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26 pages, 701 KiB

Open AccessArticle

On the Non-Local Boundary Value Problem from the Probabilistic Viewpoint

by Mirko D’Ovidio

Mathematics 2022, 10(21), 4122; https://doi.org/10.3390/math10214122 - 4 Nov 2022

Cited by 1 | Viewed by 1211

We provide a short introduction of new and well-known facts relating non-local operators and irregular domains. Cauchy problems and boundary value problems are considered in case non-local operators are involved. Such problems lead to anomalous behavior on the bulk and on the surface [...] Read more.

We provide a short introduction of new and well-known facts relating non-local operators and irregular domains. Cauchy problems and boundary value problems are considered in case non-local operators are involved. Such problems lead to anomalous behavior on the bulk and on the surface of a given domain, respectively. Such a behavior should be considered (in a macroscopic viewpoint) in order to describe regular motion on irregular domains (in the microscopic viewpoint). Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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16 pages, 1251 KiB

Open AccessArticle

Two Linearized Schemes for One-Dimensional Time and Space Fractional Differential Equations

by Victor N. Orlov, Asmaa M. Elsayed and Elsayed I. Mahmoud

Mathematics 2022, 10(19), 3651; https://doi.org/10.3390/math10193651 - 5 Oct 2022

Cited by 2 | Viewed by 1453

Abstract

This paper investigates the solution to one-dimensional fractional differential equations with two types of fractional derivative operators of orders in the range of

(1, 2)

. Two linearized schemes of the numerical method are constructed. The considered FDEs are equivalently [...] Read more.

This paper investigates the solution to one-dimensional fractional differential equations with two types of fractional derivative operators of orders in the range of

(1, 2)

. Two linearized schemes of the numerical method are constructed. The considered FDEs are equivalently transformed by the Riemann–Liouville integral into their integro-partial differential problems to reduce the requirement for smoothness in time. The analysis of stability and convergence is rigorously discussed. Finally, numerical experiments are described, which confirm the obtained theoretical analysis. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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20 pages, 346 KiB

Open AccessArticle

The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative

by Jennifer Bravo and Carlos Lizama

Mathematics 2022, 10(19), 3540; https://doi.org/10.3390/math10193540 - 28 Sep 2022

Cited by 5 | Viewed by 2988

Abstract

Given an injective closed linear operator A defined in a Banach space

X,

and writing

_{C F} D_{t}^{α}

the Caputo–Fabrizio fractional derivative of order

α \in (0, 1),

we show that the unique solution of the [...] Read more.

Given an injective closed linear operator A defined in a Banach space

X,

and writing

_{C F} D_{t}^{α}

the Caputo–Fabrizio fractional derivative of order

α \in (0, 1),

we show that the unique solution of the abstract Cauchy problem

(*)_{C F} D_{t}^{α} u (t) = A u (t) + f (t), t \geq 0,

where f is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem

u^{'} (t) = B_{α} u (t) + F_{α} (t), t \geq 0; u (0) = - A^{- 1} f (0),

where the family of bounded linear operators

B_{α}

constitutes a Yosida approximation of A and

F_{α} (t) \to f (t)

as

α \to 1 .

Moreover, if

\frac{1}{1 - α} \in ρ (A)

and the spectrum of A is contained outside the closed disk of center and radius equal to

\frac{1}{2 (1 - α)}

then the solution of

(*)

converges to zero as

t \to \infty,

in the norm of X, provided f and

f^{'}

have exponential decay. Finally, assuming a Lipchitz-type condition on

f = f (t, x)

(and its time-derivative) that depends on

α,

we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set

S : = {x \in D (A) : x = A^{- 1} f (0, x)} .

Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

19 pages, 796 KiB

Open AccessArticle

Analytical Investigation of Fractional-Order Cahn–Hilliard and Gardner Equations Using Two Novel Techniques

by Mohammed Kbiri Alaoui, Kamsing Nonlaopon, Ahmed M. Zidan, Adnan Khan and Rasool Shah

Mathematics 2022, 10(10), 1643; https://doi.org/10.3390/math10101643 - 11 May 2022

Cited by 49 | Viewed by 2080

Abstract

In this paper, we used the natural decomposition approach with non-singular kernel derivatives to find the solution to nonlinear fractional Gardner and Cahn–Hilliard equations arising in fluid flow. The fractional derivative is considered an Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF) [...] Read more.

