Caputo Operator Research Articles

A study on eco-epidemiological model with fractional operators

This paper employs fractional calculus (FC) for modeling three-dimensional prey-predator populations model. This study uses an eco-epidemiological system in which the prey disease is constructed as a susceptible-infected (SI) disease. The Caputo and Caputo-Fabrizio (CF) operators are consolidated into this model and the existence of a solution is explored. The model is evaluated for uniqueness under what conditions it provides a unique solution. Based on the singular kernel of the Caputo operator, we investigate the properties of the proposed model and show it can be stable locally. We developed maximum bifurcation diagrams to analyze the dynamics of the epidemiological model as varying transmission rates β and attack rates b1. To simulate the dynamics of proposed fractional systems, we employed the Toufik-Atangana (TA) numerical technique with the Caputo operator. Moreover, we present another numerical approach based on Adams-Bashforth (AB) technique with CF operators. Results of the numerical analysis show that diverse non-integer operator alternatives to the eco-epidemiological predator-prey model result in a range of dynamical behaviors.

On the Dynamics of a Discrete Fractional-Order Cournot–Bertrand Competition Duopoly Game

A discreet fractional-order Cournot–Bertrand competition duopoly game is introduced based on the fractional-order difference calculus of the Caputo operator. The model is designed when players can make long memory decisions. The local stability of equilibrium points is discussed for the proposed model. Some numerical simulations explore the model’s bifurcation and chaos by employing bifurcation diagrams, phase portraits, maximal Lyapunov exponents, and time series. According to our findings, the fractional-order parameter has an effect on the game’s stability and dynamics.

Application of Double Shehu Transforms to Caputo Fractional Partial Differential Equations

Recently, a new generalization of the double Laplace transform and double Sumudu transform, namely, double Shehu transform, has been introduced. This study has derived the double Shehu transform (DHT) formulas for the fractional Caputo operators. We then applied this generalized integral transform to solve fractional partial differential equations involving Caputo derivative.

On Convexity, Monotonicity and Positivity Analysis for Discrete Fractional Operators Defined Using Exponential Kernels

This article deals with analysing the positivity, monotonicity and convexity of the discrete nabla fractional operators with exponential kernels from the sense of Riemann and Caputo operators. These operators are called discrete nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional operators. Further, some of our results concern sequential nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional differences, such as ∇aCFRμ∇bCFCυh(x), for various values of start points a and b, and for orders υ and μ in different ranges. Three illustrative examples of the main lemmas in the case of Riemann–Liouville are given at the end of the article.

Constructing analytical estimates of the fuzzy fractional-order Boussinesq model and their application in oceanography

The main idea of this article is the investigation of atmospheric internal waves, often known as gravity waves. This arises within the ocean rather than at the interface. A shallow fluid assumption is illustrated by a series of nonlinear partial differential equations in the framework. Because the waves are scattered over a wide geographical region, this system can precisely replicate atmospheric internal waves. In this research, the numerical solutions to the fuzzy fourth-order time-fractional Boussinesq equation (BSe) are determined for the case of the aquifer propagation of long waves having small amplitude on the surface of water from a channel. The novel scheme, namely the generalized integral transform (proposed by H. Jafari [35]) coupled with the Adomian decomposition method (GIADM), is used to extract the fuzzy fractional BSe in R,Rn and (2nth)-order including gH-differentiability. To have a clear understanding of the physical phenomena of the projected solutions, several algebraic aspects of the generalized integral transform in the fuzzy Caputo and Atangana-Baleanu fractional derivative operators are discussed. The confrontation between the findings by Caputo and ABC fractional derivatives under generalized Hukuhara differentiability are presented with appropriate values for the fractional order and uncertainty parameters ℘∈[0,1] were depicted in diagrams. According to proposed findings, hydraulic engineers, being analysts in drainage or in water management, might access adequate storage volume quantity with an uncertainty level.

Event-Triggered State Estimation for Fractional-Order Neural Networks

This paper is concerned with the problem of event-triggered state estimation for a class of fractional-order neural networks. An event-triggering strategy is proposed to reduce the transmission frequency of the output measurement signals with guaranteed state estimation performance requirements. Based on the Lyapunov method and properties of fractional-order calculus, a sufficient criterion is established for deriving the Mittag–Leffler stability of the estimation error system. By making full use of the properties of Caputo operator and Mittag–Leffler function, the evolution dynamics of measured error is analyzed so as to exclude the unexpected Zeno phenomenon in the event-triggering strategy. Finally, two numerical examples and simulations are provided to show the effectiveness of the theoretical results.

