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Open AccessArticle

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 180

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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Open AccessArticle

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

Viewed by 362

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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Open AccessArticle

Handling a Commensurate, Incommensurate, and Singular Fractional-Order Linear Time-Invariant System

by

,

,

,

and

Axioms 2023, 12(8), 771; https://doi.org/10.3390/axioms12080771 - 09 Aug 2023

Viewed by 410

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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Open AccessArticle

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 418

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

\frac{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

\frac{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)

Open AccessArticle

Efficient Technique for Solving (3+1)-D Fourth-Order Parabolic PDEs with Time-Fractional Derivatives

by

,

,

,

and

Hamiden Abd EI-Wahed Khalifa

Axioms 2023, 12(4), 347; https://doi.org/10.3390/axioms12040347 - 31 Mar 2023

Viewed by 742

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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Open AccessArticle

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 4 | Viewed by 641

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)

Open AccessArticle

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 3 | Viewed by 765

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)

Open AccessArticle

Up and Down

h

-Pre-Invex Fuzzy-Number Valued Mappings and Some Certain Fuzzy Integral Inequalities

by

,

,

and

Axioms 2023, 12(1), 1; https://doi.org/10.3390/axioms12010001 - 20 Dec 2022

Cited by 6 | Viewed by 681

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)

Open AccessArticle

Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings

by

,

,

and

Axioms 2022, 11(11), 622; https://doi.org/10.3390/axioms11110622 - 07 Nov 2022

Cited by 7 | Viewed by 813

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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