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Fractal Fract
Special Issues
Analysis of Caputo-Type Fractional Derivatives and Differential Equations
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Analysis of Caputo-Type Fractional Derivatives and Differential Equations

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 (15 July 2023) | Viewed by 4594

Special Issue Editors

Prof. Dr. Wei Wei

E-Mail Website
Guest Editor
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
Interests: optimal control; differential equations; partical differential equation; computation for PDE
Prof. Dr. Jinrong Wang

E-Mail Website
Guest Editor
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
Interests: fractional differential equations and controls; impulsive differential equations; iterative learning control; delay systems and stability; nonlinear evolution equations; differential equations from geophysical fluid flows; periodic systems and controls; differential inclusions
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Michal Fečkan

E-Mail Website
Guest Editor
Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Bratislava, Slovakia
Interests: dynamical systems; fractional systems; functional analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer calculus, as a generalization of the traditional integer-order calculus, was mentioned already in 1695 by Leibniz and L’Hospital. The subject of fractional calculus has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics. It draws applications in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic models. Fractional calculus provides a powerful tool for the description of the hereditary properties of various materials and memory processes.

This Special Issue is devoted to the analysis of Caputo-Type fractional derivatives and differential equations. Caputo-Type fractional derivatives and differential equations are a class of important topics, which are used to characterize certain evolution processes in viscoelasticity control and physics.

Topics for this Special Issue include the existence and uniqueness, stability of solutions, Ulam’s type stability, finite time stability, periodicity, averaging principle, controllability, iterative learning controls, etc., for impulsive, or delay, quaternion-valued, and evolution equations.

For this SI, we are inviting the submission of papers concerning the theory of differential equations with both ordinary, delay, quaternion-valued, and impulsive, as well as their theoretical and practical applications. 

Prof. Dr. Wei Wei
Prof. Dr. Jinrong Wang
Prof. Dr. Michal Feckan
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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. Fractal and Fractional 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 2700 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 Hermite–Hadamard inequalities
  • existence an uniqueness
  • exponential stability
  • finite time stability
  • Ulam’s type stability
  • asymptotically periodic solutions
  • averaging principle
  • controllability
  • iterative learning controls
  • fractional order
  • delay
  • impulsive
  • multi-agent systems
  • quaternion-valued
  • evolution equations

Published Papers (5 papers)

