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Fractional Calculus and Mathematical Applications, 2nd Edition
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Fractional Calculus and Mathematical Applications, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 31 August 2025 | Viewed by 21764

Special Issue Editor

Prof. Dr. Juan Eduardo Nápoles Valdés

E-Mail Website
Guest Editor
Facultad de Ciencias Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste, Av. Libertad 5450, Corrientes 3400, Argentina
Interests: fractional calculus; generalized calculus; integral inequalities; qualitative theory of ordinary differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are delighted to announce the second volume of our Special Issue on "Fractional Calculus and Mathematical Applications". Building upon the success of the first volume, we aim to continue exploring the fascinating realm of fractional calculus and its diverse applications.

Fractional calculus possesses a unique dual nature. On the one hand, it shares a long history with ordinary (integer) calculus. On the other hand, over the past four decades, it has witnessed exponential growth in its applications across various fields and diverse topics. This interdisciplinary collaboration has resulted in an ever-increasing number of researchers and publications. From biological models to integral inequalities, from systems with delay to neutrals and hybrids, fractional calculus has found its utility and effectiveness in addressing a wide range of theoretical investigations and practical problems.

Our focus extends beyond traditional integral operators of the Riemann–Liouville type. We embrace the use of differential operators such as Caputo or Riemann–Liouville and their generalizations, allowing for the incorporation of a rich variety of mathematical tools. These tools have been extensively tested and have proven effective in tackling an array of challenging problems.

As a consequence, this vibrant field continues to produce new and exciting results. These advancements involve increasingly generalized integral operators and novel types of fractional differentials, expanding the horizons of fractional calculus to unprecedented boundaries.

We cordially invite researchers from all mathematical disciplines to contribute their latest findings and insights to this Special Issue. We aim to push the boundaries of knowledge and explore the uncharted territories of fractional calculus.

Prof. Dr. Juan Eduardo Nápoles Valdes
Guest Editor

Manuscript Submission Information

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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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

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Related Special Issue

Published Papers (21 papers)

