Fractional Operators and Their Applications

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 (31 March 2023) | Viewed by 17139

Special Issue Editors

Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania
Interests: complex analysis; applied mathematics; special functions
Department of Mathematics and Computer Science, Faculty of Informatics and Sciences, University of Oradea, Universitatii Street, 410087 Oradea, Romania
Interests: topological algebra; geometric function theory; inequalities
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The field of fractional calculus, which deals with the study and application of derivatives and integrals of arbitrary order, has a long and strong history. In recent years, certain fractional operators have been investigated and applied in various scientific fields. The complex topic of fractional operators (derivative and integral) continues to be extensively developed. It is well known that, only in recent years, different fractional operators have been included among strong tools in engineering problems. Since they are nonlinear operators, one can provide novel computation techniques.

The aim of this Special Issue is to advance novel works related to the investigation and application of fractional operators. Worth mentioning is also aspects related to the analytic or/and geometric function theory by revealing specific outcomes between certain complex valued functions and complex (differential) equations constructed by different fractional operators.

We cordially invite and welcome researchers to present their novel results and ideas comprising novel advancements in pure and applied mathematics via fractional operators. The concept of fractional order offers many possibilities of applications in mechanics, physical and chemical processes, in engineering or numerical calculations.

Dr. Adriana Catas
Prof. Dr. Alina Alb Lupas
Guest Editors

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Keywords

  • geometric function theory outcomes related to fractional operators
  • differential subordination and superordination results involving fractional operators
  • fractional operators associated with special functions
  • fractional operators involving quantum calculus
  • operators of fractional integrals and fractional derivatives and their applications
  • inequalities and identities associated with fractional-order integrals or fractional-order derivatives
  • related fractional-order ODEs and PDEs
  • fractional operators in mathematical physics
  • fractional operators in engineering and optimization
  • fractional operators in educational technologies

Published Papers (14 papers)

