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Fractional Differential Equations: Computation and Modelling with Applications

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (20 February 2025) | Viewed by 20180

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Special Issue Editors

Dr. Rajarama Mohan Jena

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Guest Editor

Department of Mathematics, C. V. Raman Global University, Bhubaneswar 752054, Odisha, India
Interests: differential equations (fractional, partial and ordinary); numerical analysis; applied mathematics; computational methods; mathematical modeling; soft computing

Prof. Dr. Snehashish Chakraverty

E-Mail Website
Guest Editor

Department of Mathematics, National Institute of Technology Rourkela, Odisha 769008, India
Interests: differential equations; numerical analysis; computational methods; structural dynamics (FGM, Nano); mathematical modeling; neural networks, robotics, and uncertainty modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nowadays, many researchers from various fields have become interested in the topic of fractional calculus based on integrals and derivatives of fractional order. It has numerous applications in the widespread field of science and engineering, including wave and fluid dynamics, mathematical biology, financial systems, structural dynamics, robotics, artificial intelligence, etc. Therefore, fractional models have become relevant when dealing with phenomena with memory effects instead of relying on ordinary or partial differential equations. Fractional calculus offers superior tools to cope with the time-dependent effects noticed compared to integer-order calculus, which forms the mathematical foundation of most mathematical systems. As a result, fractional calculus is crucial to model real-life problems and finding mathematical solutions is a great challenge. Since fractional differential equations are used to model real-life problems, many mathematical methods (numerical/analytical/exact) are being developed to obtain the solutions to fractional differential equations/models/systems.

In this Special Issue, we invite review and original research papers dealing with recent developments in fractional calculus along with all theoretical/analytical/numerical, as well as practical developments in various science and engineering, including mathematics and physics.

This Special Issue will be focused upon, but not limited to, the following:

Fractional-order differential/partial/integral equations;
New fractional-order operators and their properties;
Existence and uniqueness of solutions;
Computational efficient methods (analytical/numerical) for fractional order systems;
Special functions in fractional calculus;
Fractional models in physics, biology, medicine, engineering, etc.;
Neural computations with fractional calculus;
Bifurcation and chaos;
Artificial Intelligence;
Fuzzy fractional calculus;
Mathematical modelling of fractional complex systems;
Deterministic and stochastic fractional differential equations;
Fractional calculus with uncertainties and modelling;
Fractional delay differential equations;
Fractal-fractional models.

Dr. Rajarama Mohan Jena
Prof. Dr. Snehashish Chakraverty
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

fractional derivatives and integrals
fractional operators
fractal-fractional derivatives and integrals
mathematical modelling
computational efficient methods
uncertainty and artificial intelligence
nonsingular kernels

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Related Special Issue

Fractional Differential Equations: Computation and Modelling with Applications, 2nd Edition in Fractal and Fractional

Published Papers (17 papers)

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Research

21 pages, 476 KiB

Open AccessArticle

A New L2 Type Difference Scheme for the Time-Fractional Diffusion Equation

by Cheng-Yu Hu and Fu-Rong Lin

Fractal Fract. 2025, 9(5), 325; https://doi.org/10.3390/fractalfract9050325 - 20 May 2025

Viewed by 214

Abstract

In this paper, a new L2 (NL2) scheme is proposed to approximate the Caputo temporal fractional derivative, leading to a time-stepping scheme for the time-fractional diffusion equation (TFDE). Subsequently, the space derivative of the resulting system is discretized using a specific finite difference [...] Read more.

H^{1}

-norm stability and convergence of the time-stepping scheme on uniform meshes for the TFDE. In particular, we prove that the proposed scheme has

(3 - α)

th-order accuracy, where

α

(

0 < α < 1

) is the order of the time-fractional derivative. Finally, numerical experiments for several test problems are carried out to validate the obtained theoretical results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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12 pages, 679 KiB

Open AccessArticle

On the Laplace Residual Series Method and Its Application to Time-Fractional Fisher’s Equations

by Rawya Al-deiakeh, Sharifah Alhazmi, Shrideh Al-Omari, Mohammed Al-Smadi and Shaher Momani