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17 pages, 2913 KiB  
Article
Exploring Climate-Induced Oxygen–Plankton Dynamics Through Proportional–Caputo Fractional Modeling
by Mohamed A. Barakat, Areej A. Almoneef, Abd-Allah Hyder and Tarek Aboelenen
Mathematics 2025, 13(6), 980; https://doi.org/10.3390/math13060980 - 17 Mar 2025
Cited by 1 | Viewed by 244
Abstract
In this work, we develop and analyze a novel fractional-order framework to investigate the interactions among oxygen, phytoplankton, and zooplankton under changing climatic conditions. Unlike standard integer-order formulations, our model incorporates a Proportional–Caputo (PC) fractional derivative, allowing the system dynamics to [...] Read more.
In this work, we develop and analyze a novel fractional-order framework to investigate the interactions among oxygen, phytoplankton, and zooplankton under changing climatic conditions. Unlike standard integer-order formulations, our model incorporates a Proportional–Caputo (PC) fractional derivative, allowing the system dynamics to capture non-local influences and memory effects over time. Initially, we rigorously verify that a unique solution exists by suitable fixed-point theorems, demonstrating that the proposed fractional system is both well-defined and robust. We then derive stability criteria to ensure Ulam–Hyers stability (UHS), confirming that small perturbations in initial states lead to bounded variations in long-term behavior. Additionally, we explore extended UHS to assess sensitivity against time-varying parameters. Numerical simulations illustrate the role of fractional-order parameters in shaping oxygen availability and plankton populations, highlighting critical shifts in system trajectories as the order of differentiation approaches unity. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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21 pages, 343 KiB  
Article
Power Functions and Their Relationship with the Unified Fractional Derivative
by Manuel Duarte Ortigueira
Mathematics 2025, 13(5), 852; https://doi.org/10.3390/math13050852 - 4 Mar 2025
Viewed by 901
Abstract
The different forms of power functions will be studied in connection with the unified fractional derivative, and their Fourier transform will be computed. In particular, one-sided, even, and odd powers will be studied. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
18 pages, 497 KiB  
Article
Strict Stability of Fractional Differential Equations with a Caputo Fractional Derivative with Respect to Another Function
by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Mathematics 2025, 13(3), 452; https://doi.org/10.3390/math13030452 - 29 Jan 2025
Viewed by 732
Abstract
In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider [...] Read more.
In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider a system of two scalar fractional equations with CFDF and define a strict stability in the couple. We illustrate both definitions with several examples and, in these examples, we show that the applied function in the fractional derivative has a huge influence on the stability properties of the solutions. In addition, we use Lyapunov functions and their CFDF to obtain several sufficient conditions for strict stability. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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28 pages, 465 KiB  
Article
On the Inversion of the Mellin Convolution
by Gabriel Bengochea, Manuel Ortigueira and Fernando Arroyo-Cabañas
Mathematics 2025, 13(3), 432; https://doi.org/10.3390/math13030432 - 28 Jan 2025
Viewed by 852
Abstract
The deconvolution of the Mellin convolution is studied for a great variety of functions that are expressed in terms of α–log-exponential monomials. It is shown that the generation of pairs of functions satisfying a Sonin-like condition can be worked as a deconvolution [...] Read more.
The deconvolution of the Mellin convolution is studied for a great variety of functions that are expressed in terms of α–log-exponential monomials. It is shown that the generation of pairs of functions satisfying a Sonin-like condition can be worked as a deconvolution process. Applications of deconvolution to scale-invariant linear systems are presented. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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21 pages, 2620 KiB  
Article
AGTM Optimization Technique for Multi-Model Fractional-Order Controls of Spherical Tanks
by Sabavath Jayaram, Cristiano Maria Verrelli and Nithya Venkatesan
Mathematics 2025, 13(3), 351; https://doi.org/10.3390/math13030351 - 22 Jan 2025
Viewed by 912
Abstract
Spherical tanks are widely utilized in process industries due to their substantial storage capacity. These industries’ inherent challenges necessitate using highly efficient controllers to manage various process parameters, especially given their nonlinear behavior. This paper proposes the Approximate Generalized Time Moments (AGTM) optimization [...] Read more.
