilferderivative, which interpolates between the Riemann–Liouville and the Caputo deriva-tives, ismentioned. In Section 4, some other important fractional derivatives and inte-grals including the Grünwald–Letnikov derivative, the Marchaud derivative, the Weylintegrals andderivatives of periodic functions, theErdelyi–Kober integrals andderiva-tives, the Hadamard integrals and derivatives, and the Riesz and the Feller potentialsand fractional derivatives as well as the FC notation suggested by Nakhushev andPskhu, the fractional derivatives of distributed order, and the general fractional in-tegrals and derivatives are introduced and some of their most important propertiesare mentioned.

2 The Riemann–Liouville fractional integrals andderivatives

In the mathematical treatises related to FC, the Riemann–Liouville fractional inte-grals and derivatives are themost widely investigated and used forms of the fractionalintegro-differential operators. An in-depth investigationof these operators, their prop-erties, applications, and various generalizationswas presented in [46]. In this section,their definitions and some of the basic properties are briefly discussed.

Basic FC operators and their properties | 25

2.1 Riemann–Liouville fractional integrals on a finite interval

In an attempt to interpret the well-known formula for the n-fold definite integral

x

∫a

dxx

∫a

dx⋅ ⋅ ⋅x

∫a

f (x)dx = 1(n − 1)!

x

∫a

(x − t)n−1f (t) dt

for non-integer values of n, themain problem is how to extend the domain of the facto-rial function (n−1)!, n ∈ ℕ, to arbitrary real (or even complex) values of the argument.Evidently, there are infinite many candidates for such extension and some additionalconditions are required for uniqueness of solution to this problem. If we look for solu-tions of the functional equation (which uniquely defines the factorial function onℕ)

f (s + 1) = sf (s), f (1) = 1, (1)

in the class of analytical functions under the additional conditions f (s) ̸= 0, s ∈ ℂand f = f (s) is logarithmically convex for s > 0, then the solution to (1) is unique andcoincides with the Euler Gamma-function (see, e. g., [38]) defined by the convergentintegral

Γ(s) =∞

∫0

e−tts−1 ds (2)

forℜ(s) > 0 and by its analytic continuation for other values of s ∈ ℂ. Itmeans that theEuler Gamma-function is a natural extension of the factorial function and thus playsa prominent role in FC. Moreover, its integral representation (2) read from right to leftcan be interpreted as theMellin integral transform of the exponential function e−t . Forthe role of the Mellin integral transform in FC we refer the interested reader to [34].

Having defined an extension of the factorial function, we are ready to introducean α-fold definite integral. Let (a, b) ⊂ ℝ be a finite interval, f ∈ L1(a, b), and α > 0(ℜ(α) > 0). The left- and right-hand sided Riemann–Liouville fractional integrals oforder α of a function f are defined as follows:

(Iαa+f )(x) =1

Γ(α)

x

∫a

(x − t)α−1f (t) dt, x ∈ (a, b), (3)

(Iαb−f )(x) =1

Γ(α)

b

∫x

(t − x)α−1f (t) dt, x ∈ (a, b). (4)

In the case α = 0, both integrals are interpreted as the identity operators:

(I0a+f )(x) = f (x), (I0b−f )(x) = f (x). (5)


The definitions given by (5) are justified by the formulas [46]

limα→0+(Iαa+f )(x) = f (x), limα→0+(I

αb−f )(x) = f (x), (6)

which are valid for f ∈ L1(a, b) in every Lebesgue point of f , i. e., almost everywhereon [a, b].

Evidently, the left- and right-hand sided Riemann–Liouville fractional integralsare linear operators:

(Iαa+(c1f1 + c2f2))(x) = c1(Iαa+f1)(x) + c2(I

αa+f2)(x), (7)

(Iαb−(c1f1 + c2f2))(x) = c1(Iαb−f1)(x) + c2(I

αb−f2)(x), (8)

for any c1, c2 ∈ ℝ provided that Iαa+f1 and Iαa+f2 or I

αb−f1 and I

αb−f2, respectively, exist

almost everywhere on [a, b].The left- and right-hand sided Riemann–Liouville fractional integrals are con-

nected by means of a simple variable substitution. With the notation (Rf )(x) = f (a +b − x) we have the relations

RIαa+ = Iαb−R, RI

αb− = I

αa+R. (9)

The well-known formula for the Euler Beta-function

B(s, t) =1

∫0

τs−1(1 − τ)t−1 dτ = Γ(s)Γ(t)Γ(s + t)

, ℜ(s) > 0, ℜ(t) > 0,

and the Fubini theorem immediately lead to the important semigroup property of theRiemann–Liouville fractional integrals:

Iαa+Iβa+ = I

α+βa+ , I

αb−I

βb− = I

α+βb− . (10)

For applications, a formula for integration by parts plays a significant role:

b

∫a

g(x)(Iαa+f )(x) dx =b

∫a

f (x)(Iαb−g)(x). (11)

It holds true, say, for f ∈ Lp(a, b), q ∈ Lq(a, b) under the conditions p ≥ 1, q ≥ 1, and1/p + 1/q ≤ 1 + α (and p ̸= 1, q ̸= 1 in the case 1/p + 1/q = 1 + α).

