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Fractional Integrals and Derivatives • Yuri Luchko Fractional Integrals and Derivatives: “True” Versus “False”

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Fractional Integrals and Derivatives • Yuri Luchko Fractional Integrals and Derivatives: “True” Versus “False” Fractional Integrals and Derivatives • Yuri Luchko Yuri • Fractional Integrals and Derivatives “True” versus “False” Edited by Yuri Luchko Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Fractional Integrals and Derivatives: “True” versus “False” Fractional Integrals and Derivatives: “True” versus “False” Editor Yuri Luchko MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Yuri Luchko Beuth Technical University of Applied Sciences Berlin Germany Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/Fractional Integrals Derivatives). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Volume Number, Page Range. ISBN 978-3-0365-0494-0 (Hbk) ISBN 978-3-0365-0494-0 (PDF) © 2021 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editor .............................................. vii Preface to ”Fractional Integrals and Derivatives: “True” versus “False”” ............. ix Yuri Luchko and Masahiro Yamamoto The General Fractional Derivative and Related Fractional Differential Equations Reprinted from: Mathematics 2020, 8, 2115, doi:10.3390/math8122115 ................ 1 Virginia Kiryakova Unified Approach to Fractional Calculus Images of Special Functions—A Survey Reprinted from: Mathematics 2020, 8, 2260, doi:10.3390/math8122260 ................ 21 Min Cai and Changpin Li Numerical Approaches to Fractional Integrals and Derivatives: A Review Reprinted from: Mathematics 2020, 8, 43, doi:10.3390/math8010043 ................. 57 Rudolf Hilfer and Yuri Luchko Desiderata for Fractional Derivatives and Integrals Reprinted from: Mathematics 2019, 7, 149, doi:10.3390/math7020149 .................111 R. Hilfer and T. Kleiner Maximal Domains for Fractional Derivatives and Integrals Reprinted from: Mathematics 2020, 8, 1107, doi:10.3390/math8071107 ................117 Manuel Ortigueira and Jos´eMachado Fractional Derivatives: The Perspective of System Theory Reprinted from: Mathematics 2019, 7, 150, doi:10.3390/math7020150 .................121 Vasily E. Tarasov and Svetlana S. Tarasova Fractional Derivatives and Integrals: What Are They Needed For? Reprinted from: Mathematics 2020, 8, 164, doi:10.3390/math8020164 .................135 Kai Diethelm, Roberto Garrappa and Martin Stynes Good (and Not So Good) Practices in Computational Methods for Fractional Calculus Reprinted from: Mathematics 2020, 8, 324, doi:10.3390/math8030324 .................157 Dumitru Baleanu and Arran Fernandez On Fractional Operators and Their Classifications Reprinted from: Mathematics 2019, 7, 830, doi:10.3390/math7090830 .................179 Jocelyn Sabatier, Christophe Farges and Vincent Tartaglione Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours Reprinted from: Mathematics 2020, 8, 196, doi:10.3390/math8020196 .................189 Daniel Cao Labora Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations Reprinted from: Mathematics 2020, 8, 1093, doi:10.3390/math8071093 ................205 v Eyaya Fekadie Anley and Zhoushun Zheng Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term Reprinted from: Mathematics 2020, 8, 1878, doi:10.3390/math8111878 ................223 Christopher N. Angstmann, Byron A. Jacobs, Bruce I. Henry, and Zhuang Xu Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators Reprinted from: Mathematics 2020, 8, 2023, doi:10.3390/math8112023 ................251 vi About the Editor Yuri Luchko is a Full Professor at the Faculty of Mathematics - Physics - Chemistry of the Technical University of Applied Sciences Berlin. He studied Mathematics at the Belarussian State University in Minsk and received his PhD degree from this University. In 1994, he was awarded a DAAD research grant at the Free University of Berlin. The main field of his research is Applied Mathematics with a special focus on Fractional Calculus and its applications. Yuri Luchko published more than 100 papers in the international peer-reviewed scientific journals and about 20 books and books chapters as author or editor. He is associate editor of the international journal “Fractional Calculus and Applied Analysis” and editor of a dozen of other reputable mathematical journals. vii Preface to ”Fractional Integrals and Derivatives: “True” versus “False”” Even if the Fractional Calculus (FC) is nearly as old as the conventional calculus, for long time it was addressed and used just sporadically and only by few scientists. Within the last few decades, the situation changed dramatically and nowadays we observe an exponential growth of FC publications, conferences, and scientists involved into this topic. One of the explanations for this phenomenon is in active attempts to introduce a new kind of mathematical models containing fractional order operators into physics, chemistry, engineering, biology, medicine, and other sciences. This speeds up the development of the mathematical theory of FC, including fractional ordinary and partial differential equations, fractional calculus of variations, inverse problems for the fractional differential equations, fractional stochastic models, etc. Unfortunately, most of these new models and results are just formal “fractionalisations” of the known conventional theories, often without any justification and motivation. Additionally, a recent trend in FC is in introducing “new fractional derivatives and integrals” and considering classical equations and models with these fractional order operators in place of the conventional integrals and derivatives. This development led to an uncontrolled flood of FC publications both in mathematical and physical journals. Some of these publications contain trivial, well-known, and sometimes even wrong results that threaten the image of FC in the scientific community. Thus, the FC researches have to think about and to answer questions like “What are the fractional integrals and derivatives?”, “What are their decisive mathematical properties?”, “What fractional operators make sense in applications and why?”, etc. These and similar questions were mostly unanswered until now and the main aim of this Special Issue is a contribution to resolving of some of these questions. The Special Issue opens with three surveys. The review article [1], by Yuri Luchko and Masahiro Yamamoto, presents an in deep discussion of the general fractional derivatives and integrals as well as the fractional ordinary and partial differential equations with the general fractional derivatives. For the fractional partial differential equations with the general fractional derivatives, both direct and inverse problems are discussed. The survey paper [2], by Virginia Kiryakova, addresses another important topic in FC: a unified approach to evaluation of images of the special functions under action of different FC operators. The general scheme proposed in the paper is based on a few classical results combined with ideas and developments from more than 30 years of author’s research. The review article [3], by Min Cai and Changpin Li, focuses on numerical approximations to fractional integrals and derivatives. Almost all relevant results known up to now are presented, discussed, and compared each to other. The research part of the Special Issue contains a series of important contributions that provide some partial answers to the questions already mentioned above and others as e.g., “What are the FC operators needed for?”. This series starts with the article [4], by Rudolf Hilfer and Yuri Luchko, which proposes desiderata for calling an operator a fractional derivative or a fractional integral. The desiderata are based on a small number of time honored and well established criteria. However, they are not axioms and do not define fractional derivatives or integrals uniquely. The short communication [5], by Rudolf Hilfer and Tillmann Kleiner, announces existence of fractional calculi on precisely specified domains of distributions that satisfy the desiderata proposed in [4]. The contribution [6], by Manuel Duarte Ortigueira and Jose´ Tenreiro Machado, aims to provide the ix answers to the same questions as the ones considered in [4]. However, by doing so, the authors act from the perspective of the classical system theory. The article [7], by Vasily E. Tarasov and Svetlana S. Tarasova, addresses usefulness of the FC operators from the viewpoint of applied mathematics. The key idea of the authors is an attempt to introduce a correspondence between the kernel properties of the FC operators and the types of the phenomena that they can adequately describe. The next three contributions
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