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Fractal Fract
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Advances in Fractional Differential Operators and Their Applications
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Advances in Fractional Differential Operators and Their Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: closed (30 May 2023) | Viewed by 25217

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Angelo B. Mingarelli

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
Interests: celestial mechanics; spectral theory of differential operators; fuzzy cellular automata; irrationality questions in number theory
Special Issues, Collections and Topics in MDPI journals
Dr. Leila Gholizadeh Zivlaei

E-Mail Website
Co-Guest Editor
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
Interests: functional analysis; mathematical analysis; real analysis; measure theory; differential equations
Dr. Mohammad Dehghan

E-Mail Website
Co-Guest Editor
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
Interests: numerical methods; differential equations

Special Issue Information

This Special Issue of Fractal and Fractional is dedicated to the theory and practice of fractional differential operators and corresponding equations. Although the field of fractional derivatives is quite old, there are still many unsolved problems, and there is much significant research left to be done. Thus, we trust that more papers in the area of hybrid fractional equations (e.g., mixed Riemann–Liouville/Caputo derivatives and other combinations of such derivatives) will be published.

The areas of focused interest are as follows:

  • Generalized and fractional derivatives and integrals;
  • Riemann–Liouville derivatives and integrals;
  • Caputo derivatives and integrals;
  • Spectral and asymptotic theory;
  • Qualitative theory;
  • Variational principles;
  • Applications of fractional derivatives to any area of the science or the humanities.

We are calling on experts in this field to contribute their significant research to this Special Issue so that it can be used to lay the groundwork for future research in the specified areas. We encourage authors to share open questions within their submissions in order to attract more attention to specific problems considered of importance. The submission deadline is December 31, 2022.

Prof. Dr. Angelo Mingarelli
Dr. Leila Gholizadeh Zivlaei
Dr. Mohammad Dehghan
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

  • generalized derivatives
  • Riemann–Liouville
  • Caputo derivatives
  • spectral theory
  • asymptotic theory
  • qualitative theory
  • variational theorems
  • applications

Related Special Issue

Published Papers (20 papers)