Fractal Fract. 2025, 9(5), 275; https://doi.org/10.3390/fractalfract9050275 - 24 Apr 2025

Viewed by 280

Abstract

In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisher’s equation using a right starting space function and a unique analytic-numeric technique referred to as the Laplace residual power series approach. The generalized Taylor’s formula and the Laplace transform operator are coupled in the aforementioned method, where the coefficients, obtained through fractional expansion in the Laplace space, are determined by applying the limit concept. In order to validate and illustrate the theoretical methodology of the LRPS technique, as well as to show its effectiveness, adaptability, and superiority in solving various types of nonlinear time and space fractional differential equations, numerical experiments are generated. The obtained analytical solutions are compatible with the precise solutions and concur with those proposed by the other approaches. The outcomes show that the Laplace residual power series strategy is incredibly successful, straightforward to implement, and well suited for handling the complexity of nonlinear problems. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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24 pages, 6901 KiB

Open AccessArticle

A Suitable Algorithm to Solve a Nonlinear Fractional Integro-Differential Equation with Extended Singular Kernel in (2+1) Dimensions

by Sameeha Ali Raad and Mohamed Abdella Abdou

Fractal Fract. 2025, 9(4), 239; https://doi.org/10.3390/fractalfract9040239 - 10 Apr 2025

Viewed by 227

Abstract

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space

L_{2} [(a, b) \times (c, d)] \times C [0, T], T < 1

. In terms of time [...] Read more.

In this paper, the authors consider a problem with comprehensive properties in terms of form and content in the space

L_{2} [(a, b) \times (c, d)] \times C [0, T], T < 1

. In terms of time form, we assume that the time phase delay is implicitly contained in a nonlinear differential integral equation. The positional part is considered in two dimensions, and the position’s kernel is a general singular kernel, many different forms of which will be derived. In terms of content, all of the previously established numerical techniques are only appropriate for studying special cases of the kernel separately but are not suitable for studying the general kernel. This led to the use of the Toeplitz matrix method, which deals with the kernel in its extended nonlinear form and the special kernels will be studied as applications of the method. Moreover, this method has the advantage of converting all single integrals into regular integrals that can be easily solved. Additionally, the researchers examine the solution’s existence, uniqueness, and convergence in this paper. The error and its stability are also studied. At the end of the research, the authors studied some numerical applications of some of the singular kernels derived from the general kernel, examining the approximation error in each application separately. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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17 pages, 2414 KiB

Open AccessArticle

Analysis of Large Membrane Vibrations Using Fractional Calculus

by Nihar Ranjan Mallick, Snehashish Chakraverty and Rajarama Mohan Jena

Fractal Fract. 2025, 9(4), 219; https://doi.org/10.3390/fractalfract9040219 - 31 Mar 2025

Viewed by 284

Abstract

The study of vibration equations of large membranes is crucial in various scientific and engineering fields. Analyzing the vibration equations of bridges, roofs, and spacecraft structures helps in designing structures that resist excessive oscillations and potential failures. Aircraft wings, parachutes, and satellite components often behave like large membranes. Understanding their vibration characteristics is essential for stability, efficiency, and durability. Studying large membrane vibration involves solving partial differential equations and eigenvalue problems, contributing to advancements in numerical methods and computational physics. In this paper, the Elzaki transformation decomposition method and the Shehu transformation decomposition method, along with inverse Elzaki and inverse Shehu transformations, are used to investigate the fractional vibration equation of a large membrane. The solutions are obtained in terms of Mittag–Leffler functions. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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20 pages, 391 KiB

Open AccessArticle

Stability Analysis of a Fractional Epidemic Model Involving the Vaccination Effect

by Sümeyye Çakan

Fractal Fract. 2025, 9(4), 206; https://doi.org/10.3390/fractalfract9040206 - 27 Mar 2025

Viewed by 247

Abstract

This paper, by constructing a fractional epidemic model, analyzes the transmission dynamics of some infectious diseases under the effect of vaccination, which is one of the most effective and common control measures. In the model, considering that antibody formation by vaccination may not cause permanent immunity, it has been taken into account that the protection period provided by the vaccine may be finite, in addition to the fact that this period may change according to individuals. The model differs from other