In this paper, we used the natural decomposition approach with non-singular kernel derivatives to find the solution to nonlinear fractional Gardner and Cahn–Hilliard equations arising in fluid flow. The fractional derivative is considered an Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF) throughout this paper. We implement natural transform with the aid of the suggested derivatives to obtain the solution of nonlinear fractional Gardner and Cahn–Hilliard equations followed by inverse natural transform. To show the accuracy and validity of the proposed methods, we focused on two nonlinear problems and compared it with the exact and other method results. Additionally, the behavior of the results is demonstrated through tables and figures that are in strong agreement with the exact solutions. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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16 pages, 941 KiB

Open AccessArticle

An Efficient Technique of Fractional-Order Physical Models Involving ρ-Laplace Transform

by Nehad Ali Shah, Ioannis Dassios, Essam R. El-Zahar and Jae Dong Chung

Mathematics 2022, 10(5), 816; https://doi.org/10.3390/math10050816 - 4 Mar 2022

Cited by 13 | Viewed by 2719

Abstract

In this article, the

ρ

-Laplace transform is paired with a new iterative method to create a new hybrid methodology known as the new iterative transform method (NITM). This method is applied to analyse fractional-order third-order dispersive partial differential equations. The suggested technique [...] Read more.

In this article, the

ρ

-Laplace transform is paired with a new iterative method to create a new hybrid methodology known as the new iterative transform method (NITM). This method is applied to analyse fractional-order third-order dispersive partial differential equations. The suggested technique procedure is straightforward and appealing, and it may be used to solve non-linear fractional-order partial differential equations effectively. The Caputo operator is used to express the fractional derivatives. Four numerical problems involving fractional-order third-order dispersive partial differential equations are presented with their analytical solutions. The graphs determined that their findings are in excellent agreement with the precise answers to the targeted issues. The solution to the problems at various fractional orders is achieved and found to be correct while comparing the exact solutions at integer-order problems. Although both problems are the non-linear fractional system of partial differential equations, the present technique provides its solution sophisticatedly. Including both integer and fractional order issues, solution graphs are carefully drawn. The fact that the issues’ physical dynamics completely support the solutions at both fractional and integer orders is significant. Moreover, despite using very few terms of the series solution attained by the present technique, higher accuracy is observed. In light of the various and authentic features, it can be customized to solve different fractional-order non-linear systems in nature. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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17 pages, 1272 KiB

Open AccessArticle

Novel Analysis of the Fractional-Order System of Non-Linear Partial Differential Equations with the Exponential-Decay Kernel

by Meshari Alesemi, Naveed Iqbal and Thongchai Botmart

Mathematics 2022, 10(4), 615; https://doi.org/10.3390/math10040615 - 17 Feb 2022

Cited by 32 | Viewed by 2467

Abstract

This article presents a homotopy perturbation transform method and a variational iterative transform method for analyzing the fractional-order non-linear system of the unsteady flow of a polytropic gas. In this method, the Yang transform is combined with the homotopy perturbation transformation method and [...] Read more.

This article presents a homotopy perturbation transform method and a variational iterative transform method for analyzing the fractional-order non-linear system of the unsteady flow of a polytropic gas. In this method, the Yang transform is combined with the homotopy perturbation transformation method and the variational iterative transformation method in the sense of Caputo–Fabrizio. A numerical simulation was carried out to verify that the suggested methodologies are accurate and reliable, and the results are revealed using graphs and tables. Comparing the analytical and actual solutions demonstrates that the proposed approaches are effective and efficient in investigating complicated non-linear models. Furthermore, the proposed methodologies control and manipulate the achieved numerical solutions in a very useful way, and this provides us with a simple process to adjust and control the convergence regions of the series solution. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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11 pages, 411 KiB

Open AccessArticle

Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)

by Ervin K. Lenzi, Haroldo V. Ribeiro, Marcelo K. Lenzi, Luiz R. Evangelista and Richard L. Magin

Mathematics 2022, 10(3), 389; https://doi.org/10.3390/math10030389 - 27 Jan 2022

Cited by 6 | Viewed by 2186

Abstract

We investigate diffusion in three dimensions on a comb-like structure in which the particles move freely in a plane, but, out of this plane, are constrained to move only in the perpendicular direction. This model is an extension of the two-dimensional version of [...] Read more.

