This Special Issue of the scientific journal Axioms, entitled “Recent Advances in Fractional Calculus”, is dedicated to one of the most dynamic areas of mathematical sciences today [...

Kórus Péter

Nápoles Valdés Juan Eduardo

English

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Citation: Kórus, P.; Nápoles Valdés,J.E. Recent Advances in FractionalCalculus. Axioms 2024, 13, 310.https://doi.org/10.3390/axioms13050310Received: 6 May 2024Accepted: 7 May 2024Published: 8 May 2024Copyright: © 2024 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).axiomsEditorialRecent Advances in Fractional CalculusPéter Kórus 1,* and Juan Eduardo Nápoles Valdés 2,*1 Institute of Applied Pedagogy, Juhász Gyula Faculty of Education, University of Szeged,H-6725 Szeged, Hungary2 Facultad de Ciencias Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste,Av. Libertad 5450, Corrientes 3400, Argentina* Correspondence: korus.peter@szte.hu (P.K.); jnapoles@exa.unne.edu.ar (J.E.N.V.)1. IntroductionThis Special Issue of the scientific journal Axioms, entitled “Recent Advances in Frac-tional Calculus”, is dedicated to one of the most dynamic areas of mathematical sciencestoday. For 50 years, the number of researchers and scientific productions dealing with thistopic has been increasing day by day, which clearly demonstrates the growing interest infractional calculus, both from a practical and a theoretical point of view.Fractional calculus is important because it expands the scope of classical calculus,enabling the modeling and analysis of a wide range of complex phenomena in fields suchas physics, engineering, biology, economics, and others. Its flexibility and explanatorypower make it an invaluable tool in scientific research and practical application.The diversity of fractional calculus, and thus of this Special Issue, is well illustratedby the various types of fractional operators considered in the published contributions,such as Caputo-type, Hilfer-type, and Riemann–Liouville-type, and the various types ofinequalities presented, such as Bullen-type, Jensen–Mercer-type, and Hermite–Hadamard-type, in addition to the examined fractional-order differential equations and boundaryvalue problems.2. Overview of the Published PapersThis Special Issue contains 10 articles that were accepted for publication after a rigorousreview process.Ogbu F. Imaga, Samuel A. Iyase, and Peter O. Ogunniyi (Contribution 1) consider theexistence of solutions for a mixed fractional-order boundary value problem at resonanceon the half line, in which Caputo and Riemann–Liouville fractional derivatives appear.Conditions for the existence of solutions to the problem are given using Mawhin’s coinci-dence degree theory when the dimensions of the kernel of the linear fractional differentialoperator are two. At the end of the paper, the main result is applied to an example boundaryvalue problem.Sheza M. El-Deeb and Luminiţa-Ioana Cotîrlă (Contribution 2) introduce and investi-gate the properties of some new subclasses of the class of meromorphic p-valent functions inthe punctured open unit disk. To define these subclasses, a new linear differential operatoris presented by using the combination of q-derivative and convolution. Various propertiesare studied, and results are given for coefficient estimation, distortion bounds, convexfamily, and the concept of neighborhoods and partial sums of analytic functions for theclass in question.Ayub Samadi, Sotiris K. Ntouyas, Bashir Ahmad, and Jessada Tariboon (Contribu-tion 3) investigate a non-linear, non-local, and fully coupled boundary value problemcontaining a generalized Hilfer fractional derivative and generalized Riemann–Liouvillefractional integral operators. Existence and uniqueness results are established by trans-forming the given problem into a fixed-point problem, which facilitates the application ofAxioms 2024, 13, 310. https://doi.org/10.3390/axioms13050310 https://www.mdpi.com/journal/axiomsAxioms 2024, 13, 310 2 of 3fixed-point theorems. The main results are accompanied by three examples. The paperconcludes with some new results arising from the findings as special cases.Ahmed Salem and Kholoud N. Alharbi (Contribution 4) analyze an infinitely delayingsystem of Caputo fractional evolution equations with an infinitesimal generator operator.The authors examine a moderate controllability solution based on two different arguments,one using compactness technology and the other using non-compactness. The first argu-ment is based on Krasnoselskii’s theorem, while the second one is rooted in the Kuratowskimeasure of non-compactness and the Sadovskii fixed-point theorem. They achieve themild solution by assuming that the generator is an infinitesimal generator of a stronglycontinuous cosine family of uniformly bounded linear operators. Finally, the results areillustrated with a numerical example.Isa A. Baba, Usa W. Humphries, Fathalla A. Rihan, and Juan E. Nápoles Valdés (Con-tribution 5) construct a fractional-order COVID-19 model consisting of six compartments inCaputo sense. The model integrates the indirect mode of transmission of the virus, whicha result of the shedding effect. The main achievement of the article is the mathematicaldemonstration of the fact that an uninfected population can become infected via both directand indirect methods by the exposed or infected class. In addition to the analysis of model’smathematical properties (positivity and boundedness, computation of equilibria, basicreproduction number, existence and uniqueness analysis of the solution of the model, localstability analysis), optimal control analysis and numerical simulations are provided.Constantin Fetecău and Costică Moroşanu (Contribution 6) address two main topicsin their paper. The first topic is a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inho-mogeneous dynamic boundary conditions. They extend previously known results byenabling new