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Research

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24 pages, 567 KiB  
Article
Stability of Time Series Models Based on Fractional-Order Weakening Buffer Operators
by Chong Li, Yingjie Yang and Xinping Zhu
Fractal Fract. 2023, 7(7), 554; https://doi.org/10.3390/fractalfract7070554 - 17 Jul 2023
Viewed by 696
Abstract
Different weakening buffer operators in a time-series model analysis usually result in different model sensitivities, which sometimes affect the effectiveness of relevant operator-based methods. In this paper, the stability of two classic fractional-order weakening buffer operator-based series models is studied; then, a new [...] Read more.
Different weakening buffer operators in a time-series model analysis usually result in different model sensitivities, which sometimes affect the effectiveness of relevant operator-based methods. In this paper, the stability of two classic fractional-order weakening buffer operator-based series models is studied; then, a new data preprocessing method based on a novel fractional-order bidirectional weakening buffer operator is provided, whose effect in improving the model’s stability is tested and utilized in prediction problems. Practical examples are employed to demonstrate the efficiency of the proposed method in improving the model’s stability in noise scenarios. The comparison indicates that the proposed method overcomes the disadvantage of many weakening buffer operators in the subjectively biased weighting of the new or old information in forecasting. These expand the application of the proposed method in time series analysis. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
26 pages, 500 KiB  
Article
Coupled System of Fractional Impulsive Problem Involving Power-Law Kernel with Piecewise Order
by Arshad Ali, Khursheed J. Ansari, Hussam Alrabaiah, Ahmad Aloqaily and Nabil Mlaiki
Fractal Fract. 2023, 7(6), 436; https://doi.org/10.3390/fractalfract7060436 - 29 May 2023
Cited by 3 | Viewed by 2093
Abstract
In this research paper, we study a coupled system of piecewise-order differential equations (DEs) with variable kernel and impulsive conditions. DEs with variable kernel have high flexibility due to the freedom of changing the kernel. We study existence and stability theory and derive [...] Read more.
In this research paper, we study a coupled system of piecewise-order differential equations (DEs) with variable kernel and impulsive conditions. DEs with variable kernel have high flexibility due to the freedom of changing the kernel. We study existence and stability theory and derive sufficient conditions for main results of the proposed problem. We apply Scheafer’s fixed point theorem and Banach fixed point theorem for the result of at least one and unique solution, respectively. In addition, stability results based on the Ulam–Hyers concept are derived. Being a coupled system of piecewise fractional-order DEs with variable kernel and impulsive effects, the obtained results have multi-dimension applications. To demonstrate the applications, we apply the derived results to a numerical problem. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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14 pages, 956 KiB  
Article
Influences of the Order of Derivative on the Dynamical Behavior of Fractional-Order Antisymmetric Lotka–Volterra Systems
by Mengrui Xu
Fractal Fract. 2023, 7(5), 360; https://doi.org/10.3390/fractalfract7050360 - 28 Apr 2023
Viewed by 728
Abstract
This paper studies the dynamic behavior of a class of fractional-order antisymmetric Lotka–Volterra systems. The influences of the order of derivative on the boundedness and stability are characterized by analyzing the first-order and 0<α<1-order antisymmetric Lotka–Volterra systems separately. [...] Read more.
This paper studies the dynamic behavior of a class of fractional-order antisymmetric Lotka–Volterra systems. The influences of the order of derivative on the boundedness and stability are characterized by analyzing the first-order and 0<α<1-order antisymmetric Lotka–Volterra systems separately. We show that the order does not affect the boundedness but affects the stability. All solutions of the first-order system are periodic, while the 0<α<1-order system has no non-trivial periodic solution. Furthermore, the 0<α<1-order system can be reduced on a two-dimensional space and the reduced system is asymptotically stable, regardless of how close to zero the order of the derivative used is. Some numerical simulations are presented to better verify the theoretical analysis. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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16 pages, 569 KiB  
Article
Fractional-Order Nonlinear Multi-Agent Systems: A Resilience-Based Approach to Consensus Analysis with Distributed and Input Delays
by Asad Khan, Azmat Ullah Khan Niazi, Waseem Abbasi, Faryal Awan and Anam Khan
Fractal Fract. 2023, 7(4), 322; https://doi.org/10.3390/fractalfract7040322 - 11 Apr 2023
Cited by 2 | Viewed by 1065
Abstract
In this article, a resilient consensus analysis of fractional-order nonlinear leader and follower systems with input and distributed delays is assumed. To make controller design more practical, it is considered that the controller is not implemented as it is, and a disturbance term [...] Read more.
In this article, a resilient consensus analysis of fractional-order nonlinear leader and follower systems with input and distributed delays is assumed. To make controller design more practical, it is considered that the controller is not implemented as it is, and a disturbance term is incorporated into the controller part. A multi-agent system’s topology ahead to a weighted graph which may be directed or undirected is used. The article examines a scenario of leader–follower consensus through the application of algebraic graph theory and the fractional-order Razumikhin method. Numerical simulations are also provided to show the effectiveness of the proposed design for the leader–follower consensus. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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18 pages, 2273 KiB  
