Fractal and Fractional

Research

66 pages, 4599 KiB

Open AccessArticle

Conformal and Non-Minimal Couplings in Fractional Cosmology

by Kevin Marroquín, Genly Leon, Alfredo D. Millano, Claudio Michea and Andronikos Paliathanasis

Fractal Fract. 2024, 8(5), 253; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8050253 - 25 Apr 2024

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Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus [...] Read more.

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field,

ϕ

, to create a non-minimal interaction theory with the coupling,

ξ R ϕ^{2}

, between gravity and the scalar field, where

ξ

is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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34 pages, 495 KiB

Open AccessArticle

Fundamental Matrix, Integral Representation and Stability Analysis of the Solutions of Neutral Fractional Systems with Derivatives in the Riemann—Liouville Sense

by Hristo Kiskinov, Mariyan Milev, Slav Ivanov Cholakov and Andrey Zahariev

Fractal Fract. 2024, 8(4), 195; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract8040195 - 28 Mar 2024

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Abstract

The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a [...] Read more.

The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a unique solution. The assumptions used for the proof are similar to those used in the case of systems with first-order derivatives. Then, with the obtained result, we derive the existence and uniqueness of a fundamental matrix and a generalized fundamental matrix for the homogeneous system. In the linear case, via these fundamental matrices we obtain integral representations of the solutions of the homogeneous system and the corresponding inhomogeneous system. Furthermore, for the fractional systems with Riemann–Liouville derivatives we introduce a new concept for weighted stabilities in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, which coincides with the classical stability concepts for the cases of integer-order or Caputo-type derivatives. It is proved that the zero solution of the homogeneous system is weighted stable if and only if all its solutions are weighted bounded. In addition, for the homogeneous system it is established that the weighted stability in the Lyapunov and Ulam–Hyers senses are equivalent if and only if the inequality appearing in the Ulam–Hyers definition possess only bounded solutions. Finally, we derive natural sufficient conditions under which the property of weighted global asymptotic stability of the zero solution of the homogeneous system is preserved under nonlinear disturbances. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

11 pages, 368 KiB

Open AccessArticle

Some Results on Fractional Boundary Value Problem for Caputo-Hadamard Fractional Impulsive Integro Differential Equations

by Ymnah Alruwaily, Kuppusamy Venkatachalam and El-sayed El-hady

Fractal Fract. 2023, 7(12), 884; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7120884 - 14 Dec 2023

Cited by 1 | Viewed by 1025

Abstract

The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve [...] Read more.

The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

26 pages, 787 KiB

Open AccessArticle

Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

by Yuri Dimitrov, Slavi Georgiev and Venelin Todorov

Fractal Fract. 2023, 7(10), 750; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7100750 - 11 Oct 2023

Cited by 1 | Viewed by 1662

Abstract

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is [...] Read more.

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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18 pages, 468 KiB

Open AccessArticle

On Stability of a Fractional Discrete Reaction–Diffusion Epidemic Model

by Omar Alsayyed, Amel Hioual, Gharib M. Gharib, Mayada Abualhomos, Hassan Al-Tarawneh, Maha S. Alsauodi, Nabeela Abu-Alkishik, Abdallah Al-Husban and Adel Ouannas

Fractal Fract. 2023, 7(10), 729; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7100729 - 02 Oct 2023

Cited by 1 | Viewed by 988

Abstract

This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global [...] Read more.

This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global asymptotic stability analysis has been undertaken. We conducted a global stability analysis using a specialized Lyapunov function that captures the system’s historical data, distinguishing it from the integer-order version. This approach significantly advanced our comprehension of the complex stability properties within discrete fractional reaction–diffusion epidemic models. To substantiate the theoretical underpinnings, this paper is accompanied by numerical examples. These examples serve a dual purpose: not only do they validate the theoretical findings, but they also provide illustrations of the practical implications of the proposed discrete fractional system. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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14 pages, 325 KiB

Open AccessArticle

A Study on Generalized Degenerate Form of 2D Appell Polynomials via Fractional Operators

by Mohra Zayed and Shahid Ahmad Wani

Fractal Fract. 2023, 7(10), 723; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7100723 - 30 Sep 2023

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Abstract

This paper investigates the significance of generating expressions, operational principles, and defining characteristics in the study and development of special polynomials. The focus is on a novel generalized family of degenerate 2D Appell polynomials, which were defined using a fractional operator. Motivated by [...] Read more.

This paper investigates the significance of generating expressions, operational principles, and defining characteristics in the study and development of special polynomials. The focus is on a novel generalized family of degenerate 2D Appell polynomials, which were defined using a fractional operator. Motivated by inquiries into degenerate 2D bivariate Appell polynomials, this research reveals that well-known 2D Appell polynomials and simple Appell polynomials can be regarded as specific instances within this new family for certain values. This paper presents the operational rule, generating relation, determinant form, and recurrence relations for this generalized family. Furthermore, it explores the practical applications of these degenerate 2D Appell polynomials and establishes their connections with equivalent results for the generalized family of degenerate 2D Bernoulli, Euler, and Genocchi polynomials. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

20 pages, 4222 KiB

Open AccessArticle

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method

by A. R. Jan, M. A. Abdou and M. Basseem

Fractal Fract. 2023, 7(9), 656; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7090656 - 31 Aug 2023

Cited by 1 | Viewed by 760

Abstract

In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space [...] Read more.

In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space

L_{2} [Ω] \times C [0, T], T < 1

. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS). Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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16 pages, 2844 KiB

Open AccessArticle

Dynamical Behaviour, Control, and Boundedness of a Fractional-Order Chaotic System

by Lei Ren, Sami Muhsen, Stanford Shateyi and Hassan Saberi-Nik

Fractal Fract. 2023, 7(7), 492; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7070492 - 21 Jun 2023

Cited by 3 | Viewed by 1091

Abstract

In this paper, the fractional-order chaotic system form of a four-dimensional system with cross-product nonlinearities is introduced. The stability of the equilibrium points of the system and then the feedback control design to achieve this goal have been analyzed. Furthermore, further dynamical behaviors [...] Read more.

In this paper, the fractional-order chaotic system form of a four-dimensional system with cross-product nonlinearities is introduced. The stability of the equilibrium points of the system and then the feedback control design to achieve this goal have been analyzed. Furthermore, further dynamical behaviors including, phase portraits, bifurcation diagrams, and the largest Lyapunov exponent are presented. Finally, the global Mittag–Leffler attractive sets (MLASs) and Mittag–Leffler positive invariant sets (MLPISs) of the considered fractional order system are presented. Numerical simulations are provided to show the effectiveness of the results. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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32 pages, 11512 KiB

Open AccessArticle

A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

by Raheel Kamal, Kamran, Saleh M. Alzahrani and Talal Alzahrani

Fractal Fract. 2023, 7(5), 381; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7050381 - 01 May 2023

Viewed by 1461

Abstract

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order

0 < α < 1,

and

1 < β < 2

. The numerical method is based on [...] Read more.

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order

0 < α < 1,

and

1 < β < 2

. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered. Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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