Research

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 1 | Viewed by 1035

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 [...] Read more.

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

Viewed by 894

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.

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 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 - 09 Nov 2023

Cited by 2 | Viewed by 947

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 [...] Read more.

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 - 04 Nov 2023

Viewed by 933

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 1 | Viewed by 1070

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 [...] Read more.

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

Viewed by 703

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 [...] Read more.

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 1 | Viewed by 751

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 [...] Read more.

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 819

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 [...] Read more.

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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