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Fractal Fract
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Advanced Trends of Special Functions and Analysis of PDEs
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Advanced Trends of Special Functions and Analysis of PDEs

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 (25 March 2022) | Viewed by 20203

Special Issue Editors

Prof. Dr. Praveen Agarwal

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Guest Editor
1. Department of Mathematics, Anand International College of Engineering, Near Kanota, Agra Road, Jaipur 303012, India
2. Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman AE 346, United Arab Emirates
Interests: special functions; differential Equations; fractional calculus; integral transforms
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hari Mohan Srivastava

grade E-Mail Website
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
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Taekyun Kim

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Guest Editor
Department of Mathematics, Kwangwoon University, Seoul 01897, Korea
Interests: polynomials; number theory; complex analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Shaher Momani

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Guest Editor
Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan
Interests: numerical analysis;differential equations; fractional Calculus; fluid mechanics; nonlinear dynamics

Special Issue Information

Dear Colleagues,

In recent years, special functions have been developed and applied in a variety of fields, such as combinatory, astronomy, applied mathematics, physics, and engineering, owing mainly to their remarkable properties. The main purpose of this Special Issue is to create a forum of recently developed theories and formulas of special functions with their possible applications to other research areas. This Special Issue provides readers with an opportunity to develop an understanding of recent trends of special functions and the skills needed to apply advanced mathematical techniques to solve complex problems in the theory of partial differential equations. Subject matters are normally related to special functions involving mathematical analysis and its numerous applications, as well as more abstract methods in the theory of partial differential equations. The main objective of this Special Issue is to highlight the importance of fundamental results and techniques of the theory of complex analysis for PDEs, and emphasize articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering—particularly those that stress analytical aspects, and novel problems and their solutions. 

Prof. Dr. Praveen Agarwal
Prof. Dr. Hari Mohan Srivastava
Prof. Dr. Taekyun Kim
Prof. Dr. Shaher Momani
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.

Keywords

  • gamma, related functions, and their extensions
  • generalized hypergeometric functions and their extensions
  • classical polynomials and their extensions
  • recently developed and new polynomials
  • zeta functions
  • orthogonalpolynomials
  • PDEs
  • analytical properties and applications of special functions
  • inequalities for special functions
  • integration of products of special functions
  • properties of ordinary and general families of special polynomials
  • fractional calculus involving special functions

Published Papers (10 papers)

