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Fractal Fract
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Fractional Calculus Operators and the Mittag-Leffler Function
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Fractional Calculus Operators and the Mittag-Leffler Function

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 (30 April 2022) | Viewed by 30984

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

Special Issue Editor

Dr. Maja Andrić

E-Mail Website
Guest Editor
Faculty of Civil Engineering, Architecture and Geodesy, University of Split, Matice hrvatske 15, 21000 Split, Croatia
Interests: fractional calculus and its applications; inequalities; convex functions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent years, considerable interest in the theory of fractional calculus has been stimulated due to its many applications in almost all applied sciences, especially in numerical analysis and various fields of physics and engineering. Fractional calculus has enabled the adoption of a theoretical model based on experimental data.

Inequalities which involve integrals of functions and their derivatives, whose study has a history of about a century, are of great importance in mathematics, with far-reaching applications in the theory of differential equations, approximations, and probability, among others.

Fractional differentiation inequalities have applications to fractional differential equations; the most important ones are in establishing uniqueness of the solution of initial problems and giving upper bounds to their solutions. These applications have motivated many researchers in the field of integral inequalities to investigate certain extensions and generalizations using different fractional differential and integral operators.

The Mittag–Leffler function with its generalizations emerges as a solution of fractional order differential or integral equations. Extensions and generalizations of the Mittag–Leffler function have enabled researchers to obtain fractional integral inequalities of different types. Consequently, new results are produced for more generalized fractional integral operators containing the Mittag–Leffler function in their kernels.

Dr. Maja Andrić
Guest Editor

Manuscript Submission Information

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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 calculus
  • Mittag–Leffler function
  • Fractional integral operator
  • Integral inequality
  • Convex function
  • Bound of operator

Published Papers (16 papers)

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Editorial

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3 pages, 193 KiB  
Editorial
Editorial for Special Issue “Fractional Calculus Operators and the Mittag–Leffler Function”
by Maja Andrić
Fractal Fract. 2022, 6(8), 442; https://doi.org/10.3390/fractalfract6080442 - 14 Aug 2022
Viewed by 975
Abstract
Among the numerous applications of the theory of fractional calculus in almost all applied sciences, applications in numerical analysis and various fields of physics and engineering stand out [...] Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)