S V I R

models given in the literature in its progressive process with a distributed delay in the loss of the protective effect provided by the vaccine. To explain this process, the model was constructed by using a system of distributed delay nonlinear fractional integro-differential equations. Thus, the model aims to present a realistic approach to following the course of the disease. Additionally, an analysis was conducted regarding the minimum vaccination ratio of new members required for the elimination of the disease in the population by using the vaccine free basic reproduction number (

R_{0}^{v f}

). After providing examples for the selection of the distribution function, the variation of

R_{0}

was simulated for a specific selection of parameters in the model. Finally, the sensitivity indices of the parameters affecting

R_{0}

were calculated, and this situation is been visually supported. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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19 pages, 491 KiB

Open AccessArticle

On the Pseudospectral Method for Solving the Fractional Klein–Gordon Equation Using Legendre Cardinal Functions

by Tao Liu, Bolin Ding, Behzad Nemati Saray, Davron Aslonqulovich Juraev and Ebrahim E. Elsayed

Fractal Fract. 2025, 9(3), 177; https://doi.org/10.3390/fractalfract9030177 - 14 Mar 2025

Viewed by 488

Abstract

This work introduces the Legendre cardinal functions for the first time. Based on Jacobi and Lobatto grids, two approaches are employed to determine these basis functions. These functions are then utilized within the pseudospectral method to solve the fractional Klein–Gordon equation (FKGE). Two numerical schemes based on the pseudospectral method are considered. The first scheme reformulates the given equation into a corresponding integral equation and solves it. The second scheme directly addresses the problem by utilizing the matrix representation of the Caputo fractional derivative operator. We provide a convergence analysis and present numerical experiments to demonstrate the convergence of the schemes. The convergence analysis shows that convergence depends on the smoothness of the unknown function. Notable features of the proposed approaches include a reduction in computations due to the cardinality property of the basis functions, matrices representing fractional derivative and integral operators, and the ease of implementation. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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17 pages, 318 KiB

Open AccessArticle

Existence and Hyers–Ulam Stability Analysis of Nonlinear Multi-Term Ψ-Caputo Fractional Differential Equations Incorporating Infinite Delay

by Yating Xiong, Abu Bakr Elbukhari and Qixiang Dong

Fractal Fract. 2025, 9(3), 140; https://doi.org/10.3390/fractalfract9030140 - 22 Feb 2025

Viewed by 457

Abstract

The aim of the paper is to prove the existence results and Hyers–Ulam stability to nonlinear multi-term

Ψ

The aim of the paper is to prove the existence results and Hyers–Ulam stability to nonlinear multi-term

Ψ

-Caputo fractional differential equations with infinite delay. Some specified assumptions are supposed to be satisfied by the nonlinear item and the delayed term. The Leray–Schauder alternative theorem and the Banach contraction principle are utilized to analyze the existence and uniqueness of solutions for infinite delay problems. Some new inequalities are presented in this paper for delayed fractional differential equations as auxiliary results, which are convenient for analyzing Hyers–Ulam stability. Some examples are discussed to illustrate the obtained results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

29 pages, 975 KiB

Open AccessArticle

Theoretical Results on the pth Moment of ϕ-Hilfer Stochastic Fractional Differential Equations with a Pantograph Term

by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat

Fractal Fract. 2025, 9(3), 134; https://doi.org/10.3390/fractalfract9030134 - 20 Feb 2025

Cited by 2 | Viewed by 522

Abstract

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the

ϕ

-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle [...] Read more.

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the

ϕ

-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for

ϕ (t)

t

a = 1

. Our findings match the Riemann–Liouville fractional operator for

ϕ (t) = t

a = 0

. They agree with the Hadamard and Caputo–Hadamard fractional operators when

ϕ (t) = ln (t)

a = 0

and

ϕ (t) = ln (t)

a = 1

, respectively. Second, regarding the space, we are make generalizations for the case

p = 2

. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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17 pages, 710 KiB