Spherical tanks are widely utilized in process industries due to their substantial storage capacity. These industries’ inherent challenges necessitate using highly efficient controllers to manage various process parameters, especially given their nonlinear behavior. This paper proposes the Approximate Generalized Time Moments (AGTM) optimization technique for designing the parameters of multi-model fractional-order controllers for regulating the output (liquid level) of a real-time nonlinear spherical tank. System identification for different regions of the nonlinear process is here innovatively conducted using a black-box model, which is determined to be nonlinear and approximated as a First Order Plus Dead Time (FOPDT) system over each region. Both model identification and controller design are performed in simulation and real-time using a National Instruments NI DAQmx 6211 Data Acquisition (DAQ) card (NI SYSTEMS INDIA PVT. LTD., Bangalore Karnataka, India) and MATLAB/SIMULINK software (MATLAB R2021a). The performance of the overall algorithm is evaluated through simulation and experimental testing, with several setpoints and load changes, and is compared to the performance of other algorithms tuned within the same framework. While traditional approaches, such as integer-order controllers or linear approximations, often struggle to provide consistent performance across the operating range of spherical tanks, it is originally shown how the combination of multi-model fractional-order controller design—AGTM optimization method—GA for expansion point selection and index minimization has benefits in specifically controlling a (difficult to be controlled) nonlinear process. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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16 pages, 309 KiB  
Article
Backward Continuation of the Solutions of the Cauchy Problem for Linear Fractional System with Deviating Argument
by Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev
Mathematics 2025, 13(1), 76; https://doi.org/10.3390/math13010076 - 28 Dec 2024
Viewed by 487
Abstract
Fractional calculus provides tools to model systems with memory effects; when coupled with delays, they model process histories inspired by two independent sources—the memory of the fractional derivative and the impact conditioned by the delays. This work considers a Cauchy (initial) problem for [...] Read more.
Fractional calculus provides tools to model systems with memory effects; when coupled with delays, they model process histories inspired by two independent sources—the memory of the fractional derivative and the impact conditioned by the delays. This work considers a Cauchy (initial) problem for a linear delayed system with derivatives in Caputo’s sense of incommensurate order, distributed delays, and piecewise initial functions. For this initial problem, we study the important problem of the backward continuation of its solutions. We consider the backward continuation of the solutions as a problem of the renewal of a process with aftereffect under given final observation. Sufficient conditions for backward continuation of the solutions of these systems have been obtained. As application, a formal (Lagrange) adjoint system for the studied homogeneous system is introduced, and using the backward continuation, it is proved that for this system there exists a unique matrix solution called by us as the formal adjoint fundamental matrix, which can play the same role as the fundamental matrix in the forward case. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
26 pages, 413 KiB  
Article
On Fractal–Fractional Simpson-Type Inequalities: New Insights and Refinements of Classical Results
by Fahad Alsharari, Raouf Fakhfakh and Abdelghani Lakhdari
Mathematics 2024, 12(24), 3886; https://doi.org/10.3390/math12243886 - 10 Dec 2024
Cited by 3 | Viewed by 836
Abstract
In this paper, we introduce a novel fractal–fractional identity, from which we derive new Simpson-type inequalities for functions whose first-order local fractional derivative exhibits generalized s-convexity in the second sense. This work introduces an approach that uses the first-order local fractional derivative, [...] Read more.
In this paper, we introduce a novel fractal–fractional identity, from which we derive new Simpson-type inequalities for functions whose first-order local fractional derivative exhibits generalized s-convexity in the second sense. This work introduces an approach that uses the first-order local fractional derivative, enabling the treatment of functions with lower regularity requirements compared to earlier studies. Additionally, we present two distinct methodological frameworks, one of which achieves greater precision by refining classical outcomes in the existing literature. The paper concludes with several practical applications that demonstrate the utility of our results. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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27 pages, 460 KiB  
Article
A New Inclusion on Inequalities of the Hermite–Hadamard–Mercer Type for Three-Times Differentiable Functions
by Talib Hussain, Loredana Ciurdariu and Eugenia Grecu
Mathematics 2024, 12(23), 3711; https://doi.org/10.3390/math12233711 - 26 Nov 2024
Cited by 1 | Viewed by 526
Abstract
The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable [...] Read more.
The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable convex functions that are connected to the right side of the H–H–M-type inequality. For these results, further remarks and observations are provided. Following that, a couple of graphical representations are shown to highlight the key findings of our study. Finally, some applications on special means are shown to demonstrate the effectiveness of our inequalities. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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29 pages, 401 KiB  
Article
Equivalence Between Fractional Differential Problems and Their Corresponding Integral Forms with the Pettis Integral
by Mieczysław Cichoń, Wafa Shammakh, Kinga Cichoń and Hussein A. H. Salem
Mathematics 2024, 12(23), 3642; https://doi.org/10.3390/math12233642 - 21 Nov 2024
Cited by 1 | Viewed by 608
Abstract
The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives [...] Read more.
