Operators of Fractional Calculus and Their Multidisciplinary Applications, 2nd Edition

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 (23 June 2023) | Viewed by 7223

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Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modelling and optimization
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Special Issue Information

Dear Colleagues,

Current widespread interest in various families of fractional-order integral and derivative operators, such as those named after Riemann–Liouville, Weyl, Hadamard, Grunwald–Letnikov, Riesz, Erdélyi–Kober, Liouville–Caputo, and so on, have stemmed essentially from their demonstrated applications in numerous diverse areas of the mathematical, physical, chemical, engineering, and statistical sciences. These fractional-order operators provide interesting and potentially useful tools for solving ordinary and partial differential equations, as well as integral, differintegral, and integro-differential equations, the fractional-calculus analogues and extensions of each of these equations, and various other problems involving special functions of mathematical physics, applicable analysis and applied mathematics, as well as their extensions and generalizations in one, two and more variables.

In this Special Issue, we invite and welcome review, expository, and original research articles dealing with recent advances in the theory of integrals and derivatives of fractional order and their multidisciplinary applications.

Prof. Dr. Hari Mohan Srivastava
Guest Editor

Manuscript Submission Information

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Keywords

  • operators of fractional integrals and fractional derivatives and their applications
  • chaos and dynamical systems based upon fractional calculus
  • fractional-order ODEs and PDEs
  • fractional-order differintegral and integro-differential equations
  • integrals and derivatives of fractional order associated with special functions of mathematical physics and applied mathematics
  • identities and inequalities involving fractional-order integrals and fractional-order derivatives

Related Special Issue

Published Papers (6 papers)

