Recent Advances in Fractional Differential Equations and Inequalities

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

Deadline for manuscript submissions: 4 June 2024 | Viewed by 4544

Special Issue Editors


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Guest Editor
Department of Mathematics, Baylor University, Waco, TX 76706, USA
Interests: boundary value problems; ordinary differential equations, finite difference equations; dynamic equations on time scales; fractional differential equations; differential inclusions

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Guest Editor
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
Interests: special functions; theory of distributions
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Special Issue Information

Dear Colleagues,

Fractional calculus is a sub-branch of applied mathematics, and fractional analysis has become an absorbing field for scientists and mathematicians due to its widespread applications in modeling, engineering, mathematical biology, financial modeling, and fluid flow.

Furthermore, the combined study of convex analysis and integral inequalities in the frame of fractional operators presents a captivating field of research within mathematics. Convexity and integral inequalities have remarkable uses in probability, optimization theory, information technology, stochastic processes, statistics, integral operator theory, and numerical integration.

We invite researchers to contribute their original and high-quality research and review papers to this Special Issue, with a focus on fractional differential equations and inequalities.

Prof. Dr. Sotiris K. Ntouyas
Prof. Dr. Johnny Henderson
Dr. Kamsing Nonlaopon
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. Axioms 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 2400 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 differential equations
  • fractional differential inclusions
  • fractional integral inequalities
  • quantum integral inequalities
  • fractional initial value problems
  • fractional boundary value problems
  • fractional order nonlinear systems
  • fractional integral equations

Published Papers (6 papers)

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Research

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16 pages, 323 KiB  
Article
A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations
by Milan Medved’, Michal Pospíšil and Eva Brestovanská
Axioms 2024, 13(5), 301; https://doi.org/10.3390/axioms13050301 - 1 May 2024
Viewed by 565
Abstract
In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n1<α<nN are introduced for Cn1–functions. A nonlinear version of the second Henry–Gronwall inequality [...] Read more.
In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n1<α<nN are introduced for Cn1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ–Caputo fractional differential equations are proved. Illustrative examples are given. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)
14 pages, 317 KiB  
Article
Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space
by Fang Li, Ling Yang and Huiwen Wang
Axioms 2024, 13(5), 284; https://doi.org/10.3390/axioms13050284 - 23 Apr 2024
Viewed by 504
Abstract
In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions [...] Read more.
In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)
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18 pages, 328 KiB  
Article
Fractional p-Laplacian Coupled Systems with Multi-Point Boundary Conditions
by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon
Axioms 2023, 12(9), 866; https://doi.org/10.3390/axioms12090866 - 7 Sep 2023
Cited by 1 | Viewed by 639
Abstract
This article is allocated to the existence and uniqueness of solutions for a system of nonlinear differential equations consisting of the Caputo fractional-order derivatives. Our main results are proved via standard tools of fixed point theory. Finally, the presented results are clarified by [...] Read more.
This article is allocated to the existence and uniqueness of solutions for a system of nonlinear differential equations consisting of the Caputo fractional-order derivatives. Our main results are proved via standard tools of fixed point theory. Finally, the presented results are clarified by constructing some examples. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)
11 pages, 1198 KiB  
Article
Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator
by Azzh Saad Alshehry, Humaira Yasmin, Muhammad Wakeel Ahmad, Asfandyar Khan and Rasool Shah
Axioms 2023, 12(9), 825; https://doi.org/10.3390/axioms12090825 - 28 Aug 2023
Cited by 3 | Viewed by 704
Abstract
This paper introduces the optimal auxiliary function method (OAFM) for solving a nonlinear system of Belousov–Zhabotinsky equations. The system is characterized by its complex dynamics and is treated using the Caputo operator and concepts from fractional calculus. The OAFM provides a systematic approach [...] Read more.
This paper introduces the optimal auxiliary function method (OAFM) for solving a nonlinear system of Belousov–Zhabotinsky equations. The system is characterized by its complex dynamics and is treated using the Caputo operator and concepts from fractional calculus. The OAFM provides a systematic approach to obtain approximate analytical solutions by constructing an auxiliary function. By optimizing the parameters of the auxiliary function, an approximate solution is derived that closely matches the behavior of the original system. The effectiveness and accuracy of the OAFM are demonstrated through numerical simulations and comparisons with existing methods. Fractional calculus enhances the understanding and modeling of the nonlinear dynamics in the Belousov–Zhabotinsky system. This study contributes to fractional calculus and nonlinear dynamics, offering a powerful tool for analyzing and solving complex systems such as the Belousov–Zhabotinsky equation. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)
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15 pages, 299 KiB  
Article
Analysis of a Fractional-Order Quadratic Functional Integro-Differential Equation with Nonlocal Fractional-Order Integro-Differential Condition
by Ahmed M. A. El-Sayed, Antisar A. A. Alhamali and Eman M. A. Hamdallah
Axioms 2023, 12(8), 788; https://doi.org/10.3390/axioms12080788 - 14 Aug 2023
Viewed by 725
Abstract
Here, we center on the solvability of a fractional-order quadratic functional integro-differential equation with a nonlocal fractional-order integro-differential condition in the class of continuous functions. The maximal and minimal solutions will be discussed. The continuous dependence of the solutions on a few parameters [...] Read more.
Here, we center on the solvability of a fractional-order quadratic functional integro-differential equation with a nonlocal fractional-order integro-differential condition in the class of continuous functions. The maximal and minimal solutions will be discussed. The continuous dependence of the solutions on a few parameters will be examined. Finally, the problems of conjugate orders and integer orders, and some other problems and remarks will be discussed and presented. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)

Review

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58 pages, 639 KiB  
Review
A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators
by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh
Axioms 2023, 12(7), 719; https://doi.org/10.3390/axioms12070719 - 24 Jul 2023
Viewed by 831
Abstract
A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, [...] Read more.
A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, (θ,hm)p-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, (kp)-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and (p,q)-calculus are also included. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)
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