The left- and right-hand sided Riemann–Liouville fractional derivatives are in-troduced as the left-inverse operators to the corresponding fractional integrals. Letn − 1 < α ≤ n, n ∈ ℕ and a function f satisfy the condition In−αa+ f ∈ AC

n[a, b], whereACn[a, b] = {f : (f ∈ Cn−1[a, b]) ∩ (f (n−1) ∈ AC[a, b])} and by AC[a, b] we denote thespace of absolutely continuous functions on the interval [a, b]. The left- and right-hand


sided Riemann–Liouville fractional derivatives of order α of the function f are definedas follows:

(Dαa+f )(x) =dn

dxn(In−αa+ f )(x), x ∈ (a, b), (12)

(Dαb−f )(x) = (−1)n dn

dxn(In−αb− f )(x), x ∈ (a, b). (13)

Due to (5), the Riemann–Liouville fractional derivatives are reduced to the nth deriva-tives for α = n ∈ ℕ:

(Dna+f )(x) =dnfdxn, (Dnb−f )(x) = (−1)

n dnfdxn, x ∈ (a, b). (14)

Because of formulas (7) and (8), the left- and right-hand sided Riemann–Liouvillefractional derivatives are linear operators.

In what follows, we mostly deal with the left-hand sided Riemann–Liouville frac-tional integrals and derivatives. The corresponding formulas for the right-hand sidedoperators can easily be obtained from the ones for the left-hand sided operators andthe relations (9).

Let us mention that the inclusion In−αa+ f ∈ ACn[a, b] follows from the inclusion f ∈

ACn[a, b]. In the last case, the Riemann–Liouville fractional derivatives exist almosteverywhere on the interval (a, b) and we get the formula

(Dαa+f )(x) =n−1∑k=0

f (k)(a)Γ(1 + k − α)

(x − a)k−α + 1Γ(n − α)

x

∫a

f (n)(t) dt(x − t)α−n+1

. (15)

The integral at the right-hand side of (15) is a composition of a fractional integral andthe nth derivative like in formula (12), but in reverse order:

1Γ(n − α)

x

∫a

f (n)(t) dt(x − t)α−n+1

= (In−αa+ f(n))(x), x ∈ (a, b). (16)

In Section 2, we deal with this object that nowadays is referred to as the Caputo frac-tional derivative.

As already mentioned, the Riemann–Liouville fractional derivative is the left-inverse operator to the corresponding Riemann–Liouville fractional integral, i. e., therelation

(Dαa+Iαa+f )(x) = f (x) (17)

holds true for any function f ∈ L1(a, b). As to the composition of the Riemann–Liouville integral and the Riemann–Liouville derivative, the formula

(Iαa+Dαa+f )(x) = f (x) −

n−1∑k=0

(x − a)α−k−1

Γ(α − k)( d

n−k−1

dxn−k−1In−αa+ f)(a) (18)


holds true for any function f ∈ L1(a, b) that satisfies the condition In−αa+ f ∈ ACn[a, b]. If

a function f can be represented as a Riemann–Liouville integral of g ∈ L1(a, b), i. e., iff (x) = (Iαa+g)(x), then it follows from (17) that

(Iαa+Dαa+f )(x) = f (x). (19)

The function space that consists of all functions f that can be represented as theRiemann–Liouville fractional integrals of functions from Lp(a, b) (f (x) = (Iαa+g)(x),g ∈ Lp(a, b)) is denoted by Iαa+(L

p), ℜ(α) > 0. The space of functions Iαb−(Lp), ℜ(α) > 0

is defined as above with the corresponding adaptations.For the Riemann–Liouville fractional derivatives, there exists a formula for inte-

gration by parts similar to (11):

b

∫a

f (x)(Dαa+g)(x) dx =b

∫a

g(x)(Dαb−f )(x). (20)

This formula follows from (11) and holds true, say, for f ∈ Iαb−(Lp) and q ∈ Iαa+(L

p) underconditions p ≥ 1, q ≥ 1, and 1/p+1/q ≤ 1+α (and p ̸= 1, q ̸= 1 in the case 1/p+1/q = 1+α).

In some cases, it is convenient to use the same symbol for the Riemann–Liouvillefractional derivatives and integrals. In the rest of this subsection, we denote by Iαa+the Riemann–Liouville fractional integral of order α if ℜ(α) ≥ 0 and the Riemann–Liouville fractional derivative of order −α if ℜ(α) < 0. Similarly, Dαa+ stands for theRiemann–Liouville fractional derivative of order α if ℜ(α) ≥ 0 and for the Riemann–Liouville fractional integral of order −α ifℜ(α) < 0.