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21 pages, 690 KiB  
Article
Approximate Controllability of Ψ-Hilfer Fractional Neutral Differential Equation with Infinite Delay
by Chandrabose Sindhu Varun Bose, Ramalingam Udhayakumar, Subramanian Velmurugan, Madhrubootham Saradha and Barakah Almarri
Fractal Fract. 2023, 7(7), 537; https://doi.org/10.3390/fractalfract7070537 - 11 Jul 2023
Cited by 6 | Viewed by 1083
Abstract
In this paper, we explain the approximate controllability of Ψ-Hilfer fractional neutral differential equations with infinite delay. The outcome is demonstrated using the infinitesimal operator, fractional calculus, semigroup theory, and the Krasnoselskii’s fixed point theorem. To begin, we emphasise the presence of [...] Read more.
In this paper, we explain the approximate controllability of Ψ-Hilfer fractional neutral differential equations with infinite delay. The outcome is demonstrated using the infinitesimal operator, fractional calculus, semigroup theory, and the Krasnoselskii’s fixed point theorem. To begin, we emphasise the presence of the mild solution and show that the Ψ-Hilfer fractional system is approximately controllable. Additionally, we present theoretical and practical examples. Full article
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19 pages, 457 KiB  
Article
Hybrid Impulsive Feedback Control for Drive–Response Synchronization of Fractional-Order Multi-Link Memristive Neural Networks with Multi-Delays
by Hongguang Fan, Jiahui Tang, Kaibo Shi and Yi Zhao
Fractal Fract. 2023, 7(7), 495; https://doi.org/10.3390/fractalfract7070495 - 22 Jun 2023
Cited by 4 | Viewed by 927
Abstract
This article addresses the issue of drive–response synchronization in fractional-order multi-link memristive neural networks (FMMNN) with multiple delays, under hybrid impulsive feedback control. To address the impact of multiple delays on system synchronization, an extended fractional-order delayed comparison principle incorporating impulses is established. [...] Read more.
This article addresses the issue of drive–response synchronization in fractional-order multi-link memristive neural networks (FMMNN) with multiple delays, under hybrid impulsive feedback control. To address the impact of multiple delays on system synchronization, an extended fractional-order delayed comparison principle incorporating impulses is established. By leveraging Laplace transform, Mittag–Leffler functions, the generalized comparison principle, and hybrid impulsive feedback control schemes, several new sufficient conditions are derived to ensure synchronization in the addressed FMMNN. Unlike existing studies on fractional-order single-link memristor-based systems, our response network is a multi-link model that considers impulsive effects. Notably, the impulsive gains αi are not limited to a small interval, thus expanding the application range of our approach (αi(2,0)(,2)(0,+)). This feature allows one to choose impulsive gains and corresponding impulsive intervals that are appropriate for the system environment and control requirements. The theoretical results obtained in this study contribute to expanding the relevant theoretical achievements of fractional-order neural networks incorporating memristive characteristics. Full article
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16 pages, 1102 KiB  
Article
Multicorn Sets of z¯k+cm via S-Iteration with h-Convexity
by Asifa Tassaddiq, Muhammad Tanveer, Khuram Israr, Muhammad Arshad, Khurrem Shehzad and Rekha Srivastava
Fractal Fract. 2023, 7(6), 486; https://doi.org/10.3390/fractalfract7060486 - 18 Jun 2023
Cited by 3 | Viewed by 1333
Abstract
Fractals represent important features of our natural environment, and therefore, several scientific fields have recently begun using fractals that employ fixed-point theory. While many researchers are working on fractals (i.e., Mandelbrot and Julia sets), only a very few have focused on multicorn sets [...] Read more.
Fractals represent important features of our natural environment, and therefore, several scientific fields have recently begun using fractals that employ fixed-point theory. While many researchers are working on fractals (i.e., Mandelbrot and Julia sets), only a very few have focused on multicorn sets and their dynamic nature. In this paper, we study the dynamics of multicorn sets of z¯k+cm, where k2, c0C, and mR, by using S-iteration with h-convexity instead of standard S-iteration. We develop escape criterion z¯k+cm for S-iteration with h-convexity. We analyse the dynamical behaviour of the proposed conjugate complex function and discuss the variation of iteration parameters along with function parameter m. Moreover, we discuss the effects of input parameters of the proposed iteration and conjugate complex functions of the behaviour of multicorn sets with numerical simulations. Full article
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10 pages, 340 KiB  
Article
High-Order Nonlinear Functional Differential Equations: New Monotonic Properties and Their Applications
by Hail S. Alrashdi, Osama Moaaz, Ghada AlNemer and Elmetwally M. Elabbasy
Fractal Fract. 2023, 7(3), 271; https://doi.org/10.3390/fractalfract7030271 - 20 Mar 2023
Viewed by 963
Abstract
We provide streamlined criteria for evaluating the oscillatory behavior of solutions to a class of higher-order functional differential equations in the non-canonical case. We use a comparison approach with first-order equations that have standard oscillation criteria. Normally, in the non-canonical situation, the oscillation [...] Read more.
We provide streamlined criteria for evaluating the oscillatory behavior of solutions to a class of higher-order functional differential equations in the non-canonical case. We use a comparison approach with first-order equations that have standard oscillation criteria. Normally, in the non-canonical situation, the oscillation test requires three independent conditions, but we provide criteria with two-conditions without checking the additional conditions. Lastly, we give examples to highlight the significance of the findings. Full article
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31 pages, 528 KiB  
Article
Almost Periodic Solutions of Abstract Impulsive Volterra Integro-Differential Inclusions
by Wei-Shih Du, Marko Kostić and Daniel Velinov
Fractal Fract. 2023, 7(2), 147; https://doi.org/10.3390/fractalfract7020147 - 3 Feb 2023
Cited by 5 | Viewed by 1215
Abstract
In this paper, we introduce and systematically analyze the classes of (pre-)(B,ρ,(tk))-piecewise continuous almost periodic functions and (pre-)(B,ρ,(tk))-piecewise continuous uniformly recurrent [...] Read more.
In this paper, we introduce and systematically analyze the classes of (pre-)(B,ρ,(tk))-piecewise continuous almost periodic functions and (pre-)(B,ρ,(tk))-piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence (tk) of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions. Full article
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8 pages, 292 KiB  
Brief Report
On a Quadratic Nonlinear Fractional Equation
by Iván Area and Juan J. Nieto
Fractal Fract. 2023, 7(6), 469; https://doi.org/10.3390/fractalfract7060469 - 12 Jun 2023
Cited by 4 | Viewed by 959
Abstract
In this paper, we study a quadratic nonlinear equation from the fractional point of view. An explicit solution is given in terms of the Lambert special function. A new phenomenon appears involving the collapsing of the solution and the blow-up of the derivative. [...] Read more.
In this paper, we study a quadratic nonlinear equation from the fractional point of view. An explicit solution is given in terms of the Lambert special function. A new phenomenon appears involving the collapsing of the solution and the blow-up of the derivative. The explicit representation of the solution reveals the non-elementary nature of the solution. Full article
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