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14 pages, 4897 KiB

Open AccessArticle

Novel Dynamic Behaviors in Fractional Chaotic Systems: Numerical Simulations with Caputo Derivatives

by Mohamed A. Abdoon, Diaa Eldin Elgezouli, Borhen Halouani, Amr M. Y. Abdelaty, Ibrahim S. Elshazly, Praveen Ailawalia and Alaa H. El-Qadeem

Axioms 2024, 13(11), 791; https://doi.org/10.3390/axioms13110791 - 16 Nov 2024

Cited by 1 | Viewed by 995

Abstract

Over the last several years, there has been a considerable improvement in the possible methods for solving fractional-order chaotic systems; however, achieving high accuracy remains a challenge. This work proposes a new precise numerical technique for fractional-order chaotic systems. Through simulations, we obtain [...] Read more.

Over the last several years, there has been a considerable improvement in the possible methods for solving fractional-order chaotic systems; however, achieving high accuracy remains a challenge. This work proposes a new precise numerical technique for fractional-order chaotic systems. Through simulations, we obtain new types of complex and previously undiscussed dynamic behaviors.These phenomena, not recognized in prior numerical results or theoretical estimations, underscore the unique dynamics present in fractional systems. We also study the effects of the fractional parameters

β_{1}

,

β_{2}

, and

β_{3}

on the system’s behavior, comparing them to integer-order derivatives. It has been demonstrated via the findings that the suggested technique is consistent with conventional numerical methods for integer-order systems while simultaneously providing an even higher level of precision. It is possible to demonstrate the efficacy and precision of this technique through simulations, which demonstrates that this method is useful for the investigation of complicated chaotic models. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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16 pages, 963 KiB

Open AccessArticle

A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

by Arman Dabiri, Behrouz Parsa Moghaddam, Elham Taghizadeh and Alexandra Galhano

Axioms 2024, 13(10), 695; https://doi.org/10.3390/axioms13100695 - 7 Oct 2024

Viewed by 1042

Abstract

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a [...] Read more.

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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16 pages, 326 KiB

Open AccessArticle

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

by Michal Pospíšil and Lucia Pospíšilová Škripková

Axioms 2024, 13(10), 680; https://doi.org/10.3390/axioms13100680 - 1 Oct 2024

Viewed by 828

Abstract

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered

Ψ

–Caputo fractional derivative. These include [...] Read more.

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered

Ψ

–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

19 pages, 1007 KiB

Open AccessArticle

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

by Wenxiu Gong, Zuoliang Xu and Yesen Sun

Axioms 2024, 13(10), 674; https://doi.org/10.3390/axioms13100674 - 29 Sep 2024

Cited by 1 | Viewed by 742

Abstract

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described [...] Read more.

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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25 pages, 773 KiB

Open AccessArticle

An Efficient and Stable Caputo-Type Inverse Fractional Parallel Scheme for Solving Nonlinear Equations

by Mudassir Shams and Bruno Carpentieri

Axioms 2024, 13(10), 671; https://doi.org/10.3390/axioms13100671 - 27 Sep 2024

Cited by 1 | Viewed by 680

Abstract

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable [...] Read more.

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable Caputo-type inverse numerical fractional scheme for simultaneously approximating all roots of nonlinear equations, with a convergence order of

2 ψ + 2

. The scheme is applied to various nonlinear problems, utilizing dynamical analysis to determine efficient initial values for a single root-finding Caputo-type fractional scheme, which is further employed in inverse fractional parallel schemes to accelerate convergence rates. Several sets of random initial vectors demonstrate the global convergence behavior of the proposed method. The newly developed scheme outperforms existing methods in terms of accuracy, consistency, validation, computational CPU time, residual error, and stability. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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23 pages, 1588 KiB

Open AccessArticle

Two-Dimensional Time Fractional River-Pollution Model and Its Remediation by Unsteady Aeration

by Priti V. Tandel, Manan A. Maisuria and Trushitkumar Patel

Axioms 2024, 13(9), 654; https://doi.org/10.3390/axioms13090654 - 23 Sep 2024

Viewed by 899

Abstract

This study contains a mathematical model for river pollution and its remediation for an unsteady state and investigates the effect of aeration on the degradation of pollutants. The governing equation is a pair of nonlinear time-fractional two-dimensional advection-diffusion equations for pollutant and dissolved [...] Read more.

This study contains a mathematical model for river pollution and its remediation for an unsteady state and investigates the effect of aeration on the degradation of pollutants. The governing equation is a pair of nonlinear time-fractional two-dimensional advection-diffusion equations for pollutant and dissolved oxygen (DO) concentration. The coupling of these equations arises due to the chemical interactions between oxygen and pollutants, forming harmless chemicals. The Fractional Reduced Differential Transform Method (FRDTM) is applied to provide approximate solutions for the given model. Also, the convergence of solutions is checked for efficacy and accuracy. The effect of longitudinal and transverse diffusion coefficients of pollutant and DO on the concentration of pollutant and DO is analyzed numerically and graphically. Also, we checked the effect of change in the river’s longitudinal and transverse seepage velocity on pollutant and DO concentration numerically and graphically. We analyzed the comparison of change in the value of half-saturated oxygen demand concentration for pollutant decay on pollutant and DO concentration numerically and graphically. Also, numerical and graphical analysis examined the effect of fractional parameters on pollution levels. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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28 pages, 1656 KiB

Open AccessArticle

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

by Wenbin Li, Hongjun Ma and Tinggang Zhao

Axioms 2024, 13(5), 305; https://doi.org/10.3390/axioms13050305 - 4 May 2024

Viewed by 1406

Abstract

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, [...] Read more.

