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Quantum Analysis and Fractional Calculus and Their Multi-Disciplinary Applications

A special issue of Applied Sciences (ISSN 2076-3417). This special issue belongs to the section "Applied Physics General".

Deadline for manuscript submissions: closed (30 January 2023) | Viewed by 4586

Special Issue Editor


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Guest Editor
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 modeling and optimization
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Special Issue Information

Dear Colleagues,

Quantum (or q-) derivatives, quantum (or q-) integrals as well as various operators of fractional-order derivatives and fractional-order integrals are becoming increasingly important in the modeling and analysis of many problems in the mathematical sciences as well as other applied sciences.

As the Guest Editor of this Special Issue, I cordially invite and welcome your review, expository, and original research articles dealing with the recent advances in various potentially useful applications the operators of the quantum (or q-) derivatives, quantum (or q-) integrals as well as fractional-order derivatives and fractional-order integrals.

I look forward to receiving your valuable contributions to this Special Issue.

Prof. Dr. Hari Mohan Srivastava
Guest Editor

Manuscript Submission Information

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Keywords

  • Operators of quantum (or q-) calculus and their applications
  • Derivatives and integrals of fractional order integrals their applications
  • Dynamical systems via fractional calculus
  • Quantum (or q-) calculus in number-theoretic disciplines
  • Special functions of mathematical physics and other applied scientific fields
  • Integral transforms and operational calculus.

Published Papers (3 papers)

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Research

31 pages, 463 KiB  
Article
The Fading Memory Formalism with Mittag-Leffler-Type Kernels as A Generator of Non-Local Operators
by Jordan Hristov
Appl. Sci. 2023, 13(5), 3065; https://doi.org/10.3390/app13053065 - 27 Feb 2023
Cited by 3 | Viewed by 1260
Abstract
Transient heat conduction problems are systematically applied to the fading memory formalism with different Mittag-Leffler-type memory kernels. With such an approach, using various memories naturally results in definitions of various fractional operators. Six examples are given and interpreted from a common perspective, covering [...] Read more.
Transient heat conduction problems are systematically applied to the fading memory formalism with different Mittag-Leffler-type memory kernels. With such an approach, using various memories naturally results in definitions of various fractional operators. Six examples are given and interpreted from a common perspective, covering the most well-liked versions of the Mittag-Leffler function. The fading memory approach was used as a template and demonstrated that, if the constitutive equations are correctly built, it is also possible to directly determine where the hereditary terms are located in the models. Full article
17 pages, 494 KiB  
Article
Differentiation of the Wright Functions with Respect to Parameters and Other Results
by Alexander Apelblat and Francesco Mainardi
Appl. Sci. 2022, 12(24), 12825; https://doi.org/10.3390/app122412825 - 14 Dec 2022
Cited by 1 | Viewed by 991
Abstract
In this work, we discuss the derivatives of the Wright functions (of the first and the second kinds) with respect to parameters. The differentiation of these functions leads to infinite power series with the coefficients being the quotients of the digamma (psi) and [...] Read more.
In this work, we discuss the derivatives of the Wright functions (of the first and the second kinds) with respect to parameters. The differentiation of these functions leads to infinite power series with the coefficients being the quotients of the digamma (psi) and gamma functions. Only in few cases is it possible to obtain the sums of these series in a closed form. The functional form of the power series resembles those derived for the Mittag-Leffler functions. If the Wright functions are treated as generalized Bessel functions, differentiation operations can be expressed in terms of the Bessel functions and their derivatives with respect to the order. In many cases, it is possible to derive the explicit form of the Mittag-Leffler functions by performing simple operations with the Laplacian transforms of the Wright functions. The Laplacian transform pairs of both kinds of Wright functions are discussed for particular values of the parameters. Some transform pairs serve to obtain functional limits by applying the shifted Dirac delta function. We expect that the present analysis would find several applications in physics and more generally in applied sciences. These special functions of the Mittag-Leffler and Wright types have already found application in rheology and in stochastic processes where fractional calculus is relevant. Careful readers can benefit from the new results presented in this paper for novel applications. Full article
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13 pages, 505 KiB  
Article
Analysis of a Stochastic SICR Epidemic Model Associated with the Lévy Jump
by Hari M. Srivastava and Jaouad Danane
Appl. Sci. 2022, 12(17), 8434; https://doi.org/10.3390/app12178434 - 24 Aug 2022
Viewed by 1345
Abstract
We propose and study a Susceptible-Infected-Confined-Recovered (SICR) epidemic model. For the proposed model, the driving forces include (for example) the Brownian motion processes and the jump Lévy noise. Usually, in the existing literature involving epidemiology models, the Lévy noise perturbations are ignored. However, [...] Read more.
We propose and study a Susceptible-Infected-Confined-Recovered (SICR) epidemic model. For the proposed model, the driving forces include (for example) the Brownian motion processes and the jump Lévy noise. Usually, in the existing literature involving epidemiology models, the Lévy noise perturbations are ignored. However, in view of the presence of strong fluctuations in the SICR dynamics, it is worth including these perturbations in SICR epidemic models. Quite frequently, this results in several discontinuities in the processes under investigation. In our present study, we consider our SICR model after justifying its used form, namely, the component related to the Lévy noise. The existence and uniqueness of a global positive solution is established. Under some assumptions, we show the extinction and the persistence of the infection. In order to give some numerical simulations, we illustrate a new numerical method to validate our theoretical findings. Full article
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