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3 pages, 171 KiB

Open AccessEditorial

Recent Advances in Fractional Calculus

by Péter Kórus and Juan Eduardo Nápoles Valdés

Axioms 2024, 13(5), 310; https://doi.org/10.3390/axioms13050310 - 8 May 2024

Viewed by 955

Abstract

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 [...] Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

Research

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8 pages, 221 KiB

Open AccessArticle

Hermite–Hadamard–Mercer Inequalities Associated with Twice-Differentiable Functions with Applications

by Muhammad Aamir Ali, Thanin Sitthiwirattham, Elisabeth Köbis and Asma Hanif

Axioms 2024, 13(2), 114; https://doi.org/10.3390/axioms13020114 - 8 Feb 2024

Cited by 3 | Viewed by 1164

Abstract

In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain [...] Read more.

In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain pre-existing inequalities found in the literature. Finally, we provide applications to the newly established inequalities to verify their usefulness. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

26 pages, 405 KiB

Open AccessArticle

Some New Bullen-Type Inequalities Obtained via Fractional Integral Operators

by Asfand Fahad, Saad Ihsaan Butt, Bahtiyar Bayraktar, Mehran Anwar and Yuanheng Wang

Axioms 2023, 12(7), 691; https://doi.org/10.3390/axioms12070691 - 16 Jul 2023

Cited by 10 | Viewed by 1931

Abstract

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate [...] Read more.

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Hölder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

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18 pages, 339 KiB

Open AccessArticle

New Applications of Faber Polynomials and q-Fractional Calculus for a New Subclass of m-Fold Symmetric bi-Close-to-Convex Functions

by Mohammad Faisal Khan, Suha B. Al-Shaikh, Ahmad A. Abubaker and Khaled Matarneh

Axioms 2023, 12(6), 600; https://doi.org/10.3390/axioms12060600 - 16 Jun 2023

Cited by 1 | Viewed by 1017

Abstract

Using the concepts of q-fractional calculus operator theory, we first define a

(λ, q)

-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient [...] Read more.

Using the concepts of q-fractional calculus operator theory, we first define a

(λ, q)

-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient bounds for a newly established class using the Faber polynomial expansion method. In addition, the Faber polynomial method is used to examine the Fekete–Szegö problem and the unpredictable behavior of the initial coefficient bounds of the functions that belong to the newly established class of m-fold symmetric bi-close-to-convex functions. Our key results are both novel and consistent with prior research, so we highlight a few of their important corollaries for a comparison. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

17 pages, 319 KiB

Open AccessArticle

Some New Jensen–Mercer Type Integral Inequalities via Fractional Operators

by Bahtiyar Bayraktar, Péter Kórus and Juan Eduardo Nápoles Valdés

Axioms 2023, 12(6), 517; https://doi.org/10.3390/axioms12060517 - 25 May 2023

Cited by 2 | Viewed by 1743

Abstract

In this study, we present new variants of the Hermite–Hadamard inequality via non-conformable fractional integrals. These inequalities are proven for convex functions and differentiable functions whose derivatives in absolute value are generally convex. Our main results are established using the classical Jensen–Mercer inequality [...] Read more.

In this study, we present new variants of the Hermite–Hadamard inequality via non-conformable fractional integrals. These inequalities are proven for convex functions and differentiable functions whose derivatives in absolute value are generally convex. Our main results are established using the classical Jensen–Mercer inequality and its variants for

(h, m)

-convex modified functions proven in this paper. In addition to showing that our results support previously known results from the literature, we provide examples of their application. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

23 pages, 382 KiB

Open AccessArticle

Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions

by Constantin Fetecău and Costică Moroşanu

Axioms 2023, 12(4), 406; https://doi.org/10.3390/axioms12040406 - 21 Apr 2023

Cited by 3 | Viewed by 1958

Abstract

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data: [...] Read more.

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data:

g_{_{d}} (t, x)

,

g_{_{f r}} (t, x)

,

U_{0} (x)

and

ζ_{0} (x)

, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a solution in the space

W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

. Here, we extend previous results, enabling new mathematical models to be more suitable to describe the complexity of a wide class of different physical phenomena of life sciences, including moving interface problems, material sciences, digital image processing, automatic vehicle detection and tracking, the spread of an epidemic infection, semantic image segmentation including U-Net neural networks, etc. The second goal is to develop an iterative splitting scheme, corresponding to the non-linear second-order reaction–diffusion problem. Results relating to the convergence of the approximation scheme and error estimation are also established. On the basis of the proposed numerical scheme, we formulate the algorithm alg-frac_sec-ord_dbc, which represents a delicate challenge for our future works. The benefit of such a method could simplify the process of numerical computation. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

22 pages, 2402 KiB

Open AccessArticle

Fractional–Order Modeling and Control of COVID-19 with Shedding Effect

by Isa A. Baba, Usa W. Humphries, Fathalla A. Rihan and J. E. N. Valdés

Axioms 2023, 12(4), 321; https://doi.org/10.3390/axioms12040321 - 24 Mar 2023

Cited by 2 | Viewed by 1517

Abstract

A fractional order COVID-19 model consisting of six compartments in Caputo sense is constructed. The indirect transmission of the virus through susceptible populations by the shedding effect is studied. Equilibrium solutions are calculated, and basic reproduction ratio (that depends both on direct and [...] Read more.