We investigate diffusion in three dimensions on a comb-like structure in which the particles move freely in a plane, but, out of this plane, are constrained to move only in the perpendicular direction. This model is an extension of the two-dimensional version of the comb model, which allows diffusion along the backbone when the particles are not in the branches. We also consider memory effects, which may be handled with different fractional derivative operators involving singular and non-singular kernels. We find exact solutions for the particle distributions in this model that display normal and anomalous diffusion regimes when the mean-squared displacement is determined. As an application, we use this model to fit the anisotropic diffusion of water along and across the axons in the optic nerve using magnetic resonance imaging. The results for the observed diffusion times (8 to 30 milliseconds) show an anomalous diffusion both along and across the fibers. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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17 pages, 2837 KiB

Open AccessArticle

Control Techniques for a Class of Fractional Order Systems

by Mircea Ivanescu, Ioan Dumitrache, Nirvana Popescu and Decebal Popescu

Mathematics 2021, 9(19), 2357; https://doi.org/10.3390/math9192357 - 23 Sep 2021

Viewed by 1556

Abstract

The paper discusses several control techniques for a class of systems described by fractional order equations. The paper presents the unit frequency criteria that ensure the closed loop control for: Fractional Order Linear Systems, Fractional Order Linear Systems with nonlinear components, Time Delay [...] Read more.

The paper discusses several control techniques for a class of systems described by fractional order equations. The paper presents the unit frequency criteria that ensure the closed loop control for: Fractional Order Linear Systems, Fractional Order Linear Systems with nonlinear components, Time Delay Fractional Order Linear Systems, Time Delay Fractional Order Linear Systems with nonlinear components. The stability criterion is proposed for the systems composed of fractional order subsystems. These techniques are used in two applications: Soft Exoskeleton Glove Control, studied as a nonlinear model with time delay and Disabled Man-Wheelchair model, analysed as a fractional-order multi-system. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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17 pages, 323 KiB

Open AccessArticle

Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels

by Faïçal Ndaïrou and Delfim F. M. Torres

Mathematics 2021, 9(19), 2355; https://doi.org/10.3390/math9192355 - 22 Sep 2021

Cited by 2 | Viewed by 2206

Abstract

Fractional optimal control problems via a wide class of fractional operators with a general analytic kernel are introduced. Necessary optimality conditions of Pontryagin type for the considered problem are obtained after proving a Gronwall type inequality as well as results on continuity and [...] Read more.

Fractional optimal control problems via a wide class of fractional operators with a general analytic kernel are introduced. Necessary optimality conditions of Pontryagin type for the considered problem are obtained after proving a Gronwall type inequality as well as results on continuity and differentiability of perturbed trajectories. Moreover, a Mangasarian type sufficient global optimality condition for the general analytic kernel fractional optimal control problem is proved. An illustrative example is discussed. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

14 pages, 302 KiB

Open AccessArticle

General Fractional Calculus: Multi-Kernel Approach

by Vasily E. Tarasov

Mathematics 2021, 9(13), 1501; https://doi.org/10.3390/math9131501 - 26 Jun 2021

Cited by 31 | Viewed by 1843

Abstract

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one [...] Read more.

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one Sonin kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

17 pages, 2372 KiB

Open AccessArticle

A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

by Sandeep Kumar, Rajesh K. Pandey, H. M. Srivastava and G. N. Singh

Mathematics 2021, 9(9), 979; https://doi.org/10.3390/math9090979 - 27 Apr 2021

Cited by 15 | Viewed by 1797

Abstract

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the [...] Read more.

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results. Full article

(This article belongs to the Special Issue Computational, Experimental, and Theoretical Aspect of Fractional Order Operators)

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