Article
Approximate Solutions for Time-Fractional Fornberg–Whitham Equation with Variable Coefficients
by Fahad Alsidrani, Adem Kılıçman and Norazak Senu
Fractal Fract. 2023, 7(3), 260; https://doi.org/10.3390/fractalfract7030260 - 14 Mar 2023
Cited by 1 | Viewed by 1236
Abstract
In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) [...] Read more.
In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) equation by replacing the integer-order time derivative with the Caputo fractional derivative of order η=(0,1] with variable coefficients. We consider homogeneous boundary conditions to find the approximate solutions for the bounded space variable l<χ<L and l,L∈R. To confirm the effectiveness of the proposed methods of non-integer order η, the computation of two test problems was presented. A comparison is made between the obtained results of the (VIM), (ADM), and (HAM) through tables and graphs. The numerical results demonstrate the effectiveness of the three numerical methods. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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18 pages, 380 KiB  
Article
Analysis of a Fractional Variational Problem Associated with Cantilever Beams Subjected to a Uniformly Distributed Load
by Apassara Suechoei, Parinya Sa Ngiamsunthorn, Waraporn Chatanin, Chainarong Athisakul, Somchai Chucheepsakul and Danuruj Songsanga
Fractal Fract. 2023, 7(2), 141; https://doi.org/10.3390/fractalfract7020141 - 02 Feb 2023
Cited by 1 | Viewed by 1330
Abstract
In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as [...] Read more.
In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as a boundary value problem and obtain existence and uniqueness results in both L2 and L∞ settings. Additionally, we characterize the continuous dependence of the minimizers on varying loads in the energy functional. Moreover, an approximate solution is derived via the homotopy perturbation method, which is numerically demonstrated in various examples. The results show that the deformations are larger for smaller orders of the fractional derivative. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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21 pages, 384 KiB  
Article
Cauchy Problem for an Abstract Evolution Equation of Fractional Order
by Maksim V. Kukushkin
Fractal Fract. 2023, 7(2), 111; https://doi.org/10.3390/fractalfract7020111 - 23 Jan 2023
Cited by 4 | Viewed by 875
Abstract
In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in the Laurent series with [...] Read more.
In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in the Laurent series with the infinite principal part and finite regular part. Our central challenge is to improve this result having considered as a regular part an entire function satisfying the special condition of the growth regularity. As an application, we consider an opportunity to broaden the conditions imposed upon the second term not containing the time variable of the evolution equation in the abstract Hilbert space. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
13 pages, 332 KiB  
Article
Identification of the Order of the Fractional Derivative for the Fractional Wave Equation
by Ravshan Ashurov and Sergei Sitnik
Fractal Fract. 2023, 7(1), 67; https://doi.org/10.3390/fractalfract7010067 - 05 Jan 2023
Cited by 2 | Viewed by 890
Abstract
A fractional wave equation with a fractional Riemann–Liouville derivative is considered. An arbitrary self-adjoint operator A with a discrete spectrum was taken as the elliptic part. We studied the inverse problem of determining the order of the fractional time derivative. By setting the [...] Read more.
A fractional wave equation with a fractional Riemann–Liouville derivative is considered. An arbitrary self-adjoint operator A with a discrete spectrum was taken as the elliptic part. We studied the inverse problem of determining the order of the fractional time derivative. By setting the value of the projection of the solution onto the first eigenfunction at a fixed point in time as an additional condition, the order of the derivative was uniquely restored. The abstract operator A allows us to include many models. Several examples of operator A are discussed at the end of the article. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
10 pages, 966 KiB  
Article
Constructing Analytical Solutions of the Fractional Riccati Differential Equations Using Laplace Residual Power Series Method
by Aliaa Burqan, Aref Sarhan and Rania Saadeh
Fractal Fract. 2023, 7(1), 14; https://doi.org/10.3390/fractalfract7010014 - 25 Dec 2022
Cited by 6 | Viewed by 1301
Abstract
In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivative. The proposed method is implemented to [...] Read more.
In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivative. The proposed method is implemented to construct analytical series solutions of the target equation. The method is tested for eminent examples and the obtained results demonstrate the accuracy and efficiency of this technique by comparing it with other numerical methods. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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21 pages, 3721 KiB  
Article
A General Return-Mapping Framework for Fractional Visco-Elasto-Plasticity
by Jorge L. Suzuki, Maryam Naghibolhosseini and Mohsen Zayernouri
Fractal Fract. 2022, 6(12), 715; https://doi.org/10.3390/fractalfract6120715 - 01 Dec 2022
Cited by 2 | Viewed by 1332
Abstract
We develop a fractional return-mapping framework for power-law visco-elasto-plasticity. In our approach, the fractional viscoelasticity is accounted for through canonical combinations of Scott-Blair elements to construct a series of well-known fractional linear viscoelastic models, such as Kelvin–Voigt, Maxwell, Kelvin–Zener, and Poynting–Thomson. We also [...] Read more.