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Research

21 pages, 399 KiB  
Article
Investigation of the Second-Order Hankel Determinant for Sakaguchi-Type Functions Involving the Symmetric Cardioid-Shaped Domain
by Khalil Ullah, Muhammad Arif, Ibtisam Mohammed Aldawish and Sheza M. El-Deeb
Fractal Fract. 2023, 7(5), 376; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7050376 - 30 Apr 2023
Viewed by 935
Abstract
Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of [...] Read more.
Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of problems, such as the first three initial coefficient problems, the Zalcman inequalities, the Fekete–Szegö type results, and the second-order Hankel determinant for families of Sakaguchi-type functions related to the cardioid-shaped domain. Further, we study the logarithmic coefficients for both of these classes. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
16 pages, 371 KiB  
Article
Jackson Differential Operator Associated with Generalized Mittag–Leffler Function
by Adel A. Attiya, Mansour F. Yassen and Abdelhamid Albaid
Fractal Fract. 2023, 7(5), 362; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7050362 - 28 Apr 2023
Cited by 2 | Viewed by 714
Abstract
Quantum calculus plays a significant role in many different branches such as quantum physics, hypergeometric series theory, and other physical phenomena. In our paper and using quantitative calculus, we introduce a new family of normalized analytic functions in the open unit disk, which [...] Read more.
Quantum calculus plays a significant role in many different branches such as quantum physics, hypergeometric series theory, and other physical phenomena. In our paper and using quantitative calculus, we introduce a new family of normalized analytic functions in the open unit disk, which relates to both the generalized Mittag–Leffler function and the Jackson differential operator. By using a differential subordination virtue, we obtain some important properties such as coefficient bounds and the Fekete–Szegő problem. Some results that represent special cases of this family that have been studied before are also highlighted. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
16 pages, 329 KiB  
Article
Differential Subordination and Superordination Results for q-Analogue of Multiplier Transformation
by Alina Alb Lupaş and Adriana Cătaş
Fractal Fract. 2023, 7(2), 199; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7020199 - 17 Feb 2023
Cited by 4 | Viewed by 960
Abstract
The results obtained by the authors in the present paper refer to quantum calculus applications regarding the theories of differential subordination and superordination. These results are established by means of an operator defined as the q-analogue of the multiplier transformation. Interesting differential [...] Read more.
The results obtained by the authors in the present paper refer to quantum calculus applications regarding the theories of differential subordination and superordination. These results are established by means of an operator defined as the q-analogue of the multiplier transformation. Interesting differential subordination and superordination results are derived by the authors involving the functions belonging to a new class of normalized analytic functions in the open unit disc U, which is defined and investigated here by using this q-operator. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
12 pages, 320 KiB  
Article
Third-Order Differential Subordination for Meromorphic Functions Associated with Generalized Mittag-Leffler Function
by Adel A. Attiya, Tamer M. Seoudy and Abdelhamid Albaid
Fractal Fract. 2023, 7(2), 175; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7020175 - 09 Feb 2023
Cited by 4 | Viewed by 967
Abstract
Using the results of third-order differential subordination, we introduce certain families of admissible functions and discuss some applications of third-order differential subordination for meromorphic functions associated with a linear operator containing a generalized Mittag-Leffler function. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
16 pages, 333 KiB  
Article
Starlike Functions Based on Ruscheweyh q−Differential Operator defined in Janowski Domain
by Luminiţa-Ioana Cotîrlǎ and Gangadharan Murugusundaramoorthy
Fractal Fract. 2023, 7(2), 148; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7020148 - 03 Feb 2023
Cited by 5 | Viewed by 931
Abstract
In this paper, we make use of the concept of qcalculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of Janowski type starlike functions and to find the Fekete–Szegö functional. A similar results have [...] Read more.
In this paper, we make use of the concept of qcalculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of Janowski type starlike functions and to find the Fekete–Szegö functional. A similar results have been done for the function 1. Further, for functions in newly defined class we determine coefficient estimates, distortion bounds, radius problems, results related to partial sums. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
11 pages, 384 KiB  
Article
On Ψ-Hilfer Fractional Integro-Differential Equations with Non-Instantaneous Impulsive Conditions
by Ramasamy Arul, Panjayan Karthikeyan, Kulandhaivel Karthikeyan, Palanisamy Geetha, Ymnah Alruwaily, Lamya Almaghamsi and El-sayed El-hady
Fractal Fract. 2022, 6(12), 732; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6120732 - 10 Dec 2022
Cited by 3 | Viewed by 1053
Abstract
We establish sufficient conditions for the existence of solutions of an integral boundary value problem for a Ψ-Hilfer fractional integro-differential equations with non-instantaneous impulsive conditions. The main results are proved with a suitable fixed point theorem. An example is given to interpret [...] Read more.
