Fractional Calculus - Theory and Applications II

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

Deadline for manuscript submissions: 29 July 2024 | Viewed by 11262

Special Issue Editor


E-Mail Website1 Website2
Guest Editor
1. Department of Mathematics, School of Digital Technologies, Tallinn University, 10120 Tallinn, Estonia
2. Department of Mathematics and Physics, Autonomous University of Aguascalientes, Aguascalientes 20131, Mexico
Interests: fractional calculus; difference equations; differential equations
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Special Issue Information

Dear Colleagues,

In recent years, fractional calculus has witnessed tremendous progress in various areas of sciences and mathematics. On one hand, new definitions of fractional derivatives and integrals have appeared in recent years, extending the classical definitions in some sense or another. Moreover, the rigorous analysis of the functional properties of those new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated rigorously from the analytical and numerical points of view, and potential applications have been proposed in the sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications. We invite authors to submit high-quality reports on the analysis of fractional-order differential/integral equations, the analysis of new definitions of fractional derivatives, numerical methods for fractional-order equations, and applications to physical systems governed by fractional differential equations, among other interesting topics of research.

Prof. Dr. Jorge E. Macías Díaz
Guest Editor

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Keywords

  • fractional-order differential/integral equations
  • existence and regularity of solutions
  • numerical methods for fractional equations
  • analysis of convergence and stability
  • applications to science and technology

Published Papers (12 papers)