Research

Jump to: Editorial

15 pages, 587 KiB  
Article
A New Fractional Poisson Process Governed by a Recursive Fractional Differential Equation
by Zhehao Zhang
Fractal Fract. 2022, 6(8), 418; https://doi.org/10.3390/fractalfract6080418 - 29 Jul 2022
Cited by 1 | Viewed by 1137
Abstract
This paper proposes a new fractional Poisson process through a recursive fractional differential governing equation. Unlike the homogeneous Poison process, the Caputo derivative on the probability distribution of k jumps with respect to time is linked to all probability distribution functions of j [...] Read more.
This paper proposes a new fractional Poisson process through a recursive fractional differential governing equation. Unlike the homogeneous Poison process, the Caputo derivative on the probability distribution of k jumps with respect to time is linked to all probability distribution functions of j jumps, where j is a non-negative integer less than or equal to k. The distribution functions of arrival times are derived, while the inter-arrival times are no longer independent and identically distributed. Further, this new fractional Poisson process can be interpreted as a homogeneous Poisson process whose natural time flow has been randomized, and the underlying time randomizing process has been studied. Finally, the conditional distribution of the kth order statistic from random number samples, counted by this fractional Poisson process, is also discussed. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
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15 pages, 335 KiB  
Article
Fractional Integral Inequalities of Hermite–Hadamard Type for (h,g;m)-Convex Functions with Extended Mittag-Leffler Function
by Maja Andrić
Fractal Fract. 2022, 6(6), 301; https://doi.org/10.3390/fractalfract6060301 - 29 May 2022
Cited by 2 | Viewed by 1349
Abstract
Several fractional integral inequalities of the Hermite–Hadamard type are presented for the class of (h,g;m)-convex functions. Applied fractional integral operators contain extended generalized Mittag-Leffler functions as their kernel, thus enabling new fractional integral inequalities that extend [...] Read more.
Several fractional integral inequalities of the Hermite–Hadamard type are presented for the class of (h,g;m)-convex functions. Applied fractional integral operators contain extended generalized Mittag-Leffler functions as their kernel, thus enabling new fractional integral inequalities that extend and generalize the known results. As an application, the upper bounds of fractional integral operators for (h,g;m)-convex functions are given. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
12 pages, 354 KiB  
Article
Dirichlet Averages of Generalized Mittag-Leffler Type Function
by Dinesh Kumar, Jeta Ram and Junesang Choi
Fractal Fract. 2022, 6(6), 297; https://doi.org/10.3390/fractalfract6060297 - 28 May 2022
Cited by 4 | Viewed by 1452
Abstract
Since Gösta Magus Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903 and studied its features in five subsequent notes, passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and [...] Read more.
Since Gösta Magus Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903 and studied its features in five subsequent notes, passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and its various extensions (referred to as Mittag-Leffler type functions) have been researched and applied to a wide range of problems in physics, biology, chemistry, and engineering. In the context of fractional calculus, Mittag-Leffler type functions have been widely studied. Since Carlson established the notion of Dirichlet average and its different variations, these averages have been explored and used in a variety of fields. This paper aims to investigate the Dirichlet and modified Dirichlet averages of the R-function (an extended Mittag-Leffler type function), which are provided in terms of Riemann-Liouville integrals and hypergeometric functions of several variables. Principal findings in this article are (possibly) applicable. This article concludes by addressing an open problem. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
19 pages, 321 KiB  
Article
Hermite-Hadamard Fractional Integral Inequalities via Abel-Gontscharoff Green’s Function
by Yixia Li, Muhammad Samraiz, Ayesha Gul, Miguel Vivas-Cortez and Gauhar Rahman
Fractal Fract. 2022, 6(3), 126; https://doi.org/10.3390/fractalfract6030126 - 23 Feb 2022
Cited by 7 | Viewed by 1752
Abstract
The Hermite-Hadamard inequalities for κ-Riemann-Liouville fractional integrals (R-LFI) are presented in this study using a relatively novel approach based on Abel-Gontscharoff Green’s function. In this new technique, we first established some integral identities. Such identities are used to obtain new results for [...] Read more.
The Hermite-Hadamard inequalities for κ-Riemann-Liouville fractional integrals (R-LFI) are presented in this study using a relatively novel approach based on Abel-Gontscharoff Green’s function. In this new technique, we first established some integral identities. Such identities are used to obtain new results for monotonic functions whose second derivative is convex (concave) in absolute value. Some previously published inequalities are obtained as special cases of our main results. Various applications of our main consequences are also explored to special means and trapezoid-type formulae. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
14 pages, 1246 KiB  
Article
The Mittag-Leffler Function for Re-Evaluating the Chlorine Transport Model: Comparative Analysis
by Abdulrahman F. Aljohani, Abdelhalim Ebaid, Ebrahem A. Algehyne, Yussri M. Mahrous, Carlo Cattani and Hind K. Al-Jeaid
Fractal Fract. 2022, 6(3), 125; https://doi.org/10.3390/fractalfract6030125 - 22 Feb 2022
Cited by 8 | Viewed by 1648
Abstract
This paper re-investigates the mathematical transport model of chlorine used as a water treatment model, when a variable order partial derivative is incorporated for describing the chlorine transport system. This model was introduced in the literature and governed by a fractional partial differential [...] Read more.
