Research

12 pages, 7313 KiB

Open AccessArticle

Numerical Simulation of a Space-Fractional Molecular Beam Epitaxy Model without Slope Selection

by Hyun Geun Lee

Fractal Fract. 2023, 7(7), 558; https://doi.org/10.3390/fractalfract7070558 - 18 Jul 2023

Viewed by 845

In this paper, we introduce a space-fractional version of the molecular beam epitaxy (MBE) model without slope selection to describe super-diffusion in the model. Compared to the classical MBE equation, the spatial discretization is an important issue in the space-fractional MBE equation because [...] Read more.

In this paper, we introduce a space-fractional version of the molecular beam epitaxy (MBE) model without slope selection to describe super-diffusion in the model. Compared to the classical MBE equation, the spatial discretization is an important issue in the space-fractional MBE equation because of the nonlocal nature of the fractional operator. To approximate the fractional operator, we employ the Fourier spectral method, which gives a full diagonal representation of the fractional operator and achieves spectral convergence regardless of the fractional power. And, to combine with the Fourier spectral method directly, we present a linear, energy stable, and second-order method. Then, it is possible to simulate the dynamics of the space-fractional MBE equation efficiently and accurately. By using the numerical method, we investigate the effect of the fractional power in the space-fractional MBE equation. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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19 pages, 653 KiB

Open AccessArticle

by Xinxin Jiang and Lianzhong Li

Fractal Fract. 2023, 7(6), 457; https://doi.org/10.3390/fractalfract7060457 - 03 Jun 2023

Viewed by 746

Abstract

The current study presents a comprehensive Lie symmetry analysis for the time-fractional Mikhailov–Novikov–Wang (MNW) system with the Riemann–Liouville fractional derivative. The corresponding simplified equations with the Erdélyi–Kober fractional derivative are constructed by group invariant solutions. Furthermore, we obtain explicit solutions with the help [...] Read more.

The current study presents a comprehensive Lie symmetry analysis for the time-fractional Mikhailov–Novikov–Wang (MNW) system with the Riemann–Liouville fractional derivative. The corresponding simplified equations with the Erdélyi–Kober fractional derivative are constructed by group invariant solutions. Furthermore, we obtain explicit solutions with the help of the power series method and show the dynamical behavior via evolutional figures. Finally, by means of Ibragimov’s new conservation theorem, the conservation laws are derived for the system. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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19 pages, 695 KiB

Open AccessArticle

Parameters Estimation in a Time-Fractiona Parabolic System of Porous Media

by Miglena N. Koleva and Lubin G. Vulkov

Fractal Fract. 2023, 7(6), 443; https://doi.org/10.3390/fractalfract7060443 - 30 May 2023

Cited by 1 | Viewed by 748

Abstract

The simultaneous estimation of coefficients and the initial conditions for model fractional parabolic systems of porous media is reduced to the minimization of a least-squares cost functional. This inverse problem uses information about the pressures at a finite number of space time points. [...] Read more.

The simultaneous estimation of coefficients and the initial conditions for model fractional parabolic systems of porous media is reduced to the minimization of a least-squares cost functional. This inverse problem uses information about the pressures at a finite number of space time points. The Frechet gradient of the cost functional is derived. The application of the conjugate gradient method for numerical parameter estimation is also discussed. Computational results with noise-free and noisy data illustrate the efficiency and accuracy of the proposed algorithm. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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15 pages, 2560 KiB

Open AccessArticle

Analysis of a Fractional-Order Model for African Swine Fever with Effect of Limited Medical Resources

by Ruiqing Shi, Yang Li and Cuihong Wang

Fractal Fract. 2023, 7(6), 430; https://doi.org/10.3390/fractalfract7060430 - 25 May 2023

Cited by 1 | Viewed by 654

Abstract

In this paper, a fractional-order model for African swine fever with limited medical resources is proposed and analyzed. First, the existence and uniqueness of a positive solution is proven. Second, the basic reproduction number and the conditions sufficient for the existence of two [...] Read more.

