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Fractional Calculus in Natural and Social Sciences
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Fractional Calculus in Natural and Social Sciences

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 5957

Special Issue Editor

Prof. Dr. Vasily E. Tarasov

E-Mail Website
Guest Editor
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia
Interests: fractional calculus; fractional dynamics; mathematical economics; quantum theory; theoretical physics; processes with memory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is a theory of differential and integral operators of arbitrary (integer and non-integer) orders that form a calculus. Fractional derivatives and integrals of non-integer order are powerful tools to describe various processes with spatial nonlocality, long memory, distributed lag, depreciation and aging, fractional spatial, and frequency dispersion. Fractional calculus can be used to describe complex processes and systems in physics and mechanics, biology and chemistry, economics, and sociology.  

This Special Issue of the journal Mathematics (MDPI) invites works on the use of rigorous and proven mathematical results of fractional calculus to describe different types of processes in the natural and social sciences. In this Special Issue of Mathematics, works focused on mathematical problems and methods of applications of fractional calculus are solicited.

Prof. Dr. Vasily E. Tarasov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional calculus
  • fractional derivatives
  • fractional integrals
  • fractional differential equations
  • fractional integral equations
  • fractional differences
  • fractional dynamics
  • spatial nonlocality
  • long memory

Published Papers (2 papers)

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Research

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82 pages, 733 KiB  
Article
Nonlocal Probability Theory: General Fractional Calculus Approach
by Vasily E. Tarasov
Mathematics 2022, 10(20), 3848; https://doi.org/10.3390/math10203848 - 17 Oct 2022
Cited by 15 | Viewed by 1178
Abstract
Nonlocal generalization of the standard (classical) probability theory of a continuous distribution on a positive semi-axis is proposed. An approach to the formulation of a nonlocal generalization of the standard probability theory based on the use of the general fractional calculus in the [...] Read more.
Nonlocal generalization of the standard (classical) probability theory of a continuous distribution on a positive semi-axis is proposed. An approach to the formulation of a nonlocal generalization of the standard probability theory based on the use of the general fractional calculus in the Luchko form is proposed. Some basic concepts of the nonlocal probability theory are proposed, including nonlocal (general fractional) generalizations of probability density, cumulative distribution functions, probability, average values, and characteristic functions. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. Properties of the general fractional probability density function and the general fractional cumulative distribution function are described. The truncated GF probability density function, truncated GF cumulative distribution function, and truncated GF average values are defined. Examples of the general fractional (GF) probability distributions, the corresponding probability density functions, and cumulative distribution functions are described. Nonlocal (general fractional) distributions are described, including generalizations of uniform, degenerate, and exponential type distributions; distributions with the Mittag-Leffler, power law, Prabhakar, Kilbas–Saigo functions; and distributions that are described as convolutions of the operator kernels and standard probability density. Full article
(This article belongs to the Special Issue Fractional Calculus in Natural and Social Sciences)

Review

Jump to: Research

26 pages, 385 KiB  
Review
General Fractional Dynamics
by Vasily E. Tarasov
Mathematics 2021, 9(13), 1464; https://doi.org/10.3390/math9131464 - 22 Jun 2021
Cited by 39 | Viewed by 3822
Abstract
General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with [...] Read more.
General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described. Full article
(This article belongs to the Special Issue Fractional Calculus in Natural and Social Sciences)
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