This paper re-investigates the mathematical transport model of chlorine used as a water treatment model, when a variable order partial derivative is incorporated for describing the chlorine transport system. This model was introduced in the literature and governed by a fractional partial differential equation (FPDE) with prescribed boundary conditions. The obtained solution in the literature was based on implementing the Laplace transform (LT) combined with the method of residues and expressed in terms of regular exponential functions. However, the present analysis avoids such a method of residues, and thus a new analytical solution is introduced in this paper via Mittag-Leffler functions. Therefore, an effective approach is developed in this paper to solve the chlorine transport model with non-integer order derivative. In addition, our results are compared with several studies in the literature in case of integer-order derivative and the differences in results are explained. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
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20 pages, 355 KiB  
Article
Certain Integral and Differential Formulas Involving the Product of Srivastava’s Polynomials and Extended Wright Function
by Saima Naheed, Shahid Mubeen, Gauhar Rahman, Zareen A. Khan and Kottakkaran Sooppy Nisar
Fractal Fract. 2022, 6(2), 93; https://doi.org/10.3390/fractalfract6020093 - 8 Feb 2022
Cited by 3 | Viewed by 1139
Abstract
Many authors have established various integral and differential formulas involving different special functions in recent years. In continuation, we explore some image formulas associated with the product of Srivastava’s polynomials and extended Wright function by using Marichev–Saigo–Maeda fractional integral and differential operators, Lavoie–Trottier [...] Read more.
Many authors have established various integral and differential formulas involving different special functions in recent years. In continuation, we explore some image formulas associated with the product of Srivastava’s polynomials and extended Wright function by using Marichev–Saigo–Maeda fractional integral and differential operators, Lavoie–Trottier and Oberhettinger integral operators. The obtained outcomes are in the form of the Fox–Wright function. It is worth mentioning that some interesting special cases are also discussed. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
12 pages, 319 KiB  
Article
Generalized k-Fractional Integral Operators Associated with Pólya-Szegö and Chebyshev Types Inequalities
by Zhiqiang Zhang, Ghulam Farid, Sajid Mehmood, Kamsing Nonlaopon and Tao Yan
Fractal Fract. 2022, 6(2), 90; https://doi.org/10.3390/fractalfract6020090 - 6 Feb 2022
Cited by 5 | Viewed by 1344
Abstract
Inequalities related to derivatives and integrals are generalized and extended via fractional order integral and derivative operators. The present paper aims to define an operator containing Mittag-Leffler function in its kernel that leads to deduce many already existing well-known operators. By using this [...] Read more.
Inequalities related to derivatives and integrals are generalized and extended via fractional order integral and derivative operators. The present paper aims to define an operator containing Mittag-Leffler function in its kernel that leads to deduce many already existing well-known operators. By using this generalized operator, some well-known inequalities are studied. The results of this paper reproduce Chebyshev and Pólya-Szegö type inequalities for Riemann-Liouville and many other fractional integral operators. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
13 pages, 815 KiB  
Article
The Mittag–Leffler Functions for a Class of First-Order Fractional Initial Value Problems: Dual Solution via Riemann–Liouville Fractional Derivative
by Abdelhalim Ebaid and Hind K. Al-Jeaid
Fractal Fract. 2022, 6(2), 85; https://doi.org/10.3390/fractalfract6020085 - 2 Feb 2022
Cited by 13 | Viewed by 1474
Abstract
In this paper, a new approach is developed to solve a class of first-order fractional initial value problems. The present class is of practical interest in engineering science. The results are based on the Riemann–Liouville fractional derivative. It is shown that the dual [...] Read more.
In this paper, a new approach is developed to solve a class of first-order fractional initial value problems. The present class is of practical interest in engineering science. The results are based on the Riemann–Liouville fractional derivative. It is shown that the dual solution can be determined for the considered class. The first solution is obtained by means of the Laplace transform and expressed in terms of the Mittag–Leffler functions. The second solution was determined through a newly developed approach and given in terms of exponential and trigonometric functions. Moreover, the results reduce to the ordinary version as the fractional-order tends to unity. Characteristics of the dual solution are discussed in detail. Furthermore, the advantages of the second solution over the first one is declared. It is revealed that the second solution is real at certain values of the fractional-order. Such values are derived theoretically and accordingly, and the behavior of the real solution is shown through several plots. The present analysis may be introduced for obtaining the solution in a straightforward manner for the first time. The developed approach can be further extended to include higher-order fractional initial value problems of oscillatory types. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
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15 pages, 345 KiB  
Article
Fejér–Hadamard Type Inequalities for (α, h-m)-p-Convex Functions via Extended Generalized Fractional Integrals
by Ghulam Farid, Muhammad Yussouf and Kamsing Nonlaopon
Fractal Fract. 2021, 5(4), 253; https://doi.org/10.3390/fractalfract5040253 - 2 Dec 2021
Cited by 4 | Viewed by 1901
Abstract
Integral operators of a fractional order containing the Mittag-Leffler function are important generalizations of classical Riemann–Liouville integrals. The inequalities that are extensively studied for fractional integral operators are the Hadamard type inequalities. The aim of this paper is to find new versions of [...] Read more.
Integral operators of a fractional order containing the Mittag-Leffler function are important generalizations of classical Riemann–Liouville integrals. The inequalities that are extensively studied for fractional integral operators are the Hadamard type inequalities. The aim of this paper is to find new versions of the Fejér–Hadamard (weighted version of the Hadamard inequality) type inequalities for (α, h-m)-p-convex functions via extended generalized fractional integrals containing Mittag-Leffler functions. These inequalities hold simultaneously for different types of well-known convexities as well as for different kinds of fractional integrals. Hence, the presented results provide more generalized forms of the Hadamard type inequalities as compared to the inequalities that already exist in the literature. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
