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Fractal Fract
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Mittag-Leffler Function: Generalizations and Applications
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Mittag-Leffler Function: Generalizations and 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: 30 September 2024 | Viewed by 3030

Special Issue Editor

Prof. Dr. Juan Luis González-Santander

E-Mail Website
Guest Editor
Department de Mathematics, University of Oviedo, C Leopoldo Calvo Sotelo 18, 33007 Oviedo, Spain
Interests: Mittag-Leffler function; special functions; fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Since the seminal works of the Swedish mathematician Gösta Magnus Mittag-Leffler (1902–1905), where he introduced his famous function as a power series, many generalizations and applications of the Mittag-Leffler function have been published in the literature. This function provides the simplest non-trivial generalization of the exponential function. However, many other special functions have been introduced in the literature as generalizations of the Mittag-Leffler function.

Among the most important generalizations, we found the two-parametric Mittag-Leffler function (introduced and studied by Agarval and Humbert in 1953, and independently by Djrbashyan in 1954), the three parametric Mittag-Leffler function (introduced by Prabhakar in 1971), and other generalized Mittag-Leffler functions (introduced more recently by Kilbas and Saigo in 1995).

Among the most important applications, it is worth mentioning the solution of different types of integral equations in terms of the Mittag-Leffler function. Nevertheless, the most relevant applications come from the special role of the Mittag-Leffler function in Fractional Calculus (known as “The Queen Function of the Fractional Calculus”). Therefore, we found the Mittag-Leffler function in physical fractional models such as linear viscoelasticity, fractional Newton equations, fractional Ohm law, and fractional equations for heat transfer.

The main scope of this Special Issue is to publish recent research papers about new generalizations and applications of the Mittag-Leffler function, Also, papers describing the new analytic and asymptotic properties of this function, as well as its connection to other special functions in order to solve applied problems in Mathematics and Physical Sciences, are welcome. 

Prof. Dr. Juan Luis González-Santander
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

  • the Mittag-Leffler function in fractional calculus
  • the analytical properties of the Mittag-Leffler function
  • the mittag-leffler function in physical models
  • generalizations and extensions of the Mittag-Leffler function
  • the mittag-leffler function’s connection to other special functions
  • applications of the Mittag-Leffler function in engineering problems
  • applications of the Mittag-Leffler function in mathematical problems

Published Papers (4 papers)

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Research

14 pages, 553 KiB  
Article
Calculation of the Relaxation Modulus in the Andrade Model by Using the Laplace Transform
by Juan Luis González-Santander, Giorgio Spada, Francesco Mainardi and Alexander Apelblat
Fractal Fract. 2024, 8(8), 439; https://doi.org/10.3390/fractalfract8080439 - 26 Jul 2024
Viewed by 359
Abstract
In the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model Gαt for the case of rational parameter α=m/n∈(0,1) in [...] Read more.
In the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model Gαt for the case of rational parameter α=m/n∈(0,1) in terms of Mittag–Leffler functions from its Laplace transform G˜αs. It turns out that the expression obtained can be rewritten in terms of Rabotnov functions. Moreover, for the original parameter α=1/3 in the Andrade model, we obtain an expression in terms of Miller-Ross functions. The asymptotic behaviours of Gαt for t→0+ and t→+∞ are also derived applying the Tauberian theorem. The analytical results obtained have been numerically checked by solving the Volterra integral equation satisfied by Gαt by using a successive approximation approach, as well as computing the inverse Laplace transform of G˜αs by using Talbot’s method. Full article
(This article belongs to the Special Issue Mittag-Leffler Function: Generalizations and Applications)
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55 pages, 622 KiB  
Article
Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function
by Marc Jornet
Fractal Fract. 2024, 8(7), 411; https://doi.org/10.3390/fractalfract8070411 - 13 Jul 2024
Cited by 1 | Viewed by 589
Abstract
The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We [...] Read more.
The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. Full article
(This article belongs to the Special Issue Mittag-Leffler Function: Generalizations and Applications)
18 pages, 1193 KiB  
Article
Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function
by Aljowhara H. Honain, Khaled M. Furati, Ibrahim O. Sarumi and Abdul Q. M. Khaliq
Fractal Fract. 2024, 8(6), 319; https://doi.org/10.3390/fractalfract8060319 - 27 May 2024
Viewed by 442
Abstract
The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that [...] Read more.
The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α>1, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α∈(0,1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for Eα,β(−t), t≥0, with α∈(1,2). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants. Full article
(This article belongs to the Special Issue Mittag-Leffler Function: Generalizations and Applications)
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13 pages, 1068 KiB  
Article
Properties of a Class of Analytic Functions Influenced by Multiplicative Calculus
by Kadhavoor R. Karthikeyan and Gangadharan Murugusundaramoorthy
Fractal Fract. 2024, 8(3), 131; https://doi.org/10.3390/fractalfract8030131 - 23 Feb 2024
Cited by 1 | Viewed by 1114
Abstract
Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of [...] Read more.
Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained. Full article
(This article belongs to the Special Issue Mittag-Leffler Function: Generalizations and Applications)
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