The problem of equivalence between differential and integral problems is absolutely crucial when applying solution methods based on operators and their properties in function spaces. In this paper, we complement the solution of this important problem by considering the case of general derivatives and integrals of fractional order for vector functions for weak topology. Even if a Caputo differential fractional order problem has a right-hand side that is weakly continuous, the equivalence between the differential and integral forms may be affected. In this paper, we present a complete solution to this problem using fractional order Pettis integrals and suitably defined pseudo-derivatives, taking care to construct appropriate Hölder-type spaces on which the operators under study are mutually inverse. In this paper, we prove, in a number of cases, the equivalence of differential and integral problems in Hölder spaces and, by means of appropriate counter-examples, investigate cases where this property of the problems is absent. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
12 pages, 316 KiB  
Article
Singular Cauchy Problem for a Nonlinear Fractional Differential Equation
by Victor Orlov
Mathematics 2024, 12(22), 3629; https://doi.org/10.3390/math12223629 - 20 Nov 2024
Viewed by 599
Abstract
The paper studies a nonlinear equation including both fractional and ordinary derivatives. The singular Cauchy problem is considered. The theorem of the existence of uniqueness of a solution in the neighborhood of a fixed singular point of algebraic type is proved. An analytical [...] Read more.
The paper studies a nonlinear equation including both fractional and ordinary derivatives. The singular Cauchy problem is considered. The theorem of the existence of uniqueness of a solution in the neighborhood of a fixed singular point of algebraic type is proved. An analytical approximate solution is built, an a priori estimate is obtained. A formula for calculating the area where the proven theorem works is obtained. The theoretical results are confirmed by a numerical experiment in both digital and graphical versions. The technology of optimizing an a priori error using an a posteriori error is demonstrated. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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48 pages, 447 KiB  
Article
Approximation by Symmetrized and Perturbed Hyperbolic Tangent Activated Convolution-Type Operators
by George A. Anastassiou
Mathematics 2024, 12(20), 3302; https://doi.org/10.3390/math12203302 - 21 Oct 2024
Cited by 1 | Viewed by 867
Abstract
In this article, for the first time, the univariate symmetrized and perturbed hyperbolic tangent activated convolution-type operators of three kinds are introduced. Their approximation properties are presented, i.e., the quantitative convergence to the unit operator via the modulus of continuity. It follows the [...] Read more.
In this article, for the first time, the univariate symmetrized and perturbed hyperbolic tangent activated convolution-type operators of three kinds are introduced. Their approximation properties are presented, i.e., the quantitative convergence to the unit operator via the modulus of continuity. It follows the global smoothness preservation of these operators. The related iterated approximation as well as the simultaneous approximation and their combinations, are also extensively presented. Including differentiability and fractional differentiability into our research produced higher rates of approximation. Simultaneous global smoothness preservation is also examined. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
10 pages, 267 KiB  
Article
Strong Sandwich-Type Results for Fractional Integral of the Extended q-Analogue of Multiplier Transformation
by Alina Alb Lupaş
Mathematics 2024, 12(18), 2830; https://doi.org/10.3390/math12182830 - 12 Sep 2024
Viewed by 592
Abstract
In this research, we obtained several strong differential subordinations and strong differential superordinations, which gave sandwich-type results for the fractional integral of the extended q-analogue of multiplier transformation. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
12 pages, 315 KiB  
Article
Fractional Calculus for Non-Discrete Signed Measures
by Vassili N. Kolokoltsov and Elina L. Shishkina
Mathematics 2024, 12(18), 2804; https://doi.org/10.3390/math12182804 - 10 Sep 2024
Viewed by 938
Abstract
In this paper, we suggest a first-ever construction of fractional integral and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann–Stieltjes integral with a signed measure, using semigroup theory. The main [...] Read more.
In this paper, we suggest a first-ever construction of fractional integral and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann–Stieltjes integral with a signed measure, using semigroup theory. The main result is a theorem that provides the exact form of a semigroup for the Riemann–Stieltjes integral with a measure having a countable number of extrema. This article provides examples of semigroups based on integral operators with signed measures and discusses the fractional powers of differential operators with partial derivatives. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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20 pages, 345 KiB  
Article
A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives
by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran
Mathematics 2024, 12(11), 1654; https://doi.org/10.3390/math12111654 - 24 May 2024
Cited by 6 | Viewed by 1363
Abstract
Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We [...] Read more.
Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p=2 to results in the Lp space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
15 pages, 296 KiB  
Article
On Some Multipliers Related to Discrete Fractional Integrals