Open AccessArticle

Modeling the Dispersion of Waves in a Multilayered Inhomogeneous Membrane with Fractional-Order Infusion

by Ali M. Mubaraki, Rahmatullah Ibrahim Nuruddeen, Rab Nawaz and Tayyab Nawaz

Fractal Fract. 2024, 8(8), 445; https://doi.org/10.3390/fractalfract8080445 - 29 Jul 2024

Cited by 2 | Viewed by 1107

Abstract

The dispersion of elastic shear waves in multilayered bodies is a topic of extensive research due to its significance in contemporary science and engineering. Anti-plane shear motion, a two-dimensional mathematical model in solid mechanics, effectively captures shear wave propagation in elastic bodies with relative mathematical simplicity. This study models the vibration of elastic waves in a multilayered inhomogeneous circular membrane using the Helmholtz equation with fractional-order infusion, effectively leveraging the anti-plane shear motion equation to avoid the computational complexity of universal plane motion equations. The method of the separation of variables and the conformable Bessel equation are utilized for the analytical examination of the model’s resulting vibrational displacements, as well as the dispersion relation. Additionally, the influence of various wave phenomena, including the dependencies of the wavenumber on the frequency and the phase speed on the wavenumber, respectively, with the variational effect of the fractional order on wave dispersion is considered. Numerical simulations of prototypical cases validate the formulated model, illustrating its applicability and effectiveness. The study reveals that fractional-order infusion significantly impacts the dispersion of elastic waves in both single- and multilayer membranes. The effects vary depending on the membrane’s structure and the wave propagation regime (long-wave vs. short-wave). These findings underscore the potential of fractional-order parameters in tailoring wave behavior for diverse scientific and engineering applications. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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41 pages, 3964 KiB

Open AccessArticle

Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation

by Mehboob Alam, Sirajul Haq, Ihteram Ali, M. J. Ebadi and Soheil Salahshour

Fractal Fract. 2023, 7(12), 882; https://doi.org/10.3390/fractalfract7120882 - 13 Dec 2023

Cited by 7 | Viewed by 1871

Abstract

In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including

L_{2}

L_{\infty}

, and

L_{r m s}

. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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16 pages, 320 KiB

Open AccessArticle

Qualitative Aspects of a Fractional-Order Integro-Differential Equation with a Quadratic Functional Integro-Differential Constraint

by Ahmed M. A. El-Sayed, Antisar A. A. Alhamali, Eman M. A. Hamdallah and Hanaa R. Ebead

Fractal Fract. 2023, 7(12), 835; https://doi.org/10.3390/fractalfract7120835 - 24 Nov 2023

Cited by 9 | Viewed by 1514

Abstract

This manuscript investigates a constrained problem of an arbitrary (fractional) order quadratic functional integro-differential equation with a quadratic functional integro-differential constraint. We demonstrate that there is at least one solution

x \in C [0, T]

to the problem. Moreover, we [...] Read more.

x \in C [0, T]

to the problem. Moreover, we outline the necessary demands for the solution’s uniqueness. In addition, the continuous dependence of the solution and the Hyers–Ulam stability of the problem are analyzed. In order to illustrate our results, we provide some particular cases and instances. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

20 pages, 1176 KiB

Open AccessArticle

Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform

by Mashael M. AlBaidani, Abdul Hamid Ganie and Adnan Khan

Fractal Fract. 2023, 7(11), 812; https://doi.org/10.3390/fractalfract7110812 - 9 Nov 2023

Cited by 9 | Viewed by 1611

Abstract

The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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18 pages, 578 KiB

Open AccessArticle

Qualitative Analysis of RLC Circuit Described by Hilfer Derivative with Numerical Treatment Using the Lagrange Polynomial Method

by Naveen S., Parthiban V. and Mohamed I. Abbas

Fractal Fract. 2023, 7(11), 804; https://doi.org/10.3390/fractalfract7110804 - 4 Nov 2023

Cited by 6 | Viewed by 1739

Abstract

This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order

ω \in (1, 2)

for the RLC model. Based on Schaefer’s fixed point theorem and [...] Read more.

