Fractal and Fractional

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29 pages, 1634 KiB

Open AccessArticle

Global Stability of Fractional Order HIV/AIDS Epidemic Model under Caputo Operator and Its Computational Modeling

by Ashfaq Ahmad, Rashid Ali, Ijaz Ahmad, Fuad A. Awwad and Emad A. A. Ismail

Fractal Fract. 2023, 7(9), 643; https://doi.org/10.3390/fractalfract7090643 - 23 Aug 2023

Cited by 5 | Viewed by 1540

Abstract

The human immunodeficiency virus (HIV) causes acquired immunodeficiency syndrome (AIDS), which is a chronic and sometimes fatal illness. HIV reduces an individual’s capability against infection and illness by demolishing his or her immunity. This paper presents a new model that governs the dynamical [...] Read more.

The human immunodeficiency virus (HIV) causes acquired immunodeficiency syndrome (AIDS), which is a chronic and sometimes fatal illness. HIV reduces an individual’s capability against infection and illness by demolishing his or her immunity. This paper presents a new model that governs the dynamical behavior of HIV/AIDS by integrating new compartments, i.e., the treatment class T. The steady-state solutions of the model are investigated, and accordingly, the threshold quantity

R_{0}

is calculated, which describes the global dynamics of the proposed model. It is proved that for

R_{0}

less than one, the infection-free state of the model is globally asymptotically stable. However, as the threshold number increases by one, the endemic equilibrium becomes globally asymptotically stable, and in such case, the disease-free state is unstable. At the end of the paper, the analytic conclusions obtained from the analysis of the ordinary differential equation (ODE) model are supported through numerical simulations. The paper also addresses a comprehensive analysis of a fractional-order HIV model utilizing the Caputo fractional differential operator. The model’s qualitative analysis is investigated, and computational modeling is used to examine the system’s long-term behavior. The existence/uniqueness of the solution to the model is determined by applying some results from the fixed points of the theory. The stability results for the system are established by incorporating the Ulam–Hyers method. For numerical treatment and simulations, we apply Newton’s polynomial and the Toufik–Atangana numerical method. Results demonstrate the effectiveness of the fractional-order approach in capturing the dynamics of the HIV/AIDS epidemic and provide valuable insights for designing effective control strategies. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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16 pages, 359 KiB

Open AccessArticle

Asymptotics for Time-Fractional Venttsel’ Problems in Fractal Domains

by Raffaela Capitanelli, Simone Creo and Maria Rosaria Lancia

Fractal Fract. 2023, 7(6), 479; https://doi.org/10.3390/fractalfract7060479 - 16 Jun 2023

Viewed by 1034

Abstract

In this study, we consider fractional-in-time Venttsel’ problems in fractal domains of the Koch type. Well-posedness and regularity results are given. In view of numerical approximation, we consider the associated approximating pre-fractal problems. Our main result is the convergence of the solutions of [...] Read more.

In this study, we consider fractional-in-time Venttsel’ problems in fractal domains of the Koch type. Well-posedness and regularity results are given. In view of numerical approximation, we consider the associated approximating pre-fractal problems. Our main result is the convergence of the solutions of such problems towards the solution of the fractional-in-time Venttsel’ problem in the corresponding fractal domain. This is achieved via the convergence (in the Mosco–Kuwae–Shioya sense) of the approximating energy forms in varying Hilbert spaces. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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32 pages, 4095 KiB

Open AccessArticle

Sub-Diffusion Two-Temperature Model and Accurate Numerical Scheme for Heat Conduction Induced by Ultrashort-Pulsed Laser Heating

by Cuicui Ji and Weizhong Dai

Fractal Fract. 2023, 7(4), 319; https://doi.org/10.3390/fractalfract7040319 - 8 Apr 2023

Viewed by 1491

Abstract

In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (

K_{n}

) and two Caputo fractional derivatives (

0 < α, β < 1

) in time into the parabolic [...] Read more.