A fractional order COVID-19 model consisting of six compartments in Caputo sense is constructed. The indirect transmission of the virus through susceptible populations by the shedding effect is studied. Equilibrium solutions are calculated, and basic reproduction ratio (that depends both on direct and indirect mode of transmission), existence and uniqueness, as well as stability analysis of the solution of the model, are studied. The paper studies the effect of optimal control policy applied to shedding effect. The control is the observation of standard hygiene practices and chemical disinfectants in public spaces. Numerical simulations are carried out to support the analytic result and to show the significance of the fractional order from the biological viewpoint. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

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17 pages, 359 KiB

Open AccessArticle

Controllability for Fractional Evolution Equations with Infinite Time-Delay and Non-Local Conditions in Compact and Noncompact Cases

by Ahmed Salem and Kholoud N. Alharbi

Axioms 2023, 12(3), 264; https://doi.org/10.3390/axioms12030264 - 3 Mar 2023

Cited by 3 | Viewed by 1505

Abstract

The goal of this dissertation is to explore a system of fractional evolution equations with infinitesimal generator operators and an infinite time delay with non-local conditions. It turns out that there are two ways to regulate the solution. To demonstrate the presence of [...] Read more.

The goal of this dissertation is to explore a system of fractional evolution equations with infinitesimal generator operators and an infinite time delay with non-local conditions. It turns out that there are two ways to regulate the solution. To demonstrate the presence of the controllability of mild solutions, it is usual practice to apply Krasnoselskii’s theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness. A fractional Caputo approach of order between 1 and 2 was used to construct our model. The families of linear operators cosine and sine, which are strongly continuous and uniformly bounded, are used to achieve the mild solution. To make our results seem to be applicable, a numerical example is provided. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

16 pages, 350 KiB

Open AccessArticle

On a Coupled Differential System Involving (k,ψ)-Hilfer Derivative and (k,ψ)-Riemann–Liouville Integral Operators

by Ayub Samadi, Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon

Axioms 2023, 12(3), 229; https://doi.org/10.3390/axioms12030229 - 22 Feb 2023

Cited by 2 | Viewed by 3603

Abstract

We investigate a nonlinear, nonlocal, and fully coupled boundary value problem containing mixed

(k, \hat{ψ})

-Hilfer fractional derivative and

(k, \hat{ψ})

-Riemann–Liouville fractional integral operators. Existence and uniqueness results for the given problem are proved [...] Read more.

We investigate a nonlinear, nonlocal, and fully coupled boundary value problem containing mixed

(k, \hat{ψ})

-Hilfer fractional derivative and

(k, \hat{ψ})

-Riemann–Liouville fractional integral operators. Existence and uniqueness results for the given problem are proved with the aid of standard fixed point theorems. Examples illustrating the main results are presented. The paper concludes with some interesting findings. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

12 pages, 323 KiB

Open AccessArticle

Basic Properties for Certain Subclasses of Meromorphic p-Valent Functions with Connected q-Analogue of Linear Differential Operator

by Sheza M. El-Deeb and Luminiţa-Ioana Cotîrlă

Axioms 2023, 12(2), 207; https://doi.org/10.3390/axioms12020207 - 15 Feb 2023

Cited by 2 | Viewed by 1336

Abstract

In this paper, we define three subclasses

M_{p, α}^{n, q} (η, A, B), I_{p, α}^{n} (λ, μ, γ),

, R [...] Read more.

In this paper, we define three subclasses

M_{p, α}^{n, q} (η, A, B), I_{p, α}^{n} (λ, μ, γ),

, R

_{p}^{n, q} (λ, μ, γ)

connected with a q-analogue of linear differential operator

D_{α, p, G}^{n, q}

which consist of functions

F

of the form

F (ζ) = ζ^{- p} + \sum_{j = 1 - p}^{\infty} a_{j} ζ^{j}

(p \in N)

satisfying the subordination condition

p - \frac{1}{η} \{\frac{ζ {(D_{α, p, G}^{n, q} F (ζ))}^{^{'}}}{D_{α, p, G}^{n, q} F (ζ)} + p\} ≺ p \frac{1 + A ζ}{1 + B ζ} .

Also, we study the various properties and characteristics of this subclass

M_{p, α}^{n, q, *} (η, A, B)

such as coefficients estimate, distortion bounds and convex family. Also the concept of

δ

neighborhoods and partial sums of analytic functions to the class

M_{p, α}^{n, q} (η, A, B)

. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

13 pages, 297 KiB

Open AccessArticle

Existence Results for an m-Point Mixed Fractional-Order Problem at Resonance on the Half-Line

by Ogbu F. Imaga, Samuel A. Iyase and Peter O. Ogunniyi

Axioms 2022, 11(11), 630; https://doi.org/10.3390/axioms11110630 - 9 Nov 2022

Cited by 3 | Viewed by 1558

Abstract

This work considers the existence of solutions for a mixed fractional-order boundary value problem at resonance on the half-line. The Mawhin’s coincidence degree theory will be used to prove existence results when the dimension of the kernel of the linear fractional differential operator [...] Read more.

This work considers the existence of solutions for a mixed fractional-order boundary value problem at resonance on the half-line. The Mawhin’s coincidence degree theory will be used to prove existence results when the dimension of the kernel of the linear fractional differential operator is equal to two. An example is given to demonstrate the main result obtained. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

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