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Research

17 pages, 411 KiB  
Article
Abstract Fractional Cauchy Problem: Existence of Propagators and Inhomogeneous Solution Representation
by Dmytro Sytnyk and Barbara Wohlmuth
Fractal Fract. 2023, 7(10), 698; https://doi.org/10.3390/fractalfract7100698 - 22 Sep 2023
Viewed by 784
Abstract
We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator A and the Caputo fractional derivative of order α∈(0,2) in time. The previously known representation of the mild solution [...] Read more.
We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator A and the Caputo fractional derivative of order α∈(0,2) in time. The previously known representation of the mild solution to such a problem does not have a conventional variation-of-constants like form, with the propagator derived from the associated homogeneous problem. Instead, it relies on the existence of two propagators with different analytical properties. This fact limits theoretical and especially numerical applicability of the existing solution representation. Here, we propose an alternative representation of the mild solution to the given problem that consolidates the solution formulas for sub-parabolic, parabolic and sub-hyperbolic equations with a positive sectorial operator A and non-zero initial data. The new representation is solely based on the propagator of the homogeneous problem and, therefore, can be considered as a more natural fractional extension of the solution to the classical parabolic Cauchy problem. By exploiting a trade-off between the regularity assumptions on the initial data in terms of the fractional powers of A and the regularity assumptions on the right-hand side in time, we show that the proposed solution formula is strongly convergent for t≥0 under considerably weaker assumptions compared to the standard results from the literature. Crucially, the achieved relaxation of space regularity assumptions ensures that the new solution representation is practically feasible for any α∈(0,2) and is amenable to the numerical evaluation using uniformly accurate quadrature-based algorithms. Full article
(This article belongs to the Special Issue Analysis of Caputo-Type Fractional Derivatives and Differential Equations)
16 pages, 322 KiB  
Article
A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay
by Man Li, Yujun Niu and Jing Zou
Fractal Fract. 2023, 7(8), 622; https://doi.org/10.3390/fractalfract7080622 - 15 Aug 2023
Cited by 1 | Viewed by 846
Abstract
Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the [...] Read more.
Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results. Full article
(This article belongs to the Special Issue Analysis of Caputo-Type Fractional Derivatives and Differential Equations)
13 pages, 333 KiB  
Article
On Stability of Second Order Pantograph Fractional Differential Equations in Weighted Banach Space
by Ridha Dida, Hamid Boulares, Abdelkader Moumen, Jehad Alzabut, Mohamed Bouye and Yamina Laskri
Fractal Fract. 2023, 7(7), 560; https://doi.org/10.3390/fractalfract7070560 - 20 Jul 2023
Viewed by 789
Abstract
This work investigates a weighted Banach space second order pantograph fractional differential equation. The considered equation is of second order, expressed in terms of the Caputo–Hadamard fractional operator, and constructed in a general manner to accommodate many specific situations. The asymptotic stability of [...] Read more.
This work investigates a weighted Banach space second order pantograph fractional differential equation. The considered equation is of second order, expressed in terms of the Caputo–Hadamard fractional operator, and constructed in a general manner to accommodate many specific situations. The asymptotic stability of the main equation’s trivial solution has been given. The primary theorem was demonstrated in a unique manner by employing the Krasnoselskii’s fixed point theorem. We provide a concrete example that supports the theoretical findings. Full article
(This article belongs to the Special Issue Analysis of Caputo-Type Fractional Derivatives and Differential Equations)
15 pages, 327 KiB  
Article
Solvability for the ψ-Caputo-Type Fractional Differential System with the Generalized p-Laplacian Operator
by Yankai Li, Dongping Li, Yi Jiang and Xiaozhou Feng
Fractal Fract. 2023, 7(6), 450; https://doi.org/10.3390/fractalfract7060450 - 31 May 2023
Cited by 1 | Viewed by 746
Abstract
In this article, by combining a recent critical point theorem and several theories of the ψ-Caputo fractional operator, the multiplicity results of at least three distinct weak solutions are obtained for a new ψ-Caputo-type fractional differential system including the generalized p [...] Read more.
In this article, by combining a recent critical point theorem and several theories of the ψ-Caputo fractional operator, the multiplicity results of at least three distinct weak solutions are obtained for a new ψ-Caputo-type fractional differential system including the generalized p-Laplacian operator. It is noted that the nonlinear functions do not need to adapt certain asymptotic conditions in the paper, but, instead, are replaced by some simple algebraic conditions. Moreover, an evaluation criterion of the equation without solutions is also provided. Finally, two examples are given to demonstrate that the ψ-Caputo fractional operator is more accurate and can adapt to deal with complex system modeling problems by changing different weight functions. Full article
(This article belongs to the Special Issue Analysis of Caputo-Type Fractional Derivatives and Differential Equations)
14 pages, 1630 KiB  
Article
P-Bifurcation Analysis for a Fractional Damping Stochastic Nonlinear Equation with Gaussian White Noise
by Yujie Tang, Yun Peng, Guitian He, Wenjie Liang and Weiting Zhang
Fractal Fract. 2023, 7(5), 408; https://doi.org/10.3390/fractalfract7050408 - 18 May 2023
Viewed by 751
Abstract
This work aims to address the P-bifurcation of a stochastic nonlinear system with fractional damping driven by Gaussian white noise. Based on a stochastic averaging method, a fractional damping stochastic nonlinear equation has been studied. Furthermore, the expressions of drift and diffusion coefficients [...] Read more.
This work aims to address the P-bifurcation of a stochastic nonlinear system with fractional damping driven by Gaussian white noise. Based on a stochastic averaging method, a fractional damping stochastic nonlinear equation has been studied. Furthermore, the expressions of drift and diffusion coefficients of the Fokker–Planck equation (FPKE) have been obtained. The probability density function (PDF), the steady solution of FPKE, has also been derived. Then, PDFs of two fractional damping Morse oscillators have been obtained. One can note that the analytical results coincide with the results of numerical simulation. Importantly, stochastic P-bifurcation of a fractional damping stochastic nonlinear Morse oscillator has been further addressed and analyzed. Full article
(This article belongs to the Special Issue Analysis of Caputo-Type Fractional Derivatives and Differential Equations)
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