In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (

K_{n}

) and two Caputo fractional derivatives (

0 < α, β < 1

) in time into the parabolic two-temperature model of the diffusive type. We prove that the obtained sub-diffusion two-temperature model is well posed. The numerical scheme is obtained based on the

L 1

approximation for the Caputo fractional derivatives and the second-order finite difference for the spatial derivatives. Using the discrete energy method, we prove the numerical scheme to be unconditionally stable and convergent with

O (τ^{min {2 - α, 2 - β}} + h^{2})

, where

τ, h

are time and space steps, respectively. The accuracy and applicability of the present numerical scheme are tested in two examples. Results show that the numerical solutions are accurate, and the present model and its numerical scheme could be used as a tool by changing the values of the Knudsen number and fractional-order derivatives as well as the parameter in the boundary condition for analyzing the heat conduction in porous media, such as porous thin metal films exposed to ultrashort-pulsed lasers, where the energy transports in phonons and electrons may be ultraslow at different rates. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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15 pages, 1628 KiB

Open AccessArticle

An Application of the Homotopy Analysis Method for the Time- or Space-Fractional Heat Equation

by Rafał Brociek, Agata Wajda, Marek Błasik and Damian Słota

Fractal Fract. 2023, 7(3), 224; https://doi.org/10.3390/fractalfract7030224 - 1 Mar 2023

Cited by 6 | Viewed by 1518

Abstract

This paper focuses on the usage of the homotopy analysis method (HAM) to solve the fractional heat conduction equation. In the presented mathematical model, Caputo-type fractional derivatives over time or space are considered. In the HAM, it is not necessary to discretize the [...] Read more.

This paper focuses on the usage of the homotopy analysis method (HAM) to solve the fractional heat conduction equation. In the presented mathematical model, Caputo-type fractional derivatives over time or space are considered. In the HAM, it is not necessary to discretize the considered domain, which is its great advantage. As a result of the method, a continuous function is obtained, which can be used for further analysis. For the first time, for the considered equations, we proved that if the series created in the method converges, then the sum of the series is a solution of the equation. A sufficient condition for this convergence is provided, as well as an estimation of the error of the approximate solution. This paper also presents examples illustrating the accuracy and stability of the proposed algorithm. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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18 pages, 332 KiB

Open AccessArticle

Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels

by Fátima Cruz, Ricardo Almeida and Natália Martins

Fractal Fract. 2022, 6(12), 731; https://doi.org/10.3390/fractalfract6120731 - 10 Dec 2022

Cited by 2 | Viewed by 1267

Abstract

In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. [...] Read more.

In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examples. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

17 pages, 350 KiB

Open AccessArticle

Boundary Value Problem for Multi-Term Nonlinear Delay Generalized Proportional Caputo Fractional Differential Equations

by Ravi P. Agarwal and Snezhana Hristova

Fractal Fract. 2022, 6(12), 691; https://doi.org/10.3390/fractalfract6120691 - 22 Nov 2022

Cited by 5 | Viewed by 1311

Abstract

A nonlocal boundary value problem for a couple of two scalar nonlinear differential equations with several generalized proportional Caputo fractional derivatives and a delay is studied. The exact solution of the scalar nonlinear differential equation with several generalized proportional Caputo fractional derivatives with [...] Read more.

A nonlocal boundary value problem for a couple of two scalar nonlinear differential equations with several generalized proportional Caputo fractional derivatives and a delay is studied. The exact solution of the scalar nonlinear differential equation with several generalized proportional Caputo fractional derivatives with different orders is obtained. A mild solution of the boundary value problem for the multi-term nonlinear couple of the given fractional equations is defined. The connection between the mild solution and the solution of the studied problem is discussed. As a partial case, several results for the nonlocal boundary value problem for the linear and non-linear multi-term Caputo fractional differential equations are provided. The results generalize several known results in the literature. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

17 pages, 401 KiB

Open AccessArticle

Zener Model with General Fractional Calculus: Thermodynamical Restrictions

by Teodor M. Atanackovic and Stevan Pilipovic

Fractal Fract. 2022, 6(10), 617; https://doi.org/10.3390/fractalfract6100617 - 21 Oct 2022

Cited by 19 | Viewed by 2404

Abstract

We studied a Zener-type model of a viscoelastic body within the context of general fractional calculus and derived restrictions on coefficients that follow from the dissipation inequality, which is the entropy inequality under isothermal conditions. We showed, for a stress relaxation and a [...] Read more.

