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Mathematical Economics • Vasily E Mathematical Economics • Vasily E. • Vasily Tarasov Mathematical Economics Application of Fractional Calculus Edited by Vasily E. Tarasov Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Mathematical Economics Mathematical Economics Application of Fractional Calculus Special Issue Editor Vasily E. Tarasov MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Special Issue Editor Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University Information Technologies and Applied Mathematics Faculty, Moscow Aviation Institute (National Research University) Russia 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/Mathematical Economics). 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 978-3-03936-118-2 (Pbk) ISBN 978-3-03936-119-9 (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 Editor ...................................... vii Vasily E. Tarasov Mathematical Economics: Application of Fractional Calculus Reprinted from: Mathematics 2020, 8, 660, doi:10.3390/math8050660 ................. 1 Vasily E. Tarasov On History of Mathematical Economics: Application of Fractional Calculus Reprinted from: Mathematics 2019, 7, 509, doi:10.3390/math7060509 ................. 5 Francesco Mainardi On the Advent of Fractional Calculus in Econophysics via Continuous-Time Random Walk Reprinted from: Mathematics 2020, 8, 641, doi:10.3390/math8040641 ................. 33 Vasily E. Tarasov Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models Reprinted from: Mathematics 2019, 7, 554, doi:10.3390/math7060554 ................. 43 Anatoly N. Kochubei and Yuri Kondratiev Growth Equation of the General Fractional Calculus Reprinted from: Mathematics 2019, 7, 615, doi:10.3390/math7070615 ................. 93 Jean-Philippe Aguilar, Jan Korbel and Yuri Luchko Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations Reprinted from: Mathematics 2019, 7, 796, doi:10.3390/math7090796 .................101 Jonathan Blackledge, Derek Kearney, Marc Lamphiere, Raja Rani and Paddy Walsh Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction Reprinted from: Mathematics 2019, 7, 1057, doi:10.3390/math7111057 ................125 Tomas Skovranek The Mittag-Leffler Fitting of the Phillips Curve Reprinted from: Mathematics 2019, 7, 589, doi:10.3390/math7070589 .................183 Yingkang Xie, Zhen Wang and Bo Meng Stability and Bifurcation of a Delayed Time-Fractional Order Business Cycle Model with a General Liquidity Preference Function and Investment Function Reprinted from: Mathematics 2019, 7, 846, doi:10.3390/math7090846 .................195 Jos´e A. Tenreiro Machado, Maria Eug´enia Mata and Ant´onio M. Lopes Fractional Dynamics and Pseudo-Phase Space of Country Economic Processes Reprinted from: Mathematics 2020, 8, 81, doi:10.3390/math8010081 ..................205 In´es Tejado, Emiliano P´erez and Duarte Val´erio Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction Reprinted from: Mathematics 2020, 8, 50, doi:10.3390/math8010050 ..................223 v Hao Ming, JinRong Wang and Michal Feˇckan The Application of Fractional Calculus in Chinese Economic Growth Models Reprinted from: Mathematics 2019, 7, 665, doi:10.3390/math7080665 .................245 Ertu˘grul Kara¸cuha, Vasil Tabatadze, Kamil Kara¸cuha, Nisa Ozge¨ Onal¨ and Esra Ergun ¨ Deep Assessment Methodology Using Fractional Calculus on Mathematical Modeling and Prediction of Gross Domestic Product per Capita of Countries Reprinted from: Mathematics 2020, 8, 633, doi:10.3390/math8040633 .................251 vi About the Special Issue Editor Vasily E. Tarasov is a leading researcher with the Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, and Professor with the Information Technologies and Applied Mathematics Faculty, Moscow Aviation Institute (National Research University). His fields of research include several topics of applied mathematics, theoretical physics, mathematical economics. He has published over 200 papers in international peer-reviewed scientific journals, including over 50 papers in mathematical economics. He has published 12 scientific books, including “Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media” (Springer, 2010) and “Quantum Mechanics of Non-Hamiltonian and Dissipative Systems” (Elsevier, 2008). He was an Editor of “Handbook of Fractional Calculus with Applications. Volumes 4 and 5” (De Gruyter, 2019). Vasily E. Tarasov is a member of the Editorial Boards of 12 journals, including Fractional Calculus and Applied Analysis (De Gruyter), The European Physical Journal Plus (Springer), Communications in Nonlinear Science and Numerical Simulations (Elsevier), Entropy (MDPI), Computational and Applied Mathematics (Springer), International Journal of Applied and Computational Mathematics (Springer), Mathematics (MDPI), Journal of Applied Nonlinear Dynamics (LH Scientific Publishing LLC), Fractional Differential Calculus (Ele-Math), Fractal and Fractional (MDPI), and others. At present, his h-index is 35 in Web of Science and Scopus with an h-index of 45 on Google Scholar Citations. Web of Science ResearcherID: D-6851-2012; Scopus Author ID: 7202004582; ORCID: 0000-0002-4718-6274; Google Scholar Citations ID: sfFN5B4AAAAJ; IstinaResearcher ID (IRID): 394056. vii mathematics Editorial Mathematical Economics: Application of Fractional Calculus Vasily E. Tarasov 1,2 1 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia; [email protected]; Tel.: +7-495-939-5989 2 Faculty “Information Technologies and Applied Mathematics”, Moscow Aviation Institute (National Research University), 125993 Moscow, Russia Received: 22 April 2020; Accepted: 24 April 2020; Published: 27 April 2020 Mathematical economics is a theoretical and applied science in which economic objects, processes, and phenomena are described by using mathematically formalized language. In this science, models are formulated on the basis of mathematical formalizations of economic concepts and notions. An important purpose of mathematical economics is the formulation of notions and concepts in form, which will be mathematically adequate and self-consistent, and then, on their basis, to construct models of processes and phenomena of economy. The standard mathematical language, which is actively used in mathematical modeling of economy, is the calculus of derivatives and integrals of integer orders, the differential and difference equations. These operators and equations allowed economists to formulate models in mathematical form, and, on this basis, to describe a wide range of processes and phenomena in economy. It is known that the integer-order derivatives of functions are determined by the properties of these functions in an infinitely small neighborhood of the point, in which the derivatives are considered. As a result, economic models, which are based on differential equations of integer orders, cannot describe processes with memory and non-locality. As a result, this mathematical language cannot take into account important aspects of economic processes and phenomena. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. The methods of fractional calculus are powerful tools for describing the processes and systems with memory and nonlocality. There are various types of fractional integral and differential operators that are proposed by Riemann, Liouville, Grunwald, Letnikov, Sonine, Marchaud, Weyl, Riesz, Hadamard, Kober, Erdelyi, Caputo and other mathematicians. The fractional derivatives have a set of nonstandard properties such as a violation of the standard product and chain rules. The violation of the standard form of the product rule is a main characteristic property of derivatives of non-integer orders that allows us to describe complex properties of processes and systems. Recently, fractional integro-differential equations are actively used to describe a wide class of economical processes with power-law memory and spatial nonlocality. Generalizations of basic economic concepts and notions of the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe the economic dynamics with a long memory. The purpose of this Special Issue is to create a collection of articles reflecting the latest mathematical
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