In this paper, a fractional-order model for African swine fever with limited medical resources is proposed and analyzed. First, the existence and uniqueness of a positive solution is proven. Second, the basic reproduction number and the conditions sufficient for the existence of two equilibriums are obtained. Third, the local stability of the two equilibriums is studied. Next, some numerical simulations are performed to verify the theoretical results. The mathematical and simulation results show that the values of some parameters, such as fractional order and medical resources, are critical for the stability of the equilibriums. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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12 pages, 1506 KiB

Open AccessArticle

Efficient Solution of Fractional System Partial Differential Equations Using Laplace Residual Power Series Method

by Ahmad Shafee, Yousuf Alkhezi and Rasool Shah

Fractal Fract. 2023, 7(6), 429; https://doi.org/10.3390/fractalfract7060429 - 25 May 2023

Cited by 9 | Viewed by 1336

Abstract

In this paper, we present an efficient solution method for solving fractional system partial differential equations (FSPDEs) using the Laplace residual power series (LRPS) method. The LRPS method is a powerful technique for solving FSPDEs, as it allows for the efficient computation of [...] Read more.

In this paper, we present an efficient solution method for solving fractional system partial differential equations (FSPDEs) using the Laplace residual power series (LRPS) method. The LRPS method is a powerful technique for solving FSPDEs, as it allows for the efficient computation of the solution in the form of a power series expansion. The method is based on the Laplace transform and the residual power series, and is applied to a system of coupled FSPDEs. The method is validated using several test problems, and the results show that the LRPS method is a reliable and efficient method for solving FSPDEs. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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16 pages, 353 KiB

Open AccessArticle

A Time-Fractional Schrödinger Equation with Singular Potentials on the Boundary

by Ibtehal Alazman, Mohamed Jleli and Bessem Samet

Fractal Fract. 2023, 7(6), 417; https://doi.org/10.3390/fractalfract7060417 - 23 May 2023

Viewed by 885

Abstract

A Schrödinger equation with a time-fractional derivative, posed in

(0, \infty) \times I

, where

I =] a, b]

, is investigated in this paper. The equation involves a singular Hardy potential of the form [...] Read more.

A Schrödinger equation with a time-fractional derivative, posed in

(0, \infty) \times I

, where

I =] a, b]

, is investigated in this paper. The equation involves a singular Hardy potential of the form

\frac{λ}{{(x - a)}^{2}}

, where the parameter

λ

belongs to a certain range, and a nonlinearity of the form

μ {(x - a)}^{- ρ} {| u |}^{p}

, where

ρ \geq 0

. Using some a priori estimates, necessary conditions for the existence of weak solutions are obtained. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

18 pages, 349 KiB

Open AccessArticle

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

by Moldir Muratbekova, Bakhtiyar Kadirkulov, Maira Koshanova and Batirkhan Turmetov

Fractal Fract. 2023, 7(5), 404; https://doi.org/10.3390/fractalfract7050404 - 16 May 2023

Cited by 1 | Viewed by 840

Abstract

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of [...] Read more.

The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

16 pages, 2308 KiB

Open AccessArticle

High-Dimensional Chaotic Lorenz System: Numerical Treatment Using Changhee Polynomials of the Appell Type

by Mohamed Adel, Mohamed M. Khader and Salman Algelany

Fractal Fract. 2023, 7(5), 398; https://doi.org/10.3390/fractalfract7050398 - 13 May 2023

Cited by 9 | Viewed by 947

Abstract

Presenting and simulating the numerical treatment of the nine-dimensional fractional chaotic Lorenz system is the goal of this work. The spectral collocation method (SCM), which makes use of Changhee polynomials of the Appell type, is the suggested approximation technique to achieve this goal. [...] Read more.