17 pages, 373 KiB  
Article
Certain Recurrence Relations of Two Parametric Mittag-Leffler Function and Their Application in Fractional Calculus
by Dheerandra Shanker Sachan, Shailesh Jaloree and Junesang Choi
Fractal Fract. 2021, 5(4), 215; https://doi.org/10.3390/fractalfract5040215 - 12 Nov 2021
Cited by 5 | Viewed by 3928
Abstract
The purpose of this paper is to develop some new recurrence relations for the two parametric Mittag-Leffler function. Then, we consider some applications of those recurrence relations. Firstly, we express many of the two parametric Mittag-Leffler functions in terms of elementary functions by [...] Read more.
The purpose of this paper is to develop some new recurrence relations for the two parametric Mittag-Leffler function. Then, we consider some applications of those recurrence relations. Firstly, we express many of the two parametric Mittag-Leffler functions in terms of elementary functions by combining suitable pairings of certain specific instances of those recurrence relations. Secondly, by applying Riemann–Liouville fractional integral and differential operators to one of those recurrence relations, we establish four new relations among the Fox–Wright functions, certain particular cases of which exhibit four relations among the generalized hypergeometric functions. Finally, we raise several relevant issues for further research. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
25 pages, 393 KiB  
Article
Hilfer–Hadamard Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions
by Bashir Ahmad and Sotiris K. Ntouyas
Fractal Fract. 2021, 5(4), 195; https://doi.org/10.3390/fractalfract5040195 - 3 Nov 2021
Cited by 13 | Viewed by 1798
Abstract
This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping [...] Read more.
This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying the fixed point theorems due to Krasnoselskiĭ and Schaefer and Leray–Schauder nonlinear alternatives. We demonstrate the application of the main results by presenting numerical examples. We also derive the existence results for the cases of convex and non-convex multifunctions involved in the multi-valued analogue of the problem at hand. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
23 pages, 409 KiB  
Article
Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications
by Junesang Choi, Mohd Idris Qureshi, Aarif Hussain Bhat and Javid Majid
Fractal Fract. 2021, 5(4), 150; https://doi.org/10.3390/fractalfract5040150 - 1 Oct 2021
Cited by 13 | Viewed by 1594
Abstract
In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) [...] Read more.
In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) and 2F1(1/2), we establish six classes of generalized summation formulas for p+2Fp+1 with arguments −1 and 1/2 for any positive integer p. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of 4F3(−1) and 4F3(1/2). Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
9 pages, 326 KiB  
Article
Mittag–Leffler Function as an Approximant to the Concentrated Ferrofluid’s Magnetization Curve
by Petr A. Ryapolov and Eugene B. Postnikov
Fractal Fract. 2021, 5(4), 147; https://doi.org/10.3390/fractalfract5040147 - 30 Sep 2021
Cited by 10 | Viewed by 1392
Abstract
In this work, we show that the static magnetization curve of high-concentrated ferrofluids can be accurately approximated by the Mittag–Leffler function of the inverse external magnetic field. The dependence of the Mittag–Leffler function’s fractional index on physical characteristics of samples is analysed and [...] Read more.
In this work, we show that the static magnetization curve of high-concentrated ferrofluids can be accurately approximated by the Mittag–Leffler function of the inverse external magnetic field. The dependence of the Mittag–Leffler function’s fractional index on physical characteristics of samples is analysed and its growth with the growing degree of system’s dilution is revealed. These results provide a certain background for revealing mechanisms of hindered fluctuations in concentrated solutions of strongly interacting of the magnetic nanoparticles as well as a simple tool for an explicit specification of macroscopic force fields in ferrofluid-based technical systems. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
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12 pages, 304 KiB  
Article
Cesaro Limits for Fractional Dynamics
by Yuri Kondratiev and José da Silva
Fractal Fract. 2021, 5(4), 133; https://doi.org/10.3390/fractalfract5040133 - 22 Sep 2021
Cited by 4 | Viewed by 1509
Abstract
We study the asymptotic behavior of random time changes of dynamical systems. As random time changes we propose three classes which exhibits different patterns of asymptotic decays. The subordination principle may be applied to study the asymptotic behavior of the random time dynamical [...] Read more.
We study the asymptotic behavior of random time changes of dynamical systems. As random time changes we propose three classes which exhibits different patterns of asymptotic decays. The subordination principle may be applied to study the asymptotic behavior of the random time dynamical systems. It turns out that for the special case of stable subordinators explicit expressions for the subordination are known and its asymptotic behavior are derived. For more general classes of random time changes explicit calculations are essentially more complicated and we reduce our study to the asymptotic behavior of the corresponding Cesaro limit. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
13 pages, 512 KiB  
Article
Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method
by Hari M. Srivastava, Abedel-Karrem N. Alomari, Khaled M. Saad and Waleed M. Hamanah
Fractal Fract. 2021, 5(3), 131; https://doi.org/10.3390/fractalfract5030131 - 20 Sep 2021
Cited by 21 | Viewed by 1949
Abstract
Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method [...] Read more.
Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
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