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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21 pages, 748 KiB

Open AccessArticle

Fractional Order Mathematical Modelling of HFMD Transmission via Caputo Derivative

by Aakash Mohandoss, Gunasundari Chandrasekar, Mutum Zico Meetei and Ahmed H. Msmali

Axioms 2024, 13(4), 213; https://doi.org/10.3390/axioms13040213 - 25 Mar 2024

Cited by 2 | Viewed by 1272

Abstract

This paper studies a nonlinear fractional mathematical model for hand, foot, and mouth Disease (HFMD), incorporating a vaccinated compartment. Our initial focus involves establishing the non-negativity and boundedness of the fractional order dynamical model. The existence and uniqueness of the system are discussed [...] Read more.

This paper studies a nonlinear fractional mathematical model for hand, foot, and mouth Disease (HFMD), incorporating a vaccinated compartment. Our initial focus involves establishing the non-negativity and boundedness of the fractional order dynamical model. The existence and uniqueness of the system are discussed using the Caputo derivative operator formulation. Applying a fixed-point approach, we obtain results that confirm the presence of at least one solution. We analyze the stability behavior at the two equilibrium points (disease-free and endemic states) of the model and derive the basic reproduction number. Numerical simulations are conducted using the fractional Euler approach, and the simulation results confirm our analytical conclusions. This comprehensive approach enhances the understanding of HFMD dynamics and facilitates the policy making of health care centers to control the further spread of this disease. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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16 pages, 4276 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Time-Domain Fractional Behaviour Modelling with Rational Non-Singular Kernels

by Jocelyn Sabatier and Christophe Farges

Axioms 2024, 13(2), 99; https://doi.org/10.3390/axioms13020099 - 31 Jan 2024

Cited by 3 | Viewed by 1358

Abstract

This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power [...] Read more.

This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power law function that can be approximated in the time domain by a rational function whose pole and zeros are interlaced and linked by geometric laws. The Laplace transform and frequency response of this class of kernel is given and compared with an approximation found in the literature. The comparison reveals less phase oscillation with the solution proposed by the authors. A parameter estimation method is finally proposed to obtain the rational kernel model for general fractional behaviour. An application performed with this estimation method demonstrates the interest in non-singular rational kernels to model fractional behaviours. Another interest is the physical interpretation fractional behaviours that can be implemented with delay distributions. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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14 pages, 300 KiB

Open AccessArticle

General Fractional Calculus Operators of Distributed Order

by Mohammed Al-Refai and Yuri Luchko

Axioms 2023, 12(12), 1075; https://doi.org/10.3390/axioms12121075 - 24 Nov 2023

Cited by 5 | Viewed by 1417

Abstract

In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric [...] Read more.

In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric Sonin kernels with power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann–Liouville and Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

14 pages, 1046 KiB

Open AccessArticle

Solution of Two-Dimensional Solute Transport Model for Heterogeneous Porous Medium Using Fractional Reduced Differential Transform Method

by Manan A. Maisuria, Priti V. Tandel and Trushitkumar Patel

Axioms 2023, 12(11), 1039; https://doi.org/10.3390/axioms12111039 - 8 Nov 2023

Cited by 4 | Viewed by 1414

Abstract

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining [...] Read more.

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining exponential function in both the longitudinal and transverse directions. The governing equation is a time-fractional two-dimensional advection–dispersion equation with a variable form of dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. The solution of the present problem is obtained by the fractional reduced differential transform method (FRDTM). The analysis of the initial retardation factor has been carried out via plots. Also, the influence of initial longitudinal and transverse dispersion coefficients and velocities has been examined by graphical analysis. The impact of fractional parameters on pollution levels is also analyzed numerically and graphically. The study of convergence for the FRDTM technique has been conducted to assess its efficacy and accuracy. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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15 pages, 692 KiB

Open AccessArticle

New Results Achieved for Fractional Differential Equations with Riemann–Liouville Derivatives of Nonlinear Variable Order

by Hallouz Abdelhamid, Gani Stamov, Mohammed Said Souid and Ivanka Stamova

Axioms 2023, 12(9), 895; https://doi.org/10.3390/axioms12090895 - 20 Sep 2023

Cited by 3 | Viewed by 1779

Abstract

This paper proposes new existence and uniqueness results for an initial value problem (IVP) of fractional differential equations of nonlinear variable order. Riemann–Liouville-type fractional derivatives are considered in the problem. The new fundamental results achieved in this work are obtained by using the [...] Read more.

This paper proposes new existence and uniqueness results for an initial value problem (IVP) of fractional differential equations of nonlinear variable order. Riemann–Liouville-type fractional derivatives are considered in the problem. The new fundamental results achieved in this work are obtained by using the inequalities technique and the fixed point theory. In addition, uniform stability criteria for the solutions are derived. The accomplished results are new and complement the scientific research in the field. A numerical example is composed to show the efficacy and potency of the proposed criteria. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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13 pages, 1294 KiB

Open AccessArticle

Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation

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

Axioms 2023, 12(4), 400; https://doi.org/10.3390/axioms12040400 - 20 Apr 2023

Cited by 18 | Viewed by 1616

Abstract

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV [...] Read more.

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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