We establish sufficient conditions for the existence of solutions of an integral boundary value problem for a Ψ-Hilfer fractional integro-differential equations with non-instantaneous impulsive conditions. The main results are proved with a suitable fixed point theorem. An example is given to interpret the theoretical results. In this way, we generalize recent interesting results. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
16 pages, 339 KiB  
Article
Hankel and Symmetric Toeplitz Determinants for a New Subclass of q-Starlike Functions
by Isra Al-shbeil, Jianhua Gong, Shahid Khan, Nazar Khan, Ajmal Khan, Mohammad Faisal Khan and Anjali Goswami
Fractal Fract. 2022, 6(11), 658; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6110658 - 07 Nov 2022
Cited by 12 | Viewed by 1046
Abstract
This paper considers the basic concepts of q-calculus and the principle of subordination. We define a new subclass of q-starlike functions related to the Salagean q-differential operator. For this class, we investigate initial coefficient estimates, Hankel determinants, Toeplitz matrices, and [...] Read more.
This paper considers the basic concepts of q-calculus and the principle of subordination. We define a new subclass of q-starlike functions related to the Salagean q-differential operator. For this class, we investigate initial coefficient estimates, Hankel determinants, Toeplitz matrices, and Fekete-Szegö problem. Moreover, we consider the q-Bernardi integral operator to discuss some applications in the form of some results. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
10 pages, 318 KiB  
Article
Optimal Control for k × k Cooperative Fractional Systems
by Hassan M. Serag, Abd-Allah Hyder, Mahmoud El-Badawy and Areej A. Almoneef
Fractal Fract. 2022, 6(10), 559; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6100559 - 02 Oct 2022
Viewed by 829
Abstract
This paper discusses the optimal control issue for elliptic k×k cooperative fractional systems. The fractional operators are proposed in the Laplace sense. Because of the nonlocality of the Laplace fractional operators, we reformulate the issue as an extended issue on a [...] Read more.
This paper discusses the optimal control issue for elliptic k×k cooperative fractional systems. The fractional operators are proposed in the Laplace sense. Because of the nonlocality of the Laplace fractional operators, we reformulate the issue as an extended issue on a semi-infinite cylinder in Rk+1. The weak solution for these fractional systems is then proven to exist and be unique. Moreover, the existence and optimality conditions can be inferred as a consequence. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
12 pages, 333 KiB  
Article
Briot–Bouquet Differential Subordinations for Analytic Functions Involving the Struve Function
by Asena Çetinkaya and Luminita-Ioana Cotîrlă
Fractal Fract. 2022, 6(10), 540; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6100540 - 25 Sep 2022
Cited by 1 | Viewed by 970
Abstract
We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the [...] Read more.
We define a new class of exponential starlike functions constructed by a linear operator involving normalized form of the generalized Struve function. Making use of a technique of differential subordination introduced by Miller and Mocanu, we investigate several new results related to the Briot–Bouquet differential subordinations for the linear operator involving the normalized form of the generalized Struve function. We also obtain univalent solutions to the Briot–Bouquet differential equations and observe that these solutions are the best dominant of the Briot–Bouquet differential subordinations for the exponential starlike function class. Moreover, we give an application of fractional integral operator for a complex-valued function associated with the generalized Struve function. The significance of this paper is due to the technique employed in proving the results and novelty of these results for the Struve functions. The approach used in this paper can lead to several new problems in geometric function theory associated with special functions. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
23 pages, 1415 KiB  
Article
Discriminant and Root Trajectories of Characteristic Equation of Fractional Vibration Equation and Their Effects on Solution Components
by Jun-Sheng Duan and Yun-Yun Zhang
Fractal Fract. 2022, 6(9), 514; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6090514 - 13 Sep 2022
Cited by 3 | Viewed by 1000
Abstract
The impulsive response of the fractional vibration equation z(t)+bDtαz(t)+cz(t)=F(t), [...] Read more.
The impulsive response of the fractional vibration equation z(t)+bDtαz(t)+cz(t)=F(t), b>0,c>0,0α2, is investigated by using the complex path-integral formula of the inverse Laplace transform. Similar to the integer-order case, the roots of the characteristic equation s2+bsα+c=0 must be considered. It is proved that for any b>0, c>0 and α(0,1)(1,2), the characteristic equation always has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface. Particular attention is paid to the problem as to how the couple conjugated complex roots approach the two roots of the integer case α=1, especially to the two different real roots in the case of b24c>0. On the upper-half complex plane, the root s(α) is investigated as a function of order α and with parameters b and c, and so are the argument θ(α), modulus r(α), real part λ(α) and imaginary part ω(α) of the root s(α). For the three cases of the discriminant b24c: >0, =0 and <0, variations of the argument and modulus of the roots according to α are clarified, and the trajectories of the roots are simulated. For the case of b24c<0, the trajectories of the roots are further clarified according to the change rates of the argument, real part and imaginary part of root s(α) at α=1. The solution components, i.e., the residue contribution and the Hankel integral contribution to the impulsive response, are distinguished for the three cases of the discriminant. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
