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Fractal Fract
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Fractional Calculus and Special Functions with Applications
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Fractional Calculus and Special Functions with 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 March 2021) | Viewed by 24672

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Mehmet Ali Özarslan

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Guest Editor
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Gazimagusa, TRNC, via Mersin 10, Turkey
Interests: special functions; fractional calculus; q-calculus; Korovkin type approximation theory; statistical convergence
Dr. Arran Fernandez

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Guest Editor
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Gazimagusa, TRNC, via Mersin 10, Turkey
Interests: fractional calculus; fractional differential equations; Mittag-Leffler functions; zeta functions; asymptotic analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ivan Area

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Guest Editor
Departamento de Matematica Aplicada II, E.E. Aeronautica e do Espazo, University of Vigo, 32004-Ourense, Spain
Interests: orthogonal polynomials; special functions; fractional calculus

Special Issue Information

Dear Colleagues,

The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between operators of integer order. This field has covered the classical fractional operators such as Riemann–Liouville, Weyl, Caputo, Grunwald–Letnikov, and so on. Also, especially in the last two decades, many new operators have appeared, often defined using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and tempered, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, because of their different properties and behaviours, which are comparable to those of the classical operators.

Special functions, such as the Mittag-Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, Bessel and hyper-Bessel functions, and so on, also have some more classical and fundamental connections with fractional calculus. Some of them, such as the Mittag-Leffler function and its generalisations, appear naturally as solutions of fractional differential equations or fractional difference equations. Furthermore, many interesting relationships between different special functions may be discovered by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, such as the concept of Mittag-Leffler stability.

The aim of this Special Issue is to explore and celebrate the diverse connections between fractional calculus and special functions, as well as their associated applications. We welcome review and research papers covering any of the following topics:

  • Analytical properties of fractional-calculus operators defined using special functions in their kernels;
  • Special functions arising from the solution of fractional-order differential or difference equations;
  • Analytical properties of the special functions that arise from the use of fractional-calculus operators;
  • The application of special functions in the qualitative analysis of problems within fractional calculus;
  • Real-world applications of fractional-calculus operators with special functions in their kernels.

Prof. Dr. Mehmet Ali Ozarslan
Asst. Prof. Dr. Arran Fernandez
Prof. Dr. Ivan Area
Guest Editors

 

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.

Published Papers (10 papers)

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Editorial

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3 pages, 175 KiB  
Editorial
Editorial for Special Issue “Fractional Calculus and Special Functions with Applications”
by Mehmet Ali Özarslan, Arran Fernandez and Iván Area
Fractal Fract. 2021, 5(4), 224; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040224 - 17 Nov 2021
Viewed by 1130
Abstract
The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications due to their properties of interpolation between operators of integer order [...] Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)