We develop a fractional return-mapping framework for power-law visco-elasto-plasticity. In our approach, the fractional viscoelasticity is accounted for through canonical combinations of Scott-Blair elements to construct a series of well-known fractional linear viscoelastic models, such as Kelvin–Voigt, Maxwell, Kelvin–Zener, and Poynting–Thomson. We also consider a fractional quasi-linear version of Fung’s model to account for stress/strain nonlinearity. The fractional viscoelastic models are combined with a fractional visco-plastic device, coupled with fractional viscoelastic models involving serial combinations of Scott-Blair elements. We then develop a general return-mapping procedure, which is fully implicit for linear viscoelastic models, and semi-implicit for the quasi-linear case. We find that, in the correction phase, the discrete stress projection and plastic slip have the same form for all the considered models, although with different property and time-step-dependent projection terms. A series of numerical experiments is carried out with analytical and reference solutions to demonstrate the convergence and computational cost of the proposed framework, which is shown to be at least first-order accurate for general loading conditions. Our numerical results demonstrate that the developed framework is more flexible and preserves the numerical accuracy of existing approaches while being more computationally tractable in the visco-plastic range due to a reduction of 50% in CPU time. Our formulation is especially suited for emerging applications of fractional calculus in bio-tissues that present the hallmark of multiple viscoelastic power-laws coupled with visco-plasticity. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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16 pages, 356 KiB  
Article
On the Basic Theory of Some Generalized and Fractional Derivatives
by Leila Gholizadeh Zivlaei and Angelo B. Mingarelli
Fractal Fract. 2022, 6(11), 672; https://doi.org/10.3390/fractalfract6110672 - 14 Nov 2022
Viewed by 995
Abstract
We continue the development of the basic theory of generalized derivatives as introduced and give some of their applications. In particular, we formulate necessary conditions for extrema, Rolle’s theorem, the mean value theorem, the fundamental theorem of calculus, integration by parts, along with [...] Read more.
We continue the development of the basic theory of generalized derivatives as introduced and give some of their applications. In particular, we formulate necessary conditions for extrema, Rolle’s theorem, the mean value theorem, the fundamental theorem of calculus, integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which provides simple proofs of the corresponding version for the so-called conformable fractional derivatives considered by many. Finally, we show that for each α>1 there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
10 pages, 303 KiB  
Article
Solutions of a Nonlinear Diffusion Equation with a Regularized Hyper-Bessel Operator
by Nguyen Hoang Luc, Donal O’Regan and Anh Tuan Nguyen
Fractal Fract. 2022, 6(9), 530; https://doi.org/10.3390/fractalfract6090530 - 19 Sep 2022
Viewed by 1389
Abstract
We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness [...] Read more.
We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag–Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
11 pages, 308 KiB  
Article
Hybridization of Block-Pulse and Taylor Polynomials for Approximating 2D Fractional Volterra Integral Equations
by Davood Jabari Sabegh, Reza Ezzati, Omid Nikan, António M. Lopes and Alexandra M. S. F. Galhano
Fractal Fract. 2022, 6(9), 511; https://doi.org/10.3390/fractalfract6090511 - 12 Sep 2022
Cited by 1 | Viewed by 1198
Abstract
This paper proposes an accurate numerical approach for computing the solution of two-dimensional fractional Volterra integral equations. The operational matrices of fractional integration based on the Hybridization of block-pulse and Taylor polynomials are implemented to transform these equations into a system of linear [...] Read more.
This paper proposes an accurate numerical approach for computing the solution of two-dimensional fractional Volterra integral equations. The operational matrices of fractional integration based on the Hybridization of block-pulse and Taylor polynomials are implemented to transform these equations into a system of linear algebraic equations. The error analysis of the proposed method is examined in detail. Numerical results highlight the robustness and accuracy of the proposed strategy. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
28 pages, 444 KiB  
Article
Well-Posedness and Global Attractors for Viscous Fractional Cahn–Hilliard Equations with Memory
by Eylem Öztürk and Joseph L. Shomberg
Fractal Fract. 2022, 6(9), 505; https://doi.org/10.3390/fractalfract6090505 - 09 Sep 2022
Viewed by 886
Abstract
We examine a viscous Cahn–Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions [...] Read more.
We examine a viscous Cahn–Hilliard phase-separation model with memory and where the chemical potential possesses a nonlocal fractional Laplacian operator. The existence of global weak solutions is proven using a Galerkin approximation scheme. A continuous dependence estimate provides uniqueness of the weak solutions and also serves to define a precompact pseudometric. This, in addition to the existence of a bounded absorbing set, shows that the associated semigroup of solution operators admits a compact connected global attractor in the weak energy phase space. The minimal assumptions on the nonlinear potential allow for arbitrary polynomial growth. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
16 pages, 327 KiB  
Article
Fractional Sturm–Liouville Eigenvalue Problems, II
by Mohammad Dehghan and Angelo B. Mingarelli
Fractal Fract. 2022, 6(9), 487; https://doi.org/10.3390/fractalfract6090487 - 30 Aug 2022
Cited by 2 | Viewed by 1215
Abstract
We continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of [...] Read more.
We continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter α, 0<α<1, there is a finite set of real eigenvalues and that, for α near 1/2, there may be none at all. As α→1− we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm–Liouville problem with the composition of the operators becoming the operator of second order differentiation. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