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Research

15 pages, 813 KiB  
Article
A Note on a Fractional Extension of the Lotka–Volterra Model Using the Rabotnov Exponential Kernel
by Mohamed M. Khader, Jorge E. Macías-Díaz, Alejandro Román-Loera and Khaled M. Saad
Axioms 2024, 13(1), 71; https://doi.org/10.3390/axioms13010071 - 21 Jan 2024
Cited by 1 | Viewed by 849
Abstract
In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) [...] Read more.
In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate formula of the fractional derivative of a power function ζp in terms of the RFE kernel. Next, by using the spectral collocation method (SCM) based on the shifted Vieta–Lucas polynomials (VLPs), the fractional differential system is reduced to a set of algebraic equations. We provide a theoretical convergence analysis for the numerical approach, and the accuracy is verified by evaluating the residual error function through some concrete examples. The results are then contrasted with those derived using the fourth-order Runge-Kutta (RK4) method. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
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21 pages, 470 KiB  
Article
A Note on the Time-Fractional Navier–Stokes Equation and the Double Sumudu-Generalized Laplace Transform Decomposition Method
by Hassan Eltayeb, Imed Bachar and Said Mesloub
Axioms 2024, 13(1), 44; https://doi.org/10.3390/axioms13010044 - 11 Jan 2024
Cited by 1 | Viewed by 811
Abstract
In this work, the time-fractional Navier–Stokes equation is discussed using a calculational method, which is called the Sumudu-generalized Laplace transform decomposition method (DGLTDM). The fractional derivatives are defined in the Caputo sense. The (DGLTDM) is a hybrid of the Sumudu-generalized Laplace transform and [...] Read more.
In this work, the time-fractional Navier–Stokes equation is discussed using a calculational method, which is called the Sumudu-generalized Laplace transform decomposition method (DGLTDM). The fractional derivatives are defined in the Caputo sense. The (DGLTDM) is a hybrid of the Sumudu-generalized Laplace transform and the decomposition method. Three examples of the time-fractional Navier–Stokes equation are studied to check the validity and demonstrate the effectiveness of the current method. The results show that the suggested method succeeds remarkably well in terms of proficiency and can be utilized to study more problems in the field of nonlinear fractional differential equations (FDEs). Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
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19 pages, 348 KiB  
Article
New Majorized Fractional Simpson Estimates
by Xiaoye Ding, Xuewu Zuo, Saad Ihsan Butt, Rafia Farooq and Sanja Tipurić-Spužević
Axioms 2023, 12(10), 965; https://doi.org/10.3390/axioms12100965 - 13 Oct 2023
Viewed by 897
Abstract
Fractional calculus has been a concept used to acquire new variants of some well-known integral inequalities. In this study, our primary goal is to develop majorized fractional Simpson’s type estimates by employing a differentiable function. Practicing majorization theory, we formulate a new auxiliary [...] Read more.
Fractional calculus has been a concept used to acquire new variants of some well-known integral inequalities. In this study, our primary goal is to develop majorized fractional Simpson’s type estimates by employing a differentiable function. Practicing majorization theory, we formulate a new auxiliary identity by utilizing fractional integral operators. In order to obtain new bounds, we employ the idea of convex functions on the Niezgoda–Jensen–Mercer inequality for majorized tuples, along with some fundamental inequalities including the Hölder, power mean, and Young inequalities. Some applications to the quadrature rule and examples for special functions are provided as well. Interestingly, the main findings are the generalizations of many known results in the existing literature. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
19 pages, 551 KiB  
Article
Improvement in Some Inequalities via Jensen–Mercer Inequality and Fractional Extended Riemann–Liouville Integrals
by Abd-Allah Hyder, Areej A. Almoneef and Hüseyin Budak
Axioms 2023, 12(9), 886; https://doi.org/10.3390/axioms12090886 - 17 Sep 2023
Viewed by 603
Abstract
The primary intent of this study is to establish some important inequalities of the Hermite–Hadamard, trapezoid, and midpoint types under fractional extended Riemann–Liouville integrals (FERLIs). The proofs are constructed using the renowned Jensen–Mercer, power-mean, and Holder inequalities. Various equalities for the FERLIs and [...] Read more.
The primary intent of this study is to establish some important inequalities of the Hermite–Hadamard, trapezoid, and midpoint types under fractional extended Riemann–Liouville integrals (FERLIs). The proofs are constructed using the renowned Jensen–Mercer, power-mean, and Holder inequalities. Various equalities for the FERLIs and convex functions are construed to be the mainstay for finding new results. Some connections between our main findings and previous research on Riemann–Liouville fractional integrals and FERLIs are also discussed. Moreover, a number of examples are featured, with graphical representations to illustrate and validate the accuracy of the new findings. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
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16 pages, 410 KiB  
Article
Exploring the Efficiency of the q-Homotopy Analysis Transform Method for Solving a Fractional Initial Boundary Value Problem with a Nonlocal Condition
by Said Mesloub and Huda Alsaud
Axioms 2023, 12(8), 790; https://doi.org/10.3390/axioms12080790 - 15 Aug 2023
Cited by 1 | Viewed by 696
Abstract
This article employs the q-homotopy analysis transformation method (q-HATM) to numerically solve, subject to an integral condition, a fractional IBVP. The resulting numerical scheme is applied to solve, in which the exact solution is obtained, several test examples in order [...] Read more.
This article employs the q-homotopy analysis transformation method (q-HATM) to numerically solve, subject to an integral condition, a fractional IBVP. The resulting numerical scheme is applied to solve, in which the exact solution is obtained, several test examples in order to illustrate its efficiency. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
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19 pages, 459 KiB  
Article
Handling a Commensurate, Incommensurate, and Singular Fractional-Order Linear Time-Invariant System
by Iqbal M. Batiha, Omar Talafha, Osama Y. Ababneh, Shameseddin Alshorm and Shaher Momani
Axioms 2023, 12(8), 771; https://doi.org/10.3390/axioms12080771 - 09 Aug 2023
Cited by 3 | Viewed by 762
Abstract
From the perspective of the importance of the fractional-order linear time-invariant (FoLTI) system in plenty of applied science fields, such as control theory, signal processing, and communications, this work aims to provide certain generic solutions for commensurate and incommensurate cases of these systems [...] Read more.
