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Fractal Fract
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Advances in Variable-Order Fractional Calculus and Its Applications
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Advances in Variable-Order Fractional Calculus and Its Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 3706

Special Issue Editor

Prof. Dr. Alexander Fedotov

E-Mail Website
Guest Editor
Department of Physics, Kazan National Research Technical University, 420111 Kazan, Russia
Interests: singular integro-differential; pseude differential equations; fractional calculus

Special Issue Information

Dear Colleagues,

The problem of generalizing the concept of the derivative to fractional orders has been stated from the very beginning of the existence of differential calculus. In particular, as early as 1695, in a letter to L'Hospital, Leibniz wrote about a possible generalization of his definition of the derivative to fractional orders and about the variety of such generalizations. Since that time, the number of generalizations has grown. Many famous mathematicians were involved in this process: Euler, Laplace, Riemann, Liouville and many others.

A new wave of interest in the topic started when the applications of fractional derivatives were found in various areas of geometry, physics, mechanics and other sciences. Then, the number of articles dedicated to the different aspects of fractional order problems became immeasurable.

The focus of this Special Issue is to continue to advance research on topics relating to the theory and numerical implementation of solutions for problems concerning the variable-order fractional calculus. Topics that are invited for submission include (but are not limited to):

  1. The problems of the correct definition of the fractional and variable-order derivatives.
  2. Works devoted to the study of mathematical problems of fractional and variable-order differential equations (the existence and uniqueness of solutions, the dependence of solutions on initial and boundary conditions, the analysis of the stability of solutions, the presence of singular points, the form and meaning of initial and boundary conditions, the form and method of constructing a general solution for the main types of equations, etc.).
  3. Publications on exact methods for solving specific types of equations.
  4. Articles on the development, justification, and implementation of approximate methods for solving equations.
  5. Reviews and discussion papers concerning mainly the history of the problem, advantages and disadvantages of different definitions and their correctness.

Prof. Dr. Alexander Fedotov
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. Fractal and Fractional is an international peer-reviewed open access monthly 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 2700 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 order derivative
  • variable-order derivative
  • variable-order differential equations
  • justification of the approximate methods

Published Papers (4 papers)

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Research

18 pages, 353 KiB  
Article
Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions
by Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev
Fractal Fract. 2024, 8(3), 156; https://doi.org/10.3390/fractalfract8030156 - 10 Mar 2024
Viewed by 774
Abstract
In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the [...] Read more.
In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP. Full article
(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)
15 pages, 1025 KiB  
Article
Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations
by Donal O’Regan, Snezhana Hristova and Ravi P. Agarwal
Fractal Fract. 2024, 8(1), 11; https://doi.org/10.3390/fractalfract8010011 - 22 Dec 2023
Viewed by 988
Abstract
An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study [...] Read more.
An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative. Full article
(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)
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17 pages, 349 KiB  
Article
Lyapunov Functions and Stability Properties of Fractional Cohen–Grossberg Neural Networks Models with Delays
by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Fractal Fract. 2023, 7(10), 732; https://doi.org/10.3390/fractalfract7100732 - 02 Oct 2023
Cited by 1 | Viewed by 700
Abstract
Some inequalities for generalized proportional Riemann–Liouville fractional derivatives (RLGFDs) of convex functions are proven. As a special case, inequalities for the RLGFDs of the most-applicable Lyapunov functions such as the ones defined as a quadratic function or the ones defined by absolute values [...] Read more.
Some inequalities for generalized proportional Riemann–Liouville fractional derivatives (RLGFDs) of convex functions are proven. As a special case, inequalities for the RLGFDs of the most-applicable Lyapunov functions such as the ones defined as a quadratic function or the ones defined by absolute values were obtained. These Lyapunov functions were combined with a modification of the Razumikhin method to study the stability properties of the Cohen–Grossberg model of neural networks with both time-variable and continuously distributed delays, time-varying coefficients, and RLGFDs. The initial-value problem was set and studied. Upper bounds by exponential functions of the solutions were obtained on intervals excluding the initial time. The asymptotic behavior of the solutions of the model was studied. Some of the obtained theoretical results were applied to a particular example. Full article
(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)
33 pages, 514 KiB  
Article
Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels
by Muhammad Bilal Khan, Jorge E. Macías-Díaz, Ali Althobaiti and Saad Althobaiti
Fractal Fract. 2023, 7(7), 567; https://doi.org/10.3390/fractalfract7070567 - 24 Jul 2023
Cited by 1 | Viewed by 771
Abstract
The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known [...] Read more.
The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings (UDET-convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for UDET-convex FNVMs. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems. Full article
(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)
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