Presenting and simulating the numerical treatment of the nine-dimensional fractional chaotic Lorenz system is the goal of this work. The spectral collocation method (SCM), which makes use of Changhee polynomials of the Appell type, is the suggested approximation technique to achieve this goal. A rough formula for the Caputo fractional derivative is first derived, and it is used to build the numerical strategy for the suggested model’s solution. This procedure creates a system of algebraic equations from the model that was provided. We validate the effectiveness and precision of the provided approach by evaluating the residual error function (REF). We compare the results obtained with the fourth-order Runge–Kutta technique and other existing published work. The outcomes demonstrate that the technique used is a simple and effective tool for simulating such models. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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17 pages, 387 KiB

Open AccessArticle

Bernoulli-Type Spectral Numerical Scheme for Initial and Boundary Value Problems with Variable Order

by Zareen A. Khan, Sajjad Ahmad, Salman Zeb and Hussam Alrabaiah

Fractal Fract. 2023, 7(5), 392; https://doi.org/10.3390/fractalfract7050392 - 09 May 2023

Viewed by 945

Abstract

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration [...] Read more.

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration and differentiation are developed. With the aid of these operational matrices, the considered problems are converted to algebraic-type equations, which can be easily solved using computational software. Various examples are solved by applying the method described above, and their graphical presentation and accuracy performance are provided. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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15 pages, 934 KiB

Open AccessArticle

Bifurcation of Traveling Wave Solution of Sakovich Equation with Beta Fractional Derivative

by Munirah A. Almulhim and Muneerah Al Nuwairan

Fractal Fract. 2023, 7(5), 372; https://doi.org/10.3390/fractalfract7050372 - 29 Apr 2023

Cited by 3 | Viewed by 918

Abstract

The current work is devoted to studying the dynamical behavior of the Sakovich equation with beta derivatives. We announce the conditions of problem parameters leading to the existence of periodic, solitary, and kink solutions by applying the qualitative theory of planar dynamical systems. [...] Read more.

The current work is devoted to studying the dynamical behavior of the Sakovich equation with beta derivatives. We announce the conditions of problem parameters leading to the existence of periodic, solitary, and kink solutions by applying the qualitative theory of planar dynamical systems. Based on these conditions, we construct some new solutions by integrating the conserved quantity along the possible interval of real wave propagation in order to obtain real solutions that are significant and desirable in real-world applications. We illustrate the dependence of the solutions on the initial conditions by examining the phase plane orbit. We graphically show the fractional order beta effects on the width of the solutions and keep their amplitude approximately unchanged. The graphical representations of some 3D and 2D solutions are introduced. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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18 pages, 1337 KiB

Open AccessArticle

Studying and Simulating the Fractional COVID-19 Model Using an Efficient Spectral Collocation Approach

by Yasser F. Ibrahim, Sobhi E. Abd El-Bar, Mohamed M. Khader and Mohamed Adel

Fractal Fract. 2023, 7(4), 307; https://doi.org/10.3390/fractalfract7040307 - 02 Apr 2023

Cited by 7 | Viewed by 1188

Abstract

We give a theoretical and numerical analysis of a coronavirus (COVID-19) infection model in this research. A mathematical model of this system is provided, based on a collection of fractional differential equations (in the Caputo sense). Initially, a rough approximation formula was created [...] Read more.

We give a theoretical and numerical analysis of a coronavirus (COVID-19) infection model in this research. A mathematical model of this system is provided, based on a collection of fractional differential equations (in the Caputo sense). Initially, a rough approximation formula was created for the fractional derivative of

t^{p}

. Here, the third-kind Chebyshev approximations of the spectral collocation method (SCM) were used. To identify the unknown coefficients of the approximate solution, the proposed problem was transformed into a system of algebraic equations, which was then transformed into a restricted optimization problem. To evaluate the effectiveness and accuracy of the suggested scheme, the residual error function was computed. The objective of this research was to halt the global spread of a disease. A susceptible person may be moved immediately into the confined class after being initially quarantined or an exposed person may be transferred to one of the infected classes. The researchers adopted this strategy and considered both asymptomatic and symptomatic infected patients. Results acquired with the achieved results were contrasted with those obtained using the generalized Runge-Kutta method. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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12 pages, 329 KiB

Open AccessArticle

Solvability of Mixed Problems for a Fourth-Order Equation with Involution and Fractional Derivative

by Mokhtar Kirane and Abdissalam A. Sarsenbi

Fractal Fract. 2023, 7(2), 131; https://doi.org/10.3390/fractalfract7020131 - 30 Jan 2023

Cited by 5 | Viewed by 1249

Abstract

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems [...] Read more.