From the perspective of the importance of the fractional-order linear time-invariant (FoLTI) system in plenty of applied science fields, such as control theory, signal processing, and communications, this work aims to provide certain generic solutions for commensurate and incommensurate cases of these systems in light of the Adomian decomposition method. Accordingly, we also generate another general solution of the singular FoLTI system with the use of the same methodology. Several more numerical examples are given to illustrate the core points of the perturbations of the considered singular FoLTI systems that can ultimately generate a variety of corresponding solutions. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
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16 pages, 294 KiB  
Article
A New Hardy–Hilbert-Type Integral Inequality Involving One Multiple Upper Limit Function and One Derivative Function of Higher Order
by Bicheng Yang and Michael Th. Rassias
Axioms 2023, 12(5), 499; https://doi.org/10.3390/axioms12050499 - 19 May 2023
Viewed by 633
Abstract
Using weight functions and parameters, as well as applying real analytic techniques, we derive a new Hardy–Hilbert-type integral inequality with the homogeneous kernel 1(x+y)λ+n involving one multiple upper limit function and one derivative function of [...] Read more.
Using weight functions and parameters, as well as applying real analytic techniques, we derive a new Hardy–Hilbert-type integral inequality with the homogeneous kernel 1(x+y)λ+n involving one multiple upper limit function and one derivative function of higher order. Certain equivalent statements of the optimal constant factor related to some parameters are considered. A few particular inequalities and the case of reverses are also provided. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
13 pages, 1605 KiB  
Article
Efficient Technique for Solving (3+1)-D Fourth-Order Parabolic PDEs with Time-Fractional Derivatives
by Ramadan A. ZeinEldin, Inderdeep Singh, Gurpreet Singh, Mohammed Elgarhy and Hamiden Abd EI-Wahed Khalifa
Axioms 2023, 12(4), 347; https://doi.org/10.3390/axioms12040347 - 31 Mar 2023
Viewed by 1025
Abstract
In this presented research, a hybrid technique is proposed for solving fourth-order (3+1)-D parabolic PDEs with time-fractional derivatives. For this purpose, we utilized the Elzaki integral transform with the coupling of the homotopy perturbation method (HPM). From performing various numerical experiments, we observed [...] Read more.
In this presented research, a hybrid technique is proposed for solving fourth-order (3+1)-D parabolic PDEs with time-fractional derivatives. For this purpose, we utilized the Elzaki integral transform with the coupling of the homotopy perturbation method (HPM). From performing various numerical experiments, we observed that the presented scheme is simple and accurate with very small computational errors. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
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15 pages, 326 KiB  
Article
Third-Order Differential Subordinations Using Fractional Integral of Gaussian Hypergeometric Function
by Georgia Irina Oros, Gheorghe Oros and Lavinia Florina Preluca
Axioms 2023, 12(2), 133; https://doi.org/10.3390/axioms12020133 - 28 Jan 2023
Cited by 5 | Viewed by 896
Abstract
Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that [...] Read more.
Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
14 pages, 300 KiB  
Article
On New Estimates of q-Hermite–Hadamard Inequalities with Applications in Quantum Calculus
by Saowaluck Chasreechai, Muhammad Aamir Ali, Muhammad Amir Ashraf, Thanin Sitthiwirattham, Sina Etemad, Manuel De la Sen and Shahram Rezapour
Axioms 2023, 12(1), 49; https://doi.org/10.3390/axioms12010049 - 02 Jan 2023
Cited by 5 | Viewed by 1134
Abstract
In this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving [...] Read more.
In this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q-differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
22 pages, 2208 KiB  
Article
Up and Down h-Pre-Invex Fuzzy-Number Valued Mappings and Some Certain Fuzzy Integral Inequalities
by Muhammad Bilal Khan, Hatim Ghazi Zaini, Jorge E. Macías-Díaz and Mohamed S. Soliman
Axioms 2023, 12(1), 1; https://doi.org/10.3390/axioms12010001 - 20 Dec 2022
Cited by 7 | Viewed by 976
Abstract
The objective of the current paper is to incorporate the new class and concepts of convexity and Hermite–Hadamard inequality with the fuzzy Riemann integral operators because almost all classical single-valued and interval-valued convex functions are special cases of fuzzy-number valued convex mappings. Therefore, [...] Read more.
The objective of the current paper is to incorporate the new class and concepts of convexity and Hermite–Hadamard inequality with the fuzzy Riemann integral operators because almost all classical single-valued and interval-valued convex functions are special cases of fuzzy-number valued convex mappings. Therefore, a new class of nonconvex mapping in the fuzzy environment has been defined; up and down h-pre-invex fuzzy-number valued mappings (U.D h-pre-invex F-N∙V∙Ms). With the help of this newly defined class, some new versions of Hermite–Hadamard (HH) type inequalities have been also presented. Moreover, some related inequalities such as HH Fejér- and Pachpatte-type inequalities for U∙D h-pre-invex F-N∙V∙Ms are also introduced. Some exceptional cases have been discussed, which can be seen as applications of the main results. We have provided some nontrivial examples. Finally, we also discuss some future scopes. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
17 pages, 1210 KiB  
Article
Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings
by Muhammad Bilal Khan, Jorge E. Macías-Díaz, Mohamed S. Soliman and Muhammad Aslam Noor
Axioms 2022, 11(11), 622; https://doi.org/10.3390/axioms11110622 - 07 Nov 2022
Cited by 8 | Viewed by 1034
Abstract
In recent years, there has been a significant amount of research on the extension of convex functions which are known as preinvex functions. In this paper, we have used this approach to generalize the preinvex interval-valued function in terms of [...] Read more.
In recent years, there has been a significant amount of research on the extension of convex functions which are known as preinvex functions. In this paper, we have used this approach to generalize the preinvex interval-valued function in terms of (£1, £2)-preinvex interval-valued functions because of its extraordinary applications in both pure and applied mathematics. The idea of (£1, £2)-preinvex interval-valued functions is explained in this work. By using the Riemann integral operator, we obtain Hermite-Hadamard and Fejér-type inequalities for (£1, £2)-preinvex interval-valued functions. To discuss the validity of our main results, we provide non-trivial examples. Some exceptional cases have been discussed that can be seen as applications of main outcomes. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
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