27 pages, 436 KiB  
Article
Multiplicity of Solutions for Fractional-Order Differential Equations via the κ(x)-Laplacian Operator and the Genus Theory
by Hari M. Srivastava and Jose Vanterler da Costa Sousa
Fractal Fract. 2022, 6(9), 481; https://doi.org/10.3390/fractalfract6090481 - 29 Aug 2022
Cited by 15 | Viewed by 1274
Abstract
In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χ-fractional space Hκ(x)γ,β;χ(Δ). Using the Genus [...] Read more.
In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χ-fractional space Hκ(x)γ,β;χ(Δ). Using the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem, we show that under certain suitable assumptions the considered problem has at least k pairs of non-trivial solutions. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
11 pages, 864 KiB  
Article
A Cubic Spline Collocation Method to Solve a Nonlinear Space-Fractional Fisher’s Equation and Its Stability Examination
by Adel R. Hadhoud, Faisal E. Abd Alaal, Ayman A. Abdelaziz and Taha Radwan
Fractal Fract. 2022, 6(9), 470; https://doi.org/10.3390/fractalfract6090470 - 26 Aug 2022
Cited by 1 | Viewed by 1033
Abstract
This article seeks to show a general framework of the cubic polynomial spline functions for developing a computational technique to solve the space-fractional Fisher’s equation. The presented approach is demonstrated to be conditionally stable using the von Neumann technique. A numerical illustration is [...] Read more.
This article seeks to show a general framework of the cubic polynomial spline functions for developing a computational technique to solve the space-fractional Fisher’s equation. The presented approach is demonstrated to be conditionally stable using the von Neumann technique. A numerical illustration is given to demonstrate the proposed algorithm’s effectiveness. The novelty of the present work lies in the fact that the results suggest that the presented technique is accurate and convenient in solving such problems. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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16 pages, 358 KiB  
Article
Existence, Stability and Simulation of a Class of Nonlinear Fractional Langevin Equations Involving Nonsingular Mittag–Leffler Kernel
by Kaihong Zhao
Fractal Fract. 2022, 6(9), 469; https://doi.org/10.3390/fractalfract6090469 - 26 Aug 2022
Cited by 22 | Viewed by 1209
Abstract
The fractional Langevin equation is a very effective mathematical model for depicting the random motion of particles in complex viscous elastic liquids. This manuscript is mainly concerned with a class of nonlinear fractional Langevin equations involving nonsingular Mittag–Leffler (ML) kernel. We first investigate [...] Read more.
The fractional Langevin equation is a very effective mathematical model for depicting the random motion of particles in complex viscous elastic liquids. This manuscript is mainly concerned with a class of nonlinear fractional Langevin equations involving nonsingular Mittag–Leffler (ML) kernel. We first investigate the existence and uniqueness of the solution by employing some fixed-point theorems. Then, we apply direct analysis to obtain the Ulam–Hyers (UH) type stability. Finally, the theoretical analysis and numerical simulation of some interesting examples show that there is a great difference between the fractional Langevin equation and integer Langevin equation in describing the random motion of free particles. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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24 pages, 1440 KiB  
Article
Sinc Numeric Methods for Fox-H, Aleph (ℵ), and Saxena-I Functions
by Gerd Baumann and Norbert Südland
Fractal Fract. 2022, 6(8), 449; https://doi.org/10.3390/fractalfract6080449 - 18 Aug 2022
Viewed by 1435
Abstract
The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on [...] Read more.
The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on the functions in question. The approach’s pillars are the concept of a Mellin-Barnes integral and the Mellin representation of the given function. A Sinc quadrature is used in conjunction with a Sinc approximation of the function to achieve the numerical approximation of the Mellin-Barnes integral. The method converges exponentially and can handle endpoint singularities. We give numerical representations of the Aleph ℵ and Saxena I functions for the first time. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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16 pages, 1288 KiB  
Review
Stability Properties of Multi-Term Fractional-Differential Equations
by Oana Brandibur and Éva Kaslik
Fractal Fract. 2023, 7(2), 117; https://doi.org/10.3390/fractalfract7020117 - 26 Jan 2023
Cited by 1 | Viewed by 1053
Abstract
Necessary and sufficient stability and instability conditions are reviewed and extended for multi-term homogeneous linear fractional differential equations with Caputo derivatives and constant coefficients. A comprehensive review of the state of the art regarding the stability analysis of two-term and three-term fractional-order differential [...] Read more.
Necessary and sufficient stability and instability conditions are reviewed and extended for multi-term homogeneous linear fractional differential equations with Caputo derivatives and constant coefficients. A comprehensive review of the state of the art regarding the stability analysis of two-term and three-term fractional-order differential equations is provided, which is then extended to the case of four-term fractional-order differential equations. The stability and instability properties are characterized with respect to the coefficients of the multi-term fractional differential equations, leading to both fractional-order-dependent and fractional-order-independent characterizations. In the general case, fractional-order-independent stability and instability properties are described for fractional-order differential equations with an arbitrary number of fractional derivatives. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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