Special Functions of Fractional Calculus: Applications to Diffusion and Random Search Processes

292 Pages·2022·7.239 MB·English

by  Trifce Sandev, Alexander Iomin

SPECIAL FUNCTIONS OF
FRACTIONAL CALCULUS
Applications to Diffusion
and Random Search Processes
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B1948 Governing Asia
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SPECIAL FUNCTIONS OF
FRACTIONAL CALCULUS
Applications to Diffusion
and Random Search Processes
Trifce Sandev
Macedonian Academy of Sciences and Arts, Macedonia
Alexander Iomin
Technion-Israel Institute of Technology, Israel
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SPECIAL FUNCTIONS OF FRACTIONAL CALCULUS
Applications to Diffusion and Random Search Processes
Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd.
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To my wife Irina.
TS
To the memory of my father Miron-Meer Iomin,
son of Herchen and Bluma, may his memory be blessed.
AI
v
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Preface
Мы все учились понемногу,
Чему-нибудь и как-нибудь1
А. С. Пушкин
“Евгений Онегин”
Mathematics is the art of giving things misleading names.
The beautiful – and at first look mysterious – name the
fractional calculus is just one of the those misnomers
which are the essence of mathematics.2
I. Podlubny
Investigation of stochastic processes in complex media has been at-
tracted much attention for years. Theoretical modeling of diffusion in
heterogeneous and disordered media is a considerable part of these stud-
ies. Heterogeneousanddisorderedmaterialsincludevariousmaterialswith
defects, multi-scale amorphous composites, fractal and sparse structures,
weighted graphs and networks, and so on. Diffusion in such complicated
media with geometric constraint and random forces is often anomalous.
Furtherdevelopingofthetheoryforunderstandinganddescriptionofthese
random processes in a variety of realizations in physics, biology, social life
and finance is a part of modern studies, what we called “complex systems”.
These complex phenomena entailing complex descriptions by fractional ki-
netics and fractional calculus.
Thenewmathematicalapproachshedslightonmanyquestionsarestillex-
ist for studying and opens new ones. Such an example is a random search
1Wealllearnedalittle,Anythingandsomehow(A.S.Pushkin,“EugeneOnegin”).
2Ref.[13],Chapter1.
vii
August12,2022 16:49 ws-book9x6 SpecialFunctionsofFractionalCalculus 12743-main pageviii
viii Special Functions of Fractional Calculus
processwhosesystematicresearchstemsfromprojectsinvolvinghuntingfor
submarines during World War II, while the modern study of first-passage,
orhittingtimescoversalargeareaofsearchproblemsfromanimalfoodfor-
aging to molecular reactions and gene regulation. In many cases, an intro-
duction of a stochastic resetting in complex systems, has significant effects
on the first-passage properties. Moreover, the random search processes in
complex networks are important for various reasons, such as understand-
ing animal food search strategies and improving web search engines, or for
prolonging or speeding up the survival time in first-encounter tasks.
Many of the aforementioned processes can be described by various form
of random walk models, including fractional Fokker-Planck equations and
generalised Langevin equations, which in their turn describe a variety of
completely different problems shared common features. In particular, a
class of diffusion in heterogeneous environment is closely related to turbu-
lent diffusion represented by an inhomogeneous advection-diffusion equa-
tion, and it also relates to generalized geometric Brownian motion, used to
model stock prices.
The suggested book addresses technical issues surrounding the concept of
special functions of anomalous transport, or fractional kinetics, described
intheframeworkoffractionalcalculus. Muchattentionispaidtotechnical
details in using the Fox H-functions and the Mittag-Leffler functions as
well as functions related to them in description of fractional transport in
random composite media. The latter defines an extremely large class of
problems in life science and physics, medical physics and technology. We
attempttopresentthematerialinaself-containedway, suchthatthebook
can be a convenient reading not only for experts in the field but also for
newcomersandundergraduatestudentsinterestinginthefieldoffractional
calculus and fractional kinetics.
Skopje, Macedonia Trifce Sandev,
Haifa, Israel Alexander Iomin
April, 2022
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Acknowledgments
It is our great pleasure to thank our colleagues for collaboration, nu-
merous helpful and stimulating discussions. Their constant interest, sup-
port and encourage cannot be overestimated. Our thanks to Emad Awad
(Alexandria), Lasko Basnarkov (Skopje), Aleksei Chechkin (Potsdam &
Kharkov), Weihua Deng (Lanzhou), Viktor Domazetoski (Skopje), Sergei
Fedotov (Manchester), Katarzyna G´orska (Krak´ow), Andrzej Horzela
(Krako´w), Holger Kantz (Dresden), Ljupco Kocarev (Skopje), Ervin
Kaminski Lenzi (Ponta Grossa), Vicen¸c M´endez (Barcelona), Ralf Met-
zler (Potsdam), Alexander Milovanov (Rome), Arnab Pal (Chennai), Irina
Petreska (Skopje), Haroldo Valentin Ribeiro (Maringa), R. K. Singh (Bar-
Ilan), Igor M. Sokolov (Berlin), Viktor Stojkoski (Skopje), Zˇivorad To-
movski (Ostrava).
TS sincerely thanks Academician Ljupco Kocarev, President of the
MacedonianAcademyofSciencesandArts,forthegiventrust,greatcollab-
oration and continuous support in my research all these years, and for pro-
vidingaveryprofessionalandpositiveworkingatmosphereattheAcademy.
TS expresses his special thanks to Prof. Dr. Ralf Metzler for the un-
forgettable moments in doing research in his group at the University of
Potsdam, where a part of this book has been written, and for the great
collaboration and support.
TS acknowledges DFG (Deutsche Forschungsgemeinschaft) research
grant (DFG, Grant number ME 1535/12-1) for the project “Diffusion and
random search in heterogeneous media: theory and applications” between
Macedonian Academy of Sciences and Arts and University of Potsdam.
TS acknowledges financial support by the Alexander von Humboldt
(AvH) Foundation of the project “Statistical Mechanics of Diffusion Pro-
cesses in Disordered Media” (2020–2022), within the AvH fellowship
ix