We studied a Zener-type model of a viscoelastic body within the context of general fractional calculus and derived restrictions on coefficients that follow from the dissipation inequality, which is the entropy inequality under isothermal conditions. We showed, for a stress relaxation and a wave propagation, that the restriction that follows from the entropy inequality is sufficient to guarantee the existence and uniqueness of the solution. We presented numerical data related to the solution of a wave equation for several values of parameters. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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25 pages, 1641 KiB

Open AccessArticle

An Efficient Dissipation-Preserving Numerical Scheme to Solve a Caputo–Riesz Time-Space-Fractional Nonlinear Wave Equation

by Jorge E. Macías-Díaz and Tassos Bountis

Fractal Fract. 2022, 6(9), 500; https://doi.org/10.3390/fractalfract6090500 - 6 Sep 2022

Cited by 1 | Viewed by 1754

Abstract

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce [...] Read more.

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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13 pages, 5908 KiB

Open AccessArticle

A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme

by Darshana J. Prajapati, Shivam Rawat, Anita Tomar, Mohammad Sajid and R. C. Dimri

Fractal Fract. 2022, 6(7), 397; https://doi.org/10.3390/fractalfract6070397 - 19 Jul 2022

Cited by 8 | Viewed by 2934

Abstract

In this research, we look at the Julia set patterns that are linked to the entire transcendental function

f (z) = a e^{z^{n}} + b z + c

, where

a, b, c \in C

and [...] Read more.

In this research, we look at the Julia set patterns that are linked to the entire transcendental function

f (z) = a e^{z^{n}} + b z + c

, where

a, b, c \in C

and

n \geq 2

, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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13 pages, 296 KiB

Open AccessArticle

Minimization Problems for Functionals Depending on Generalized Proportional Fractional Derivatives

by Ricardo Almeida

Fractal Fract. 2022, 6(7), 356; https://doi.org/10.3390/fractalfract6070356 - 25 Jun 2022

Cited by 2 | Viewed by 1494

Abstract

In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with [...] Read more.

In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with and without boundary conditions, and with additional restrictions like isoperimetric and holonomic. Herglotz’s variational problem and when in presence of time delays are also considered. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

Review

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40 pages, 9182 KiB

Open AccessReview

Generalized Beta Models and Population Growth: So Many Routes to Chaos

by M. Fátima Brilhante, M. Ivette Gomes, Sandra Mendonça, Dinis Pestana and Pedro Pestana

Fractal Fract. 2023, 7(2), 194; https://doi.org/10.3390/fractalfract7020194 - 15 Feb 2023

Cited by 2 | Viewed by 1883

Abstract

Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models [...] Read more.

Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models proportional to either geo-max-stable distributions (log-logistic and backward log-logistic) or to other max-stable distributions (Fréchet or max-Weibull). We show that the former arise when in the hyper-logistic Blumberg equation, connected to the Beta

(p, q)

function, we use fractional exponents

p - 1 = 1 \mp 1 / α

and

q - 1 = 1 \pm 1 / α

, and the latter when in the hyper-Gompertz-Turner equation, the exponents of the logarithmic factor are real and eventually fractional. The use of a BetaBoop function establishes interesting connections to Probability Theory, Riemann–Liouville’s fractional integrals, higher-order monotonicity and convexity and generalized unimodality, and the logistic map paradigm inspires the investigation of the dynamics of the hyper-logistic and hyper-Gompertz maps. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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