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Research

35 pages, 949 KiB  
Article
The Sharp Bounds of Hankel Determinants for the Families of Three-Leaf-Type Analytic Functions
by Muhammad Arif, Omar Mohammed Barukab, Sher Afzal Khan and Muhammad Abbas
Fractal Fract. 2022, 6(6), 291; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6060291 - 26 May 2022
Cited by 11 | Viewed by 1471
Abstract
The theory of univalent functions has shown strong significance in the field of mathematics. It is such a vast and fully applied topic that its applications exist in nearly every field of applied sciences such as nonlinear integrable system theory, fluid dynamics, modern [...] Read more.
The theory of univalent functions has shown strong significance in the field of mathematics. It is such a vast and fully applied topic that its applications exist in nearly every field of applied sciences such as nonlinear integrable system theory, fluid dynamics, modern mathematical physics, the theory of partial differential equations, engineering, and electronics. In our present investigation, two subfamilies of starlike and bounded turning functions associated with a three-leaf-shaped domain were considered. These classes are denoted by BT3l and S3l*, respectively. For the class BT3l, we study various coefficient type problems such as the first four initial coefficients, the Fekete–Szegö and Zalcman type inequalities and the third-order Hankel determinant. Furthermore, the existing third-order Hankel determinant bounds for the second class will be improved here. All the results that we are going to prove are sharp. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
15 pages, 2692 KiB  
Article
On a Class of Partial Differential Equations and Their Solution via Local Fractional Integrals and Derivatives
by Mohammad Abdelhadi, Sharifah E. Alhazmi and Shrideh Al-Omari
Fractal Fract. 2022, 6(4), 210; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6040210 - 08 Apr 2022
Cited by 8 | Viewed by 1596
Abstract
This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave [...] Read more.
This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
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13 pages, 4229 KiB  
Article
Generalized Full Order Observer Subject to Incremental Quadratic Constraint (IQC) for a Class of Fractional Order Chaotic Systems
by Muhammad Marwan, Muhammad Zainul Abidin, Humaira Kalsoom and Maoan Han
Fractal Fract. 2022, 6(4), 189; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6040189 - 29 Mar 2022
Cited by 6 | Viewed by 1713
Abstract
In this paper, we used Lyapunov theory and Linear Matrix Inequalities (LMI) to design a generalized observer by adding more complexity in the output of the dynamic systems. Our designed observer is based on the optimization problem, minimizing error between trajectories of master [...] Read more.
In this paper, we used Lyapunov theory and Linear Matrix Inequalities (LMI) to design a generalized observer by adding more complexity in the output of the dynamic systems. Our designed observer is based on the optimization problem, minimizing error between trajectories of master and slave systems subject to the incremental quadratic constraint. Moreover, an algorithm is given in our paper used to demonstrate a method for obtaining desired observer and gain matrixes, whereas these gain matrixes are obtained with the aid of LMI and incremental multiplier matrix (IMM). Finally, discussion of two examples are an integral part of our study for the explanation of achieved analytical results using MATLAB and SCILAB. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
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31 pages, 440 KiB  
Article
Geometric Properties of a Certain Class of Mittag–Leffler-Type Functions
by Hari M. Srivastava, Anish Kumar, Sourav Das and Khaled Mehrez
Fractal Fract. 2022, 6(2), 54; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6020054 - 22 Jan 2022
Cited by 19 | Viewed by 2331
Abstract
The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are [...] Read more.
The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are discussed for these functions. Some sufficient conditions are also derived so that these functions belong to the Hardy spaces Hp and H∞. Furthermore, the inclusion properties of some modified Mittag–Leffler-type functions are discussed. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
10 pages, 284 KiB  
Article
Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas
by Sabah Iftikhar, Samet Erden, Muhammad Aamir Ali, Jamel Baili and Hijaz Ahmad
Fractal Fract. 2022, 6(1), 33; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6010033 - 10 Jan 2022
Cited by 4 | Viewed by 1557
Abstract
Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on [...] Read more.
Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
12 pages, 299 KiB  
Article
Relevance of Factorization Method to Differential and Integral Equations Associated with Hybrid Class of Polynomials
by Naeem Ahmad, Raziya Sabri, Mohammad Faisal Khan, Mohammad Shadab and Anju Gupta
Fractal Fract. 2022, 6(1), 5; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6010005 - 23 Dec 2021
Cited by 1 | Viewed by 1919
Abstract
This article has a motive to derive a new class of differential equations and associated integral equations for some hybrid families of Laguerre–Gould–Hopper-based Sheffer polynomials. We derive recurrence relations, differential equation, integro-differential equation, and integral equation for the Laguerre–Gould–Hopper-based Sheffer polynomials by using [...] Read more.
This article has a motive to derive a new class of differential equations and associated integral equations for some hybrid families of Laguerre–Gould–Hopper-based Sheffer polynomials. We derive recurrence relations, differential equation, integro-differential equation, and integral equation for the Laguerre–Gould–Hopper-based Sheffer polynomials by using the factorization method. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
19 pages, 336 KiB  
Article
Some Dynamic Inequalities via Diamond Integrals for Function of Several Variables
by Muhammad Bilal, Khuram Ali Khan, Hijaz Ahmad, Ammara Nosheen, Khalid Mahmood Awan, Sameh Askar and Mosleh Alharthi
Fractal Fract. 2021, 5(4), 207; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040207 - 11 Nov 2021
Cited by 3 | Viewed by 1350
Abstract
In this paper, Jensen’s inequality and Fubini’s Theorem are extended for the function of several variables via diamond integrals of time scale calculus. These extensions are used to generalize Hardy-type inequalities with general kernels via diamond integrals for the function of several variables. [...] Read more.
In this paper, Jensen’s inequality and Fubini’s Theorem are extended for the function of several variables via diamond integrals of time scale calculus. These extensions are used to generalize Hardy-type inequalities with general kernels via diamond integrals for the function of several variables. Some Hardy Hilbert and Polya Knop type inequalities are also discussed as special cases. Classical and new inequalities are deduced from the main results using special kernels and particular time scales. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
12 pages, 800 KiB  
Article
Certain Coefficient Estimate Problems for Three-Leaf-Type Starlike Functions
by Lei Shi, Muhammad Ghaffar Khan, Bakhtiar Ahmad, Wali Khan Mashwani, Praveen Agarwal and Shaher Momani
Fractal Fract. 2021, 5(4), 137; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040137 - 24 Sep 2021
Cited by 21 | Viewed by 1660
Abstract
In our present investigation, some coefficient functionals for a subclass relating to starlike functions connected with three-leaf mappings were considered. Sharp coefficient estimates for the first four initial coefficients of the functions of this class are addressed. Furthermore, we obtain the Fekete–Szegö inequality, [...] Read more.
In our present investigation, some coefficient functionals for a subclass relating to starlike functions connected with three-leaf mappings were considered. Sharp coefficient estimates for the first four initial coefficients of the functions of this class are addressed. Furthermore, we obtain the Fekete–Szegö inequality, sharp upper bounds for second and third Hankel determinants, bounds for logarithmic coefficients, and third-order Hankel determinants for two-fold and three-fold symmetric functions. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
15 pages, 4978 KiB  
Article
Influence of Fin Length on Magneto-Combined Convection Heat Transfer Performance in a Lid-Driven Wavy Cavity
by Md. Fayz-Al-Asad, Mehmet Yavuz, Md. Nur Alam, Md. Manirul Alam Sarker and Omar Bazighifan
Fractal Fract. 2021, 5(3), 107; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030107 - 31 Aug 2021
Cited by 14 | Viewed by 2061
Abstract
In the existent study, combined magneto-convection heat exchange in a driven enclosure having vertical fin was analyzed numerically. The finite element system-based GWR procedure was utilized to determine the flow model’s governing equations. A parametric inquiry was executed to review the influence of [...] Read more.
In the existent study, combined magneto-convection heat exchange in a driven enclosure having vertical fin was analyzed numerically. The finite element system-based GWR procedure was utilized to determine the flow model’s governing equations. A parametric inquiry was executed to review the influence of Richardson and Hartmann numbers on flow shape and heat removal features inside a frame. The problem’s resulting numerical outcomes were demonstrated graphically in terms of isotherms, streamlines, velocity sketches, local Nusselt number, global Nusselt number, and global fluid temperature. It was found that the varying lengths of the fin surface have a substantial impact on flow building and heat line sketch. Further, it was also noticed that a relatively fin length is needed to increase the heat exchange rate on the right cool wall at a high Richardson number. The fin can significantly enhance heat removal performance rate from an enclosure to adjacent fluid. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
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14 pages, 296 KiB  
Article
Degenerate Derangement Polynomials and Numbers
by Minyoung Ma and Dongkyu Lim
Fractal Fract. 2021, 5(3), 59; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5030059 - 22 Jun 2021
Cited by 3 | Viewed by 1483
Abstract
In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. [...] Read more.
In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
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