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15 pages, 342 KiB  
Article
On Sharp Estimate of Third Hankel Determinant for a Subclass of Starlike Functions
by Lei Shi, Muhammad Arif, Khalil Ullah, Naseer Alreshidi and Meshal Shutaywi
Fractal Fract. 2022, 6(8), 437; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6080437 - 11 Aug 2022
Cited by 4 | Viewed by 1035
Abstract
In our present investigation, a subclass of starlike function Sn1,L* connected with a domain bounded by an epicycloid with n1 cusps was considered. The main work is to investigate some coefficient inequalities, and second and [...] Read more.
In our present investigation, a subclass of starlike function Sn1,L* connected with a domain bounded by an epicycloid with n1 cusps was considered. The main work is to investigate some coefficient inequalities, and second and third Hankel determinants for functions belonging to this class. In particular, we calculate the sharp bounds of the third Hankel determinant for fS4L* with zf(z)f(z) bounded by a four-leaf shaped domain under the unit disk D. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
21 pages, 418 KiB  
Article
Some New Estimates on Coordinates of Generalized Convex Interval-Valued Functions
by Muhammad Bilal Khan, Adriana Cătaş and Omar Mutab Alsalami
Fractal Fract. 2022, 6(8), 415; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6080415 - 28 Jul 2022
Cited by 12 | Viewed by 1048
Abstract
The theory of convex and nonconvex mapping has a lot of applications in the field of applied mathematics and engineering. The Riemann integrals are the most significant operator of interval theory, which permits the generalization of the classical theory of integrals. This study [...] Read more.
The theory of convex and nonconvex mapping has a lot of applications in the field of applied mathematics and engineering. The Riemann integrals are the most significant operator of interval theory, which permits the generalization of the classical theory of integrals. This study considers the well-known coordinated interval-valued Hermite–Hadamard-type and associated inequalities. To full fill this mileage, we use the introduced coordinated interval left and right preinvexity (LR-preinvexity) and Riemann integrals for further extension. Moreover, we have introduced some new important classes of interval-valued coordinated LR-preinvexity (preincavity), which are known as lower coordinated preinvex (preincave) and upper preinvex (preincave) interval-valued mappings, by applying some mild restrictions on coordinated preinvex (preincave) interval-valued mappings. By using these definitions, we have acquired many classical and new exceptional cases that can be viewed as applications of the main results. We also present some examples of interval-valued coordinated LR-preinvexity to demonstrate the validity of the inclusion relations proposed in this paper. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
19 pages, 423 KiB  
Article
Generalized Fractional Integral Inequalities for p-Convex Fuzzy Interval-Valued Mappings
by Muhammad Bilal Khan, Adriana Cătaș and Tareq Saeed
Fractal Fract. 2022, 6(6), 324; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6060324 - 09 Jun 2022
Cited by 13 | Viewed by 1345
Abstract
The fuzzy order relation and fuzzy inclusion relation are two different relations in fuzzy-interval calculus. Due to the importance of p-convexity, in this article we consider the introduced class of nonconvex fuzzy-interval-valued mappings known as p-convex fuzzy-interval-valued mappings ( [...] Read more.
The fuzzy order relation and fuzzy inclusion relation are two different relations in fuzzy-interval calculus. Due to the importance of p-convexity, in this article we consider the introduced class of nonconvex fuzzy-interval-valued mappings known as p-convex fuzzy-interval-valued mappings (p-convex f-i-v-ms) through fuzzy order relation. With the support of a fuzzy generalized fractional operator, we establish a relationship between p-convex f-i-v-ms and Hermite–Hadamard (ℋ–ℋ) inequalities. Moreover, some related ℋ–ℋ inequalities are also derived by using fuzzy generalized fractional operators. Furthermore, we show that our conclusions cover a broad range of new and well-known inequalities for p-convex f-i-v-ms, as well as their variant forms as special instances. The theory proposed in this research is shown, with practical examples that demonstrate its usefulness. These findings and alternative methodologies may pave the way for future research in fuzzy optimization, modeling, and interval-valued mappings (i-v-m). Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
14 pages, 930 KiB  
Article
Bazilevič Functions of Complex Order with Respect to Symmetric Points
by Daniel Breaz, Kadhavoor R. Karthikeyan and Gangadharan Murugusundaramoorthy
Fractal Fract. 2022, 6(6), 316; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6060316 - 05 Jun 2022
Cited by 2 | Viewed by 1469
Abstract
In this paper, we familiarize a class of multivalent functions with respect to symmetric points related to the differential operator and discuss the impact of Janowski functions on conic regions. Inclusion results, the subordination property, and coefficient inequalities are obtained. Further, the applications [...] Read more.
In this paper, we familiarize a class of multivalent functions with respect to symmetric points related to the differential operator and discuss the impact of Janowski functions on conic regions. Inclusion results, the subordination property, and coefficient inequalities are obtained. Further, the applications of our results that are extensions of those given in earlier works are presented as corollaries. Full article
(This article belongs to the Special Issue Fractional Operators and Their Applications)
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