Deterministic and fractional modeling of a computer virus propagation

The dynamic behaviors of computer virus models are investigated. In the first phase, we discussed the deterministic version of the proposed model by taking into consideration the local and global stability. For global stability the Castillo-Chavez approach is taken into account. The deterministic version is numerically solved by the Runge–Kutta scheme. The model is then fractionalized by using the Atangana–Baleanu–Caputo operator. Existence uniqueness and Hyers–Ulam stability of the fractionalized model is established. The Atangana–Toufik method is used for the numerical examination of a fractional version of the proposed model.

Stability results for fractional integral pantograph differential equations involving two Caputo operators

<abstract><p>In this paper, we investigate the existence-uniqueness, and Ulam Hyers stability (UHS) of solutions to a fractional-order pantograph differential equation (FOPDE) with two Caputo operators. Banach's fixed point (BFP) and Leray-alternative Schauder's are used to prove the existence- uniqueness of solutions. In addition, we discuss and demonstrate various types of Ulam-stability for our problem. Finally, an example is provided for clarity.</p></abstract>

Analytical Investigation of Third-Order Time-Fractional Dispersive Partial Differential Equations Using Sumudu Transform Iterative Method

This paper investigates the approximate analytical solutions of third-order timefractional dispersive partial differential equations in one-and higher-dimensional spaces by employing a newly developed analytical method, the Sumudu transform iterative method. To express fractional derivatives, the Caputo operator is used. Furthermore, the results of this investigation are graphically represented, and the solution graphs reveal that the approximate solutions are closely connected to the exact solutions.

A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative

<abstract><p>In this study, the $ \psi $-Haar wavelets operational matrix of integration is derived and used to solve linear $ \psi $-fractional partial differential equations ($ \psi $-FPDEs) with the fractional derivative defined in terms of the $ \psi $-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear $ \psi $-FPDE using Haar wavelets. By combining the operational matrix and $ \psi $-fractional integration, we approximate the solution and its other $ \psi $-fractional partial derivatives. Then substituting these approximations in the given $ \psi $-FPDEs, we obtained a system of linear algebraic equations. Finally, the approximate solution is obtained by solving this system. The simplicity and effectiveness of the proposed method as a mathematical tool for solving $ \psi $-Fractional partial differential equations is one of its main advantages. The sparse nature of the operational matrices improves the ability of the proposed method to execute with less computation complexity. Numerical examples are provided to show the efficiency and effectiveness of the method.</p></abstract>

Evaluation of regularized long-wave equation via Caputo and Caputo-Fabrizio fractional derivatives

<abstract><p>The analytical solution of fractional-order regularized long waves in the context of various operators is presented in this study as a framework for the homotopy perturbation transform technique. To investigate regularized long wave equations, we first establish the Yang transform of the fractional Caputo and Caputo-Fabrizio operators. The fractional order regularized long wave equation is solved using the Yang transform as well. The accuracy of the proposed operators are verified using numerical problems, and the resulting solutions are shown in the figures. The solutions demonstrate how the suggested approach is accurate and suitable for analyzing nonlinear physical and engineering challenges.</p></abstract>

Numerical solutions of fuzzy equal width models via generalized fuzzy fractional derivative operators

<abstract><p>The Shehu homotopy perturbation transform method (SHPTM) via fuzziness, which combines the homotopy perturbation method and the Shehu transform, is the subject of this article. With the assistance of fuzzy fractional Caputo and Atangana-Baleanu derivatives operators, the proposed methodology is designed to illustrate the reliability by finding fuzzy fractional equal width (EW), modified equal width (MEW) and variants of modified equal width (VMEW) models with fuzzy initial conditions (ICs). In cold plasma, the proposed model is vital for generating hydro-magnetic waves. We investigated SHPTM's potential to investigate fractional nonlinear systems and demonstrated its superiority over other numerical approaches that are accessible. Another significant aspect of this research is to look at two significant fuzzy fractional models with differing nonlinearities considering fuzzy set theory. Evaluating various implementations verifies the method's impact, capabilities, and practicality. The level impacts of the parameter $ \hbar $ and fractional order are graphically and quantitatively presented, demonstrating good agreement between the fuzzy approximate upper and lower bound solutions. The findings are numerically examined to crisp solutions and those produced by other approaches, demonstrating that the proposed method is a handy and astonishingly efficient instrument for solving a wide range of physics and engineering problems.</p></abstract>

Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

<abstract><p>In this research, the $ \bar{\mathbf{q}} $-homotopy analysis transform method ($ \bar{\mathbf{q}} $-HATM) is employed to identify fractional-order Whitham–Broer–Kaup equation (WBKE) solutions. The WBKE is extensively employed to examine tsunami waves. With the aid of Caputo and Atangana-Baleanu fractional derivative operators, to obtain the analytical findings of WBKE, the predicted algorithm employs a combination of $ \bar{\mathbf{q}} $-HAM and the Aboodh transform. The fractional operators are applied in this work to show how important they are in generalizing the frameworks connected with kernels of singularity and non-singularity. To demonstrate the applicability of the suggested methodology, various relevant problems are solved. Graphical and tabular results are used to display and assess the findings of the suggested approach. In addition, the findings of our recommended approach were analyzed in relation to existing methods. The projected approach has fewer processing requirements and a better accuracy rate. Ultimately, the obtained results reveal that the improved strategy is both trustworthy and meticulous when it comes to assessing the influence of nonlinear systems of both integer and fractional order.</p></abstract>

Fractional-order coronavirus models with vaccination strategies impacted on Saudi Arabia's infections

<abstract><p>Several newly nonlinear models for describing dynamics of COVID-19 pandemic have been proposed and investigated in literature recently. In light of these models, we attempt to reveal the role of fractional calculus in describing the growth of COVID-19 dynamics implemented on Saudi Arabia's society over 107 days; from 17 Dec 2020 to 31 March 2021. Above is achieved by operating two fractional-order differential operators, Caputo and the Caputo-Fabrizio operators, instead of the classical one. One of expanded SEIR models is utilized for achieving our purpose. With the help of using the Generalized Euler Method (GEM) and Adams-Bashforth Method (ABM), the numerical simulations are performed respectively in view of the Caputo and Caputo-Fabrizio operators. Accordance with said, the stability analysis of the two proposed fractional-order models is discussed and explored in view of obtaining the equilibrium points, determining the reproductive number ($ R_0 $) and computing the elasticity indices of $ R_0 $. Several numerical comparisons reveal that the fractional-order COVID-19 models proposed in this work are better than that of classical one when such comparisons are performed between them and some real data collected from Saudi Arabia's society. This inference together with the cases predictions that could easily deduced from the proposed fractional-order models can allow primary decision makers and influencers to set the right plans and logic strategies that should be followed to face this pandemic.</p></abstract>

3-D Time-Domain Airborne EM Forward Modeling With IP Effect Based on Implicit Difference Discretization of Caputo Operator

As an efficient geophysical exploration method, the time-domain airborne electromagnetic (AEM) data often show sign reversal in late-time channels due to induced polarization (IP) effect. The traditional imaging and inversion methods without considering the IP effect cannot recover the true electrical structure of the earth, so it is necessary to develop 3D EM forward modeling and inversion techniques with IP effect. In this paper, we propose a 3D forward modeling method for time-domain AEM with IP effect based on unstructured finite-element (FE) method. To describe the IP effect of a medium, we introduce the Cole-Cole model and transform it into fractional derivative form using frequency-time conversion. Then, we discretize it using the difference discretization format of Caputo fractional derivative. Finally, we use the vector FE method based on unstructured tetrahedral mesh and unconditionally stable second-order backward Euler’s scheme to discretize Maxwell’s equations in space and time. In this way, we can solve the 3D forward modeling problem for AEM with IP effect in time domain. We verify the accuracy of our algorithm by comparing it with the 1D semi-analytical solution for a half-space model, and then calculate EM responses for typical abnormal models and analyze the characteristics of IP effect.

Study of Fractional Order Tri-Tropic Prey-Predator Model with Fear Effect on Prey Population

In this manuscript, we have studied a fractional-order tri-trophic model with the help of Caputo operator. The total population is divided into three parts, namely prey, intermediate predator and top predator. In addition, the predator fear impact on prey population is suggested in this paper. Existence and uniqueness along with non-negativity and