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Advanced Mathematical Methods Francesco Mainardi and Andrea Giusti Andrea Mainardi and  • Francesco • Advanced Mathematical Methods Mathematical Advanced • Francesco and Mainardi Andrea Giusti Advanced Mathematical Methods Theory and Applications Edited by Francesco Mainardi and Andrea Giusti Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Advanced Mathematical Methods Advanced Mathematical Methods Theory and Applications Special Issue Editors Francesco Mainardi Andrea Giusti MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors Francesco Mainardi Andrea Giusti University of Bologna University of Bologna Italy Italy 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) from 2018 to 2020 (available at: https://www.mdpi.com/journal/ mathematics/special issues/Advanced Mathematical Methods). 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, Article Number, Page Range. ISBN 978-3-03928-246-3 (Pbk) ISBN 978-3-03928-247-0 (PDF) c 2020 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 Special Issue Editors ..................................... vii Andrea Giusti and Francesco Mainardi Advanced Mathematical Methods: Theory and Applications Reprinted from: Mathematics 2020, 8, 107, doi:10.3390/math8010107 ................. 1 Yuri Luchko Some Schemata for Applications of the Integral Transforms of Mathematical Physics Reprinted from: Mathematics 2019, 7, 254, doi:10.3390/math7030254 ................. 3 Roberto Garrappa, Eva Kaslik and Marina Popolizio Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial Reprinted from: Mathematics 2019, 7, 407, doi:10.3390/math7050407 ................. 21 Emilia Bazhlekova and Ivan Bazhlekov Subordination Approach to Space-Time Fractional Diffusion Reprinted from: Mathematics 2019, 7, 415, doi:10.3390/math7050415 ................. 42 Silvia Vitali, Iva Budimir, Claudio Runfola, Gastone Castellani The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach Reprinted from: Mathematics 2019, 7, 1145, doi:10.3390/math7121145 ................ 54 Marina Popolizio On the Matrix Mittag–Leffler Function: Theoretical Properties and Numerical Computation Reprinted from: Mathematics 2019, 7, 1140, doi:10.3390/math7121140 ................ 63 Luisa Beghin, Roberto Garra A Note on the Generalized Relativistic Diffusion Equation Reprinted from: Mathematics 2019, 7, 1009, doi:10.3390/math7111009 ................ 75 Berardino D’Acunto, Luigi Frunzo, Vincenzo Luongo and Maria Rosaria Mattei Modeling Heavy Metal Sorption and Interaction in a Multispecies Biofilm Reprinted from: Mathematics 2019, 7, 781, doi:10.3390/math7090781 ................. 84 Ivano Colombaro, Josep Font-Segura and Alfonso Martinez An Introduction to Space–Time Exterior Calculus Reprinted from: Mathematics 2019, 7, 564, doi:10.3390/math7060564 ................. 96 Vasily E. Tarasov and Valentina V. Tarasova Dynamic Keynesian Model of Economic Growth with Memory and Lag Reprinted from: Mathematics 2019, 7, 178, doi:10.3390/math7020178 .................116 Natalie Baddour Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules Reprinted from: Mathematics 2019, 7, 698, doi:10.3390/math7080698 .................133 Giulio Starita and Alfonsina Tartaglione On the Fredholm Property of the Trace Operators Associated with the Elastic Layer Potentials Reprinted from: Mathematics 2019, 7, 134, doi:10.3390/math7020134 .................161 v Michael D. Marcozzi Probabilistic Interpretation of Solutions of Linear Ultraparabolic Equations Reprinted from: Mathematics 2018, 6, 286, doi:10.3390/math6120286 .................173 vi About the Special Issue Editors Francesco Mainardi is a retired Professor of Mathematical Physics from the University of Bologna where he taught for 40 years (he retired in November, 2013 at the age of 70). Even in his retirement, he continues to carry out teaching and research activities. His fields of research concern several topics of applied mathematics, including diffusion and wave problems, asymptotic methods, integral transforms, special functions, fractional calculus, and nonGaussian stochastic processes. At present his h-index is more than 50. He has published more than 150 refereed papers and some books as an author or editor. His homepage is http://www.fracalmo.org/mainardi. Andrea Giusti is a postdoctoral fellow at Bishop’s University. He received his PhD in Physics