Research

Jump to: Editorial

14 pages, 541 KiB  
Article
Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method
by Esmail Bargamadi, Leila Torkzadeh, Kazem Nouri and Amin Jajarmi
Fractal Fract. 2021, 5(3), 70; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030070 - 14 Jul 2021
Cited by 5 | Viewed by 1865
Abstract
In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from [...] Read more.
In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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22 pages, 342 KiB  
Article
On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators
by Mehmet Ali Özarslan and Arran Fernandez
Fractal Fract. 2021, 5(2), 45; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5020045 - 14 May 2021
Cited by 10 | Viewed by 2203
Abstract
Several extensions of the classical Mittag-Leffler function, including multi-parameter and multivariate versions, have been used to define fractional integral and derivative operators. In this paper, we consider a function of one variable with five parameters, a special case of the Fox–Wright function. It [...] Read more.
Several extensions of the classical Mittag-Leffler function, including multi-parameter and multivariate versions, have been used to define fractional integral and derivative operators. In this paper, we consider a function of one variable with five parameters, a special case of the Fox–Wright function. It turns out that the most natural way to define a fractional integral based on this function requires considering it as a function of two variables. This gives rise to a model of bivariate fractional calculus, which is useful in understanding fractional differential equations involving mixed partial derivatives. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
18 pages, 599 KiB  
Article
A Fractional SAIDR Model in the Frame of Atangana–Baleanu Derivative
by Esmehan Uçar, Sümeyra Uçar, Fırat Evirgen and Necati Özdemir
Fractal Fract. 2021, 5(2), 32; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5020032 - 15 Apr 2021
Cited by 27 | Viewed by 2231
Abstract
It is possible to produce mobile phone worms, which are computer viruses with the ability to command the running of cell phones by taking advantage of their flaws, to be transmitted from one device to the other with increasing numbers. In our day, [...] Read more.
It is possible to produce mobile phone worms, which are computer viruses with the ability to command the running of cell phones by taking advantage of their flaws, to be transmitted from one device to the other with increasing numbers. In our day, one of the services to gain currency for circulating these malignant worms is SMS. The distinctions of computers from mobile devices render the existing propagation models of computer worms unable to start operating instantaneously in the mobile network, and this is particularly valid for the SMS framework. The susceptible–affected–infectious–suspended–recovered model with a classical derivative (abbreviated as SAIDR) was coined by Xiao et al., (2017) in order to correctly estimate the spread of worms by means of SMS. This study is the first to implement an Atangana–Baleanu (AB) derivative in association with the fractional SAIDR model, depending upon the SAIDR model. The existence and uniqueness of the drinking model solutions together with the stability analysis are shown through the Banach fixed point theorem. The special solution of the model is investigated using the Laplace transformation and then we present a set of numeric graphics by varying the fractional-order θ with the intention of showing the effectiveness of the fractional derivative. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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18 pages, 636 KiB  
Article
Electrical Circuits RC, LC, and RLC under Generalized Type Non-Local Singular Fractional Operator
by Bahar Acay and Mustafa Inc
Fractal Fract. 2021, 5(1), 9; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5010009 - 12 Jan 2021
Cited by 19 | Viewed by 3048
Abstract
The current study is of interest when performing a useful extension of a crucial physical problem through a non-local singular fractional operator. We provide solutions that include three arbitrary parameters α, ρ, and γ for the Resistance-Capacitance (RC), Inductance-Capacitance (LC), and [...] Read more.
The current study is of interest when performing a useful extension of a crucial physical problem through a non-local singular fractional operator. We provide solutions that include three arbitrary parameters α, ρ, and γ for the Resistance-Capacitance (RC), Inductance-Capacitance (LC), and Resistance-Inductance-Capacitance (RLC) electric circuits utilizing a generalized type fractional operator in the sense of Caputo, called non-local M-derivative. Additionally, to keep the dimensionality of the physical parameter in the proposed model, we use an auxiliary parameter. Owing to the fact that all solutions depend on three parameters unlike the other solutions containing one or two parameters in the literature, the solutions obtained in this study have more general results. On the other hand, in order to observe the advantages of the non-local M-derivative, a comprehensive comparison is carried out in the light of experimental data. We make this comparison for the RC circuit between the non-local M-derivative and Caputo derivative. It is clearly shown on graphs that the fractional M-derivative behaves closer to the experimental data thanks to the added parameters α, ρ, and γ. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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20 pages, 347 KiB  
Article
On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative
by Övgü Gürel Yılmaz, Rabia Aktaş and Fatma Taşdelen
Fractal Fract. 2020, 4(4), 48; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4040048 - 24 Sep 2020
Cited by 7 | Viewed by 2546 | Correction
Abstract
Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on [...] Read more.
Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of F1Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F2 and F3 Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
15 pages, 296 KiB  
Article
Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus
by Arran Fernandez and Iftikhar Husain
Fractal Fract. 2020, 4(3), 45; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030045 - 12 Sep 2020
Cited by 13 | Viewed by 2514
Abstract
Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, [...] Read more.
Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
10 pages, 294 KiB  
Article
A Stochastic Fractional Calculus with Applications to Variational Principles
by Houssine Zine and Delfim F. M. Torres
Fractal Fract. 2020, 4(3), 38; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030038 - 01 Aug 2020
Cited by 9 | Viewed by 2558
Abstract
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, [...] Read more.
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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13 pages, 301 KiB  
Article
Multi-Strip and Multi-Point Boundary Conditions for Fractional Langevin Equation
by Ahmed Salem and Balqees Alghamdi
Fractal Fract. 2020, 4(2), 18; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4020018 - 28 Apr 2020
Cited by 33 | Viewed by 2375
Abstract
In the present paper, we discuss a new boundary value problem for the nonlinear Langevin equation involving two distinct fractional derivative orders with multi-point and multi-nonlocal integral conditions. The fixed point theorems for Schauder and Krasnoselskii–Zabreiko are applied to study the existence results. [...] Read more.
In the present paper, we discuss a new boundary value problem for the nonlinear Langevin equation involving two distinct fractional derivative orders with multi-point and multi-nonlocal integral conditions. The fixed point theorems for Schauder and Krasnoselskii–Zabreiko are applied to study the existence results. The uniqueness of the solution is given by implementing the Banach fixed point theorem. Some examples showing our basic results are provided. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
15 pages, 336 KiB  
Article
Existence and Uniqueness Results for a Coupled System of Caputo-Hadamard Fractional Differential Equations with Nonlocal Hadamard Type Integral Boundary Conditions
by Shorog Aljoudi, Bashir Ahmad and Ahmed Alsaedi
Fractal Fract. 2020, 4(2), 13; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4020013 - 12 Apr 2020
Cited by 19 | Viewed by 2594
Abstract
In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use [...] Read more.
In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use of the Leray-Schauder alternative and contraction mapping principle to derive the desired results. Illustrative examples for the main results are also presented. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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