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13 pages, 316 KiB

Open AccessArticle

Analysis of Hilfer Fractional Integro-Differential Equations with Almost Sectorial Operators

by Kulandhaivel Karthikeyan, Amar Debbouche and Delfim F. M. Torres

Fractal Fract. 2021, 5(1), 22; https://doi.org/10.3390/fractalfract5010022 - 08 Mar 2021

Cited by 25 | Viewed by 2216

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing [...] Read more.

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

12 pages, 425 KiB

Open AccessArticle

Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

by Cundi Han, Yiming Chen, Da-Yan Liu and Driss Boutat

Fractal Fract. 2021, 5(1), 8; https://doi.org/10.3390/fractalfract5010008 - 13 Jan 2021

Cited by 11 | Viewed by 2446

Abstract

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, [...] Read more.

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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19 pages, 2887 KiB

Open AccessArticle

The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons

by Corina S. Drapaca

Fractal Fract. 2021, 5(1), 4; https://doi.org/10.3390/fractalfract5010004 - 05 Jan 2021

Cited by 3 | Viewed by 2544

Abstract

Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The [...] Read more.

Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases such as epilepsy, multiple sclerosis, and Alzheimer’s disease, which have become more prevalent in the latest years. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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21 pages, 370 KiB

Open AccessArticle

Boundary Value Problem for Fractional Order Generalized Hilfer-Type Fractional Derivative with Non-Instantaneous Impulses

by Abdelkrim Salim, Mouffak Benchohra, John R. Graef and Jamal Eddine Lazreg

Fractal Fract. 2021, 5(1), 1; https://doi.org/10.3390/fractalfract5010001 - 22 Dec 2020

Cited by 27 | Viewed by 2427

Abstract

This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and [...] Read more.

This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and Krasnosel’skii’s fixed point theorem. To illustrate the results, an example is provided. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

21 pages, 1479 KiB

Open AccessArticle

An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations

by Moreno Concezzi and Renato Spigler

Fractal Fract. 2020, 4(4), 57; https://doi.org/10.3390/fractalfract4040057 - 14 Dec 2020

Cited by 7 | Viewed by 2995

Abstract

A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a [...] Read more.

A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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12 pages, 333 KiB

Open AccessEditor’s ChoiceArticle

Fractional Diffusion to a Cantor Set in 2D

by Alexander Iomin and Trifce Sandev

Fractal Fract. 2020, 4(4), 52; https://doi.org/10.3390/fractalfract4040052 - 07 Nov 2020

Cited by 4 | Viewed by 2484

Abstract

A random walk on a two dimensional square in

R^{2}

space with a hidden absorbing fractal set

F_{μ}

is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck [...] Read more.

A random walk on a two dimensional square in

R^{2}

space with a hidden absorbing fractal set

F_{μ}

is considered. This search-like problem is treated in the framework of a diffusion–reaction equation, when an absorbing term is included inside a Fokker–Planck equation as a reaction term. This macroscopic approach for the 2D transport in the

R^{2}

space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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13 pages, 400 KiB

Open AccessArticle

Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties

by Xuefeng Zhang and Wenkai Huang

Fractal Fract. 2020, 4(4), 50; https://doi.org/10.3390/fractalfract4040050 - 22 Oct 2020

Cited by 28 | Viewed by 2740

Abstract

This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear [...] Read more.

This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufficient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a finite time. Finally, a simulation example is proposed to verify the validity of the designed procedures. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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13 pages, 1186 KiB

Open AccessEditor’s ChoiceArticle

Numerical Simulation of the Fractal-Fractional Ebola Virus

by H. M. Srivastava and Khaled M. Saad

Fractal Fract. 2020, 4(4), 49; https://doi.org/10.3390/fractalfract4040049 - 29 Sep 2020

Cited by 38 | Viewed by 3895

Abstract

In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function [...] Read more.

In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter

ρ

is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of

ρ = k = 1

, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of

ρ

and

k .

All calculations in this work are accomplished by using the Mathematica package. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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6 pages, 580 KiB

Open AccessArticle

Integral Representation of Fractional Derivative of Delta Function

by Ming Li

Fractal Fract. 2020, 4(3), 47; https://doi.org/10.3390/fractalfract4030047 - 20 Sep 2020

Cited by 8 | Viewed by 2387

Abstract

Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the [...] Read more.

Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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8 pages, 269 KiB

Open AccessArticle

Functional Differential Equations Involving the ψ-Caputo Fractional Derivative

by Ricardo Almeida

Fractal Fract. 2020, 4(2), 29; https://doi.org/10.3390/fractalfract4020029 - 23 Jun 2020

Cited by 32 | Viewed by 2685

Abstract

This paper is devoted to the study of existence and uniqueness of solutions for fractional functional differential equations, whose derivative operator depends on an arbitrary function. The introduction of such function allows generalization of some known results, and others can be also obtained. [...] Read more.

This paper is devoted to the study of existence and uniqueness of solutions for fractional functional differential equations, whose derivative operator depends on an arbitrary function. The introduction of such function allows generalization of some known results, and others can be also obtained. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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5 pages, 218 KiB

Open AccessViewpoint

Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?

by Jocelyn Sabatier

Fractal Fract. 2020, 4(3), 40; https://doi.org/10.3390/fractalfract4030040 - 11 Aug 2020

Cited by 22 | Viewed by 2967

Abstract

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are [...] Read more.

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations. Full article

(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)

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