from the Ludwig-Maximilians-Universitat¨ Munchen,¨ en cotutelle with the University of Bologna. His research focuses on the classical and quantum aspects of gravity, as well as on more mathematical topics related to fractional calculus. He has published over 40 papers in international peer-reviewed scientific journals. In light of his research achievements, he was awarded the Augusto Righi prize by the Italian Physical Society and the Vaclav´ Votruba prize by the Doppler Institute in Prague. vii mathematics Editorial Advanced Mathematical Methods: Theory and Applications Andrea Giusti 1 and Francesco Mainardi 2,* 1 Physics & Astronomy Department, Bishop’s University, 2600 College Street, Sherbrooke, QC J1M 1Z7, Canada; [email protected] 2 Department of Physics & Astronomy and INFN, University of Bologna, Via Irnerio 46, 40126 Bologna, Italy * Correspondence: [email protected] Received: 1 January 2020; Accepted: 2 January 2020; Published: 9 January 2020 The many technical and computational problems that appear to be constantly emerging in various branches of physics and engineering beg for a more detailed understanding of the fundamental mathematics that serves as the cornerstone of our way of understanding natural phenomena. The purpose of this Special Issue is to establish a brief collection of carefully selected articles authored by promising young scientists and the world’s leading experts in pure and applied mathematics, highlighting the state-of-the-art of the various research lines focusing on the study of analytical and numerical mathematical methods for pure and applied sciences. Our collection opens with a featured review article [1], by Yuri Luchko, aimed at providing a pedagogical discussion of the role of integral transforms in mathematical physics, with particular regard for the Laplace and Mellin transforms. We continue with another survey paper [2], by Roberto Garrappa, Eva Kaslik, and Marina Popolizio, dedicated to an in-depth analysis evaluation of fractional integrals and derivatives of some elementary functions. Similarly to the first article, the work of R. Garrappa et al. is very pedagogical in nature and can serve as an effective reference to those who wish to gradually approach the study of numerical aspects of fractional calculus. This collection then continues with two important featured articles. Specifically, it starts with the work [3], by Emilia Bazhlekova and Ivan Bazhlekov, concerning a subordination approach to the multi-dimensional space–time fractional diffusion equation. In detail, the fundamental solution of this equation is studied by means of the subordination principle, which in turn provides a relation to the classical Gaussian function. We then move to the contribution [4], by Silvia Vitali, Iva Budimir, Claudio Runfola, and Gastone Castellani, dedicated to the study of the role of the central limit theorem within the framework of an heterogeneous ensemble of Brownian particles (dubbed the HEBP approach, for short). The collection then closes with a series of eight very interesting original contributions. We begin this series with the work of Marina Popolizio [5] analyzing numerical properties and theoretical features of the Mittag–Leffler function with matrix arguments. It is then followed by an interesting note [6] on a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function, by Luisa Beghin and Roberto Garra. We then move to biophysical modeling with the inspiring work [7] by Berardino D’Acunto, Luigi Frunzo, Vincenzo Luongo, and Maria Rosaria Mattei, in which the authors propose a mathematical model of heavy metal sorption and interaction in a multispecies biofilm. We then continue with a pedagogical article on space–time exterior calculus [8], and its relation to Maxwell’s theory, by Ivano Colombaro, Josep Font-Segura, and Alfonso Martinez. One then finds an interesting proposal for a mathematical model of economic growth with fading memory and a continuous distribution of time-delay. This work [9], by Vasily E. Tarasov, and Valentina V. Tarasova, represents a generalization of the standard Keynesian macroeconomic model based on Abel-type integrals and integro-differential operators involving the confluent hypergeometric Kummer function in the kernel. The collection then features a work [10] by Natalie Baddour on the discrete
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