The semigroup property (10), which was formulated for the Riemann–Liouvillefractional integrals, can be extended to some compositions of the fractional integralsand derivatives [46]. More precisely, the formula

(Iαa+Iβa+f )(x) = (I

α+βa+ f )(x) (21)

is valid in the following cases:1) ℜ(β) > 0,ℜ(α + β) > 0, f ∈ L1(a, b);2) ℜ(β) < 0,ℜ(α) > 0, f ∈ I−βa+ (L

1);3) ℜ(α) < 0,ℜ(α) < 0, f ∈ I−α−βa+ (L

1).

For analytical functions f and g, the semigroup formula

(Dαa+Dβa+f )(x) = (D

α+βa+ f )(x) (22)

is valid for α ∈ ℝ and β < 1 [46]. This property is employed to prove the Leibniz typerule for the Riemann–Liouville fractional derivative that can be written in differentforms like, e. g.,

(Dαa+f ⋅ g)(x) =∞∑k=0(αk)(Dα−ka+ f )(x)g

(k)(x), α ∈ ℝ, (23)


(Dαa+f ⋅ g)(x) =+∞∑

k=−∞( αk + β)(Dα−β−ka+ f )(x)(D

β+ka+ g)(x), (24)

α, β ∈ ℝ (α ̸= −1,−2, . . . if β ̸∈ ℤ),

with the generalized binomial coefficients ( αβ ) =Γ(α+1)

Γ(β+1)Γ(α−β+1) . In particular, formulas(23) and (24) are valid for the functions f and g analytical on (a, b). Other forms of theLeibniz rule for the Riemann–Liouville fractional derivatives including the integralform were deduced in [32, 50] by using an operational method. The Leibniz rule (23)was shown to be a direct consequence of the well-known summation theorem for theGauss hypergeometric function 2F1 saying that

2F1(a, b; c; 1) =Γ(c)Γ(c − a − b)Γ(c − a)Γ(c − b)

, ℜ(c − a − b) > 0,

where the Gauss function 2F1 is defined as the series

2F1(a, b; c; z) =Γ(c)

Γ(a)Γ(b)

∞∑n=0

Γ(a + n)Γ(b + n)Γ(c + n)

zn

n!

for |z| ≤ 1 and ℜ(c − a − b) > 0 and as an analytic continuation of this series for othervalues of z.

It is worth mentioning that violation of the standard Leibniz rule is one of thecharacteristic properties of the fractional derivatives of any kind. More precisely, theformula

Dα(f ⋅ g) = (Dαf ) ⋅ g + f (Dαg)

with any fractional derivative is valid only for α = 1 [49].For further properties of the Riemann–Liouville integrals and derivatives on a fi-

nite interval including their mapping properties on the spaces Hλ(a, b) of the Hölderfunctions, Lp(a, b) of the Lebesque integrable functions, and on the spaces Hλ(a, b)and Lp(a, b) with some weights, we refer the reader to [46].

2.2 Riemann–Liouville fractional integrals on infinite intervals

The definitions (3) and (4) of the left- and right-hand sided Riemann–Liouville frac-tional integrals can be used in the case of the semi-infinite intervals (a,+∞) (formula(3) with b = +∞) and (−∞, b) (formula (4) with a = −∞), respectively. For a = 0,the left-hand sided Riemann–Liouville fractional integral on the semi-infinite inter-val (0,+∞) is defined as follows:

(Iα0+f )(x) =1

Γ(α)

x

∫0

(x − t)α−1f (t) dt, x > 0. (25)


The only differences in the theory of these fractional integrals compared to the case ofthe Riemann–Liouville integrals defined on the finite intervals are their domains (theintegrals have to be convergent in +∞ or −∞, respectively) and the correspondingmapping properties. Usually, these integrals are treated either on Lp(ℝ) with 1 < p <1/α or on Lp(ℝ)with someweights, or on the spaces of the Hölder functions with someweights that tend to zero at infinity.

On the whole axis ℝ, the left- and right-hand sided Riemann–Liouville fractionalintegrals of order α > 0 (orℜ(α) > 0) are defined as follows:

(Iα+f )(x) =1

Γ(α)

x

∫−∞

(x − t)α−1f (t) dt, x ∈ ℝ, (26)

(Iα−f )(x) =1

Γ(α)

+∞

∫x

(t − x)α−1f (t) dt, x ∈ ℝ. (27)

In the case α = 0, both integrals are interpreted as the identity operators:

(I0+ f )(x) = f (x), (I0− f )(x) = f (x). (28)

The fractional integrals I± are well defined for f ∈ Lp(ℝ) under the conditions 0 <α < 1 and 1 ≤ p < 1/α. The properties of the Riemann–Liouville fractional integralsmentioned in the previous subsection are also valid with some suitable modificationsand restrictions for the Riemann–Liouville integrals on the infinite intervals.

Evidently, the left- and right-hand sided Riemann–Liouville fractional integralson infinite intervals are linear operators.

The left- and right-hand sided Riemann–Liouville fractional integrals Iα± are con-nected by the relation (compare to formula (9))

QIα± = Iα∓Q, (Qf )(x) = f (−x), x ∈ ℝ. (29)

The semigroup property (10) and the integration by parts formula (11) are also validfor the Riemann–Liouville fractional integrals on the infinite intervals:

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