by Jinhua Cheng
Mathematics 2024, 12(10), 1545; https://doi.org/10.3390/math12101545 - 15 May 2024
Viewed by 1202
Abstract
This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of [...] Read more.
This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓp→ℓq bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
18 pages, 799 KiB  
Article
Refinements and Applications of Hermite–Hadamard-Type Inequalities Using Hadamard Fractional Integral Operators and GA-Convexity
by Muhammad Amer Latif
Mathematics 2024, 12(3), 442; https://doi.org/10.3390/math12030442 - 30 Jan 2024
Cited by 4 | Viewed by 1153
Abstract
In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using GA-convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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11 pages, 779 KiB  
Article
The Fractional Dunkl Laplacian: Definition and Harmonization via the Mellin Transform
by Fethi Bouzeffour
Mathematics 2023, 11(22), 4668; https://doi.org/10.3390/math11224668 - 16 Nov 2023
Viewed by 1314
Abstract
In this paper, we extend the scope of the Tate and Ormerod Lemmas to the Dunkl setting, revealing a profound interconnection that intricately links the Dunkl transform and the Mellin transform. This illumination underscores the pivotal significance of the Mellin integral transform in [...] Read more.
In this paper, we extend the scope of the Tate and Ormerod Lemmas to the Dunkl setting, revealing a profound interconnection that intricately links the Dunkl transform and the Mellin transform. This illumination underscores the pivotal significance of the Mellin integral transform in the realm of fractional calculus associated with differential-difference operators. Our primary focus centers on the Dunkl–Laplace operator, which serves as a prototype of a differential-difference second-order operator within an unbounded domain. Following influential research by Pagnini and Runfola, we embark on an innovative exploration employing Bochner subordination approaches tailored for the fractional Dunkl Laplacian (FDL). Notably, the Mellin transform emerges as a robust and enlightening tool, particularly in its application to the FDL. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
13 pages, 328 KiB  
Article
Variable-Order Fractional Scale Calculus
by Duarte Valério and Manuel D. Ortigueira
Mathematics 2023, 11(21), 4549; https://doi.org/10.3390/math11214549 - 4 Nov 2023
Cited by 4 | Viewed by 1602
Abstract
General variable-order fractional scale derivatives are introduced and studied. Both the stretching and the shrinking cases are considered for definitions of the derivatives of the GL type and of the Hadamard type. Their properties are deduced and discussed. Fractional variable-order systems of autoregressive–moving-average [...] Read more.
General variable-order fractional scale derivatives are introduced and studied. Both the stretching and the shrinking cases are considered for definitions of the derivatives of the GL type and of the Hadamard type. Their properties are deduced and discussed. Fractional variable-order systems of autoregressive–moving-average type are introduced and exemplified. The corresponding transfer functions are obtained and used to find the corresponding impulse responses. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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18 pages, 318 KiB  
Article
(ω,ρ)-BVP Solution of Impulsive Hadamard Fractional Differential Equations
by Ahmad Al-Omari and Hanan Al-Saadi
Mathematics 2023, 11(20), 4370; https://doi.org/10.3390/math11204370 - 20 Oct 2023
Cited by 3 | Viewed by 1061
Abstract
The purpose of this research is to examine the uniqueness and existence of the (ω,ρ)-BVP solution for a particular solution to a class of Hadamard fractional differential equations with impulsive boundary value requirements on Banach spaces. The notion [...] Read more.
The purpose of this research is to examine the uniqueness and existence of the (ω,ρ)-BVP solution for a particular solution to a class of Hadamard fractional differential equations with impulsive boundary value requirements on Banach spaces. The notion of Banach contraction and Schaefer’s theorem are used to prove the study’s key findings. In addition, we offer the prerequisites for the set of solutions to the investigated boundary value with impulsive fractional differential issue to be convex. To enhance the comprehension and practical application of our findings, we offer two illustrative examples at the end of the paper to show how the results can be applied. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
9 pages, 260 KiB  
Article
On the Solution of the Dirichlet Problem for Second-Order Elliptic Systems in the Unit Disk
by Astamur Bagapsh and Alexandre Soldatov
Mathematics 2023, 11(20), 4360; https://doi.org/10.3390/math11204360 - 20 Oct 2023
Viewed by 1064
Abstract
The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In [...] Read more.
The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
14 pages, 287 KiB  
Article
(ω,ρ)-BVP Solutions of Impulsive Differential Equations of Fractional Order on Banach Spaces
by Michal Fečkan, Marko Kostić and Daniel Velinov
Mathematics 2023, 11(14), 3086; https://doi.org/10.3390/math11143086 - 13 Jul 2023
Cited by 2 | Viewed by 1199
Abstract
The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω,ρ)-BVP solution. The proof of the [...] Read more.
The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω,ρ)-BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer’s fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results. Full article
(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)
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