In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

16 pages, 376 KiB

Open AccessArticle

The Right Equivalent Integral Equation of Impulsive Caputo Fractional-Order System of Order ϵ∈(1,2)

by Xianmin Zhang, Zuohua Liu, Shixian Yang, Zuming Peng, Yali He and Liran Wei

Fractal Fract. 2023, 7(1), 37; https://doi.org/10.3390/fractalfract7010037 - 29 Dec 2022

Cited by 3 | Viewed by 1353

Abstract

For the impulsive fractional-order system (IFrOS) of order

ϵ \in (1, 2)

, there have appeared some conflicting equivalent integral equations in existing studies. However, we find two fractional-order properties of piecewise function and use them to verify that these [...] Read more.

For the impulsive fractional-order system (IFrOS) of order

ϵ \in (1, 2)

, there have appeared some conflicting equivalent integral equations in existing studies. However, we find two fractional-order properties of piecewise function and use them to verify that these given equivalent integral equations have some defects to not be the equivalent integral equation of the IFrOS. For the IFrOS, its limit property shows the linear additivity of the impulsive effects. For the IFrOS, we use the limit analysis and the linear additivity of the impulsive effects to find its correct equivalent integral equation, which is a combination of some piecewise functions with two arbitrary constants; that is, the solution of the IFrOS is a general solution. Finally, a numerical example is given to show the equivalent integral equation and the non-uniqueness of the solution of the IFrOS. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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20 pages, 350 KiB

Open AccessArticle

Well-Posedness and Regularity Results for Fractional Wave Equations with Time-Dependent Coefficients

by Li Peng and Yong Zhou

Fractal Fract. 2022, 6(11), 644; https://doi.org/10.3390/fractalfract6110644 - 03 Nov 2022

Cited by 1 | Viewed by 1092

Abstract

Fractional wave equations with time-dependent coefficients are natural generations of classical wave equations which can be used to characterize propagation of wave in inhomogeneous media with frequency-dependent power-law behavior. This paper discusses the well-posedness and regularity results of the weak solution for a [...] Read more.

Fractional wave equations with time-dependent coefficients are natural generations of classical wave equations which can be used to characterize propagation of wave in inhomogeneous media with frequency-dependent power-law behavior. This paper discusses the well-posedness and regularity results of the weak solution for a fractional wave equation allowing that the coefficients may have low regularity. Our analysis relies on mollification arguments, Galerkin methods, and energy arguments. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

12 pages, 305 KiB

Open AccessArticle

Maximum Principles for Fractional Differential Inequalities with Prabhakar Derivative and Their Applications

by Mohammed Al-Refai, Ameina Nusseir and Sharifa Al-Sharif

Fractal Fract. 2022, 6(10), 612; https://doi.org/10.3390/fractalfract6100612 - 20 Oct 2022

Cited by 1 | Viewed by 1053

Abstract

This paper is devoted to studying a class of fractional differential equations (FDEs) with the Prabhakar fractional derivative of Caputo type in an analytical manner. At first, an estimate of the Prabhakar fractional derivative of a function at its extreme points is obtained. [...] Read more.

This paper is devoted to studying a class of fractional differential equations (FDEs) with the Prabhakar fractional derivative of Caputo type in an analytical manner. At first, an estimate of the Prabhakar fractional derivative of a function at its extreme points is obtained. This estimate is used to formulate and prove comparison principles for related fractional differential inequalities. We then apply these comparison principles to derive pre-norm estimates of solutions and to obtain a uniqueness result for linear FDEs. The solution of linear FDEs with constant coefficients is obtained in closed form via the Laplace transform. For linear FDEs with variable coefficients, we apply the obtained comparison principles to establish an existence result using the method of lower and upper solutions. Two well-defined monotone sequences that converge uniformly to the actual solution of the problem are generated. Full article

(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)

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