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

Deadline for manuscript submissions: 31 July 2023 | Viewed by 6913

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Special Issue Editors

Dr. Péter Kórus

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Guest Editor

Institute of Applied Pedagogy, Juhász Gyula Faculty of Education, University of Szeged, H-6725 Szeged, Hungary
Interests: mathematical analysis; convex functions; fractional integrals

Prof. Dr. Juan Eduardo Nápoles Valdes

E-Mail Website
Guest Editor

1. UNNE, FaCENA, Ave Libertad 5470, 3400 Corrientes, Argentina 2. UTN, FRRE, French 414, Resistencia, 3500 Chaco, Argentina
Interests: nonlinear analysis; fractional and ordinary differential equations; fractional calculus; q-calculus; derivatives generalized; integral inequalities
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Dear Colleagues,

One of the fundamental characteristics of fractional calculus is its two-sided character: on the one hand, it is an area as old as classical calculus (of integer order), and on the other it is up-to-date, making it one of the most dynamic areas of mathematical sciences today. In fact, in recent decades there has been an increase in the number of researchers and publications related to this topic. This increase can be observed in both pure and applied mathematics, in a wide range of areas: from biological models to integral inequalities, through q-calculus, to the study of delayed, neutral, hybrid systems etc. It is interesting that the interaction between specialists from different areas and the mathematicians themselves has provided results that were unthinkable years ago.

All the above means that we can work not only with integral operators of Riemann–Liouville type, but also with differential operators of Caputo or Riemann–Liouville type and their generalizations, with q-calculus operators, with generalized local operators, which gives us the possibility of studying and analyzing phenomena of a very different nature, in a wide variety of problems.

We cordially invite researchers to contribute their original and high-quality research papers in the above topics.

Dr. Péter Kórus
Prof. Dr. Juan Eduardo Nápoles Valdes
Guest Editors

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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 1600 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 calculus
q-calculus
fractional integral and differential operators
fractional differential equation
fractional integral equation
integral inequalities

Published Papers (7 papers)

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Research

Open AccessArticle

Some New Jensen–Mercer Type Integral Inequalities via Fractional Operators

Bahtiyar Bayraktar

Péter Kórus

and

Juan Eduardo Nápoles Valdés

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

Viewed by 277

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.

(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)

Open AccessArticle

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

Constantin Fetecău

and

Costică Moroşanu

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

Viewed by 522

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.

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)

Open AccessArticle

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

and

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

Viewed by 577

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 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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Open AccessArticle

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

Ahmed Salem

and

Kholoud N. Alharbi

Axioms 2023, 12(3), 264; https://doi.org/10.3390/axioms12030264 - 03 Mar 2023

Viewed by 627

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 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)

Open AccessArticle

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

and

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

Viewed by 2869

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)

Open AccessArticle

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

Sheza M. El-Deeb

and

Luminiţa-Ioana Cotîrlă

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

Viewed by 450

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)

Open AccessArticle

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

Ogbu F. Imaga

Samuel A. Iyase

and

Peter O. Ogunniyi

Axioms 2022, 11(11), 630; https://doi.org/10.3390/axioms11110630 - 09 Nov 2022

Viewed by 725

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

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