This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order

ω \in (1, 2)

for the RLC model. Based on Schaefer’s fixed point theorem and Banach’s contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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16 pages, 442 KiB

Open AccessArticle

The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives

by Haifa Bin Jebreen

Fractal Fract. 2023, 7(10), 763; https://doi.org/10.3390/fractalfract7100763 - 17 Oct 2023

Cited by 2 | Viewed by 1830

Abstract

We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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33 pages, 7785 KiB

Open AccessArticle

Modeling and Dynamical Analysis of a Fractional-Order Predator–Prey System with Anti-Predator Behavior and a Holling Type IV Functional Response

by Baiming Wang and Xianyi Li

Fractal Fract. 2023, 7(10), 722; https://doi.org/10.3390/fractalfract7100722 - 30 Sep 2023

Cited by 6 | Viewed by 1542

Abstract

We here investigate the dynamic behavior of continuous and discrete versions of a fractional-order predator–prey system with anti-predator behavior and a Holling type IV functional response. First, we establish the non-negativity, existence, uniqueness and boundedness of solutions to the system from a mathematical analysis perspective. Then, we analyze the stability of its equilibrium points and the possibility of bifurcations using stability analysis methods and bifurcation theory, demonstrating that, under specific parameter conditions, the continuous system exhibits a Hopf bifurcation, while the discrete version exhibits a Neimark–Sacker bifurcation and a period-doubling bifurcation. After providing numerical simulations to illustrate the theoretically derived conclusions and by summarizing the various analytical results obtained, we finally present four interesting conclusions that can contribute to better management and preservation of ecological systems. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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33 pages, 678 KiB

Open AccessArticle

Exploring Families of Solitary Wave Solutions for the Fractional Coupled Higgs System Using Modified Extended Direct Algebraic Method

by Muhammad Bilal, Javed Iqbal, Rashid Ali, Fuad A. Awwad and Emad A. A. Ismail

Fractal Fract. 2023, 7(9), 653; https://doi.org/10.3390/fractalfract7090653 - 30 Aug 2023

Cited by 31 | Viewed by 1598

Abstract

In this paper, we suggest the modified Extended Direct Algebraic Method (mEDAM) to examine the existence and dynamics of solitary wave solutions in the context of the fractional coupled Higgs system, with Caputo’s fractional derivatives. The method begins with the formulation of nonlinear differential equations using a fractional complex transformation, followed by the derivation of solitary wave solutions. Two-dimensional, Three-dimensional and contour graphs are used to investigate the behavior of traveling wave solutions. The research reveals many families of solitary wave solutions as well as their deep interrelationships and dynamics. These discoveries add to a better understanding of the dynamics of the fractionally coupled Higgs system and have potential applications in areas that use nonlinear Fractional Partial Differential Equations (FPDEs). Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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24 pages, 1506 KiB

Open AccessArticle

Dynamic of Some Relapse in a Giving Up Smoking Model Described by Fractional Derivative

by Fawaz K. Alalhareth, Ahmed Boudaoui, Yacine El hadj Moussa, Noura Laksaci and Mohammed H. Alharbi

Fractal Fract. 2023, 7(7), 543; https://doi.org/10.3390/fractalfract7070543 - 14 Jul 2023

Cited by 1 | Viewed by 1777

Abstract

Smoking is associated with various detrimental health conditions, including cancer, heart disease, stroke, lung illnesses, diabetes, and fatal diseases. Motivated by the application of fractional calculus in epidemiological modeling and the exploration of memory and nonlocal effects, this paper introduces a mathematical model that captures the dynamics of relapse in a smoking cessation context and presents the dynamic behavior of the proposed model utilizing Caputo fractional derivatives. The model incorporates four compartments representing potential, persistent (heavy), temporally recovered, and permanently recovered smokers. The basic reproduction number

R_{0}

is computed, and the local and global dynamic behaviors of the free equilibrium smoking point (

Y_{0}

) and the smoking-present equilibrium point (

Y^{*}

) are analyzed. It is demonstrated that the free equilibrium smoking point (

Y_{0}

) exhibits global asymptotic stability when

R_{0} \leq 1

, while the smoking-present equilibrium point (

Y^{*}

) is globally asymptotically stable when

R_{0} > 1

. Additionally, analytical results are validated through a numerical simulation using the predictor–corrector PECE method for fractional differential equations in Matlab software. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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