Special Functions Associated with Fractional Calculus

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 March 2021) | Viewed by 14588

Special Issue Editors


grade E-Mail Website1 Website2
Guest Editor
Emeritus Research Professor of Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA
Interests: boundary value problems; nonlinear analysis; differential and difference equations; fixed point theory; general inequalities
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
1. Mathematics, Anand International College of Engineering, Jaipur, Rajasthan 303012, India
2. Nonlinear Dynamics Research Center (NDRC), Ajman University, Al Jerf 1, Ajman, United Arab Emirates
Interests: special functions; fractional calculus; integral transform; control theory
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Guest Editor
Department of Mathematics, Poornima College of Engineering, ISI-6, RI- ICO Institutional Area, Sitapura, Jaipur, Rajasthan 302022, India
Interests: special functions and fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Many important functions in applied sciences are defined via improper integrals or series (or infinite products). Those functions are generally called special functions. Special functions contain a very old branch of mathematics. For example, trigonometric functions have been studied for over a thousand years, due mainly to their numerous applications in astronomy. Nonetheless, the origins of their unified and rather complete theory date back to the nineteenth century. From an application point of view, special functions such as important mathematical tools, due to their remarkable properties, are designated so based on their usefulness for the applied scientists and engineers—as Paul Tur´an once remarked, special functions would be more appropriately labeled useful functions. Various special functions, such as Bessel and all cylindrical functions; the Gauss, Kummer, confluent, and generalized hypergeometric functions; the classical orthogonal polynomials; the incomplete Gamma and Beta functions and error functions; the Airy, Whittaker functions; etc., will provide solutions to integer-order differential equations and systems, used as mathematical models. However, there has recently been an increasing interest in and widely extended use of differential equations and systems of fractional order (that is, of arbitrary order) as better models of phenomena of physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. Today, new unified presentation and extensive development of special functions associated with fractional calculus are necessary tools related to the theory of differentiation and integration of arbitrary order (i.e., fractional calculus) and to fractional-order (or multiorder) differential and integral equations.

This Special Issue is to provide a multidisciplinary forum of discussion in diverse branches of mathematics and statistics but also physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. This issue will accept high-quality articles containing original research results and survey articles of exceptional merit. Subject matters should be strongly related to special functions involving fractional calculus. The main objective of this Special Issue is to highlight the importance of fundamental results and techniques of the theory of fractional calculus and let the readers of this issue know about the possibilities of this branch of mathematics. Potential topics include but are not limited to:

  • Fractional calculus;
  • Sequence and series in functional analysis;
  • Generalized fractional calculus and applications;
  • Fractional differential equations;
  • Fractional derivatives and special functions;
  • Various special functions related to generalized fractional calculus;
  • Special functions related to fractional (non-integer) order control systems and equations;
  • Applications of fractional calculus in mechanics;
  • Applications of fractional calculus in physics;
  • Special functions arising in the fractional diffusion-wave equations;
  • Operational method in fractional calculus;
  • Fractional integral inequalities and their q-analogues;
  • Inequalities involving the fractional integral operators;
  • Applications of inequalities for classical and fractional differential equations.

Prof. Dr. Ravi P. Agarwal
Prof. Dr. Praveen Agarwal
Prof. Dr. Shilpi Jain
Guest Editors

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Published Papers (6 papers)

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Research

8 pages, 257 KiB  
Article
Some Unified Integrals for Generalized Mittag-Leffler Functions
by Prakash Singh, Shilpi Jain and Carlo Cattani
Axioms 2021, 10(4), 261; https://doi.org/10.3390/axioms10040261 - 19 Oct 2021
Cited by 4 | Viewed by 1799
Abstract
Here, we ascertain generalized integral formulas concerning the product of the generalized Mittag-Leffler function. These integral formulas are described in the form of the generalized Lauricella series. Some special cases are also presented in terms of the Wright hypergeometric function. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
11 pages, 288 KiB  
Article
An Extension of Beta Function by Using Wiman’s Function
by Rahul Goyal, Shaher Momani, Praveen Agarwal and Michael Th. Rassias
Axioms 2021, 10(3), 187; https://doi.org/10.3390/axioms10030187 - 16 Aug 2021
Cited by 13 | Viewed by 2537
Abstract
The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this [...] Read more.
The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this extended beta function. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
6 pages, 738 KiB  
Article
Certain Unified Integrals Involving a Multivariate Mittag–Leffler Function
by Shilpi Jain, Ravi P. Agarwal, Praveen Agarwal and Prakash Singh
Axioms 2021, 10(2), 81; https://doi.org/10.3390/axioms10020081 - 2 May 2021
Cited by 12 | Viewed by 2042
Abstract
A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms [...] Read more.
A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. We also present some interesting special cases. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
9 pages, 298 KiB  
Article
Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs
by Praveen Agarwal, Jochen Merker and Gregor Schuldt
Axioms 2021, 10(2), 74; https://doi.org/10.3390/axioms10020074 - 24 Apr 2021
Cited by 11 | Viewed by 2427
Abstract
In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of [...] Read more.
In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
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9 pages, 266 KiB  
Article
Pseudo-Lucas Functions of Fractional Degree and Applications
by Clemente Cesarano, Pierpaolo Natalini and Paolo Emilio Ricci
Axioms 2021, 10(2), 51; https://doi.org/10.3390/axioms10020051 - 2 Apr 2021
Cited by 5 | Viewed by 1926
Abstract
In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show [...] Read more.
In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a r×r matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
16 pages, 273 KiB  
Article
On Fractional q-Extensions of Some q-Orthogonal Polynomials
by P. Njionou Sadjang and S. Mboutngam
Axioms 2020, 9(3), 97; https://doi.org/10.3390/axioms9030097 - 12 Aug 2020
Cited by 1 | Viewed by 1824
Abstract
In this paper, we introduce a fractional q-extension of the q-differential operator Dq1 and prove some of its main properties. Next, fractional q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties [...] Read more.
In this paper, we introduce a fractional q-extension of the q-differential operator Dq1 and prove some of its main properties. Next, fractional q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties of the newly-defined functions are given. Finally, a fractional q-difference equation of Gaussian type is introduced and solved by means of the power series method. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
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