mathematical models to be more suitable to describe the complexity of awide class of different physical phenomena in life sciences, including moving interfaceproblems, material sciences, digital image processing, and others. The second topic is thedevelopment of an iterative fractional step-type scheme which approximates the non-linearsecond-order reaction–diffusion problem. Convergence and error estimates are establishedfor the proposed numerical scheme, and a conceptual numerical algorithm is formulated.Bahtiyar Bayraktar, Péter Kórus, and Juan Eduardo Nápoles Valdés (Contribution 7)consider convex functions, general convex functions, and differentiable functions whosederivatives, in absolute value, are generally convex. They obtain various relevant Hermite–Hadamard-type fractional inequalities via non-conformable fractional integrals, using theclassical Jensen–Mercer inequality and its variants for general convex functions. In additionto showing that the main results extend previously known results from the literature, theirthree examples illustrate the scope and strength of their results.Mohammad Faisal Khan, Suha B. Al-Shaikh, Ahmad A. Abubaker, and KhaledMatarneh (Contribution 8), starting from the known theory of q-calculus, define a differin-tegral operator for m-fold symmetric functions and obtain a new class of close-to-convexbi-univalent functions. The authors estimate the general Taylor–Maclaurin coefficientbounds, the initial coefficients, and the Fekete–Szegö functional for this class of func-tions using the Faber polynomial expansion method. The results obtained are novel andconsistent with previous research, which is highlighted by some of the obtained corollaries.Asfand Fahad, Saad Ihsaan Butt, Bahtiyar Bayraktar, Mehran Anwar, and Yuan-heng Wang (Contribution 9) establish a new fractional Bullen-type identity for twice-differentiable functions in terms of fractional integral operators. Using convexity properties,the authors obtain some generalized Bullen-type inequalities, which are supplementedwith concrete examples with graphical representations. They provide an analysis of theestimates of boundaries and show that the improved Hölder and power mean inequalitiesgive better results in the upper limit than classical inequalities. Some applications with re-spect to quadrature rules, modified Bessel functions, and digamma functions are providedat the end of the article.Axioms 2024, 13, 310 3 of 3Muhammad Aamir Ali, Thanin Sitthiwirattham, Elisabeth Köbis, and Asma Hanif(Contribution 10) present an integral identity that incorporates a twice-differentiable func-tion. After presenting this equality, some new Hermite–Hadamard–Mercer-type inequali-ties are given for twice-differentiable convex functions. Furthermore, it is demonstratedthat the newly introduced inequalities serve as generalizations of certain inequalities previ-ously established in the literature. Finally, the authors provide some applications whichillustrate the scope and usefulness of their results.Acknowledgments: The Guest Editors of this Special Issue would like to thank all the authors whocontributed their high-quality research papers to this publication. Furthermore, thanks are due to thereviewers and editors, who, through their tireless work, helped make this publication a success.Conflicts of Interest: The authors declare no conflicts of interest.List of Contributions1. Imaga, O.; Iyase, S.; Ogunniyi, P. Existence Results for an m-Point Mixed Fractional-OrderProblem at Resonance on the Half-Line. Axioms 2022, 11, 630. https://doi.org/10.3390/axioms11110630.2. El-Deeb, S.; Cotîrlă, L. Basic Properties for Certain Subclasses of Meromorphic p-Valent Func-tions with Connected q-Analogue of Linear Differential Operator. Axioms 2023, 12, 207.https://doi.org/10.3390/axioms12020207.3. Samadi, A.; Ntouyas, S.; Ahmad, B.; Tariboon, J. On a Coupled Differential System Involving(k,ψ)-Hilfer Derivative and (k,ψ)-Riemann–Liouville Integral Operators. Axioms 2023, 12, 229.https://doi.org/10.3390/axioms12030229.4. Salem, A.; Alharbi, K. Controllability for Fractional Evolution Equations with Infinite Time-Delay and Non-Local Conditions in Compact and Noncompact Cases. Axioms 2023, 12, 264.https://doi.org/10.3390/axioms12030264.5. Baba, I.; Humphries, U.; Rihan, F.; Valdés, J. Fractional–Order Modeling and Control of COVID-19 with Shedding Effect. Axioms 2023, 12, 321. https://doi.org/10.3390/axioms12040321.6. Fetecău, C.; Moroşanu, C. Fractional Step Scheme to Approximate a Non-Linear Second-OrderReaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions. Axioms 2023,12, 406. https://doi.org/10.3390/axioms12040406.7. Bayraktar, B.; Kórus, P.; Nápoles Valdés, J. Some New Jensen–Mercer Type Integral Inequalitiesvia Fractional Operators. Axioms 2023, 12, 517. https://doi.org/10.3390/axioms12060517.8. Khan, M.; Al-Shaikh, S.; Abubaker, A.; Matarneh, K. New Applications of Faber Polynomi-als and q-Fractional Calculus for a New Subclass of m-Fold Symmetric bi-Close-to-ConvexFunctions. Axioms 2023, 12, 600. https://doi.org/10.3390/axioms12060600.9. Fahad, A.; Butt, S.; Bayraktar, B.; Anwar, M.; Wang, Y. Some New Bullen-Type InequalitiesObtained via Fractional Integral Operators. Axioms 2023, 12, 691. https://doi.org/10.3390/axioms12070691.10. Ali, M.; Sitthiwirattham, T.; Köbis, E.; Hanif, A. Hermite–Hadamard–Mercer InequalitiesAssociated with Twice-Differentiable Functions with Applications. Axioms 2024, 13, 114. https://doi.org/10.3390/axioms13020114.Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individualauthor(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury topeople or property resulting from any ideas, methods, instructions or products referred to in the content.

Recent Advances in Fractional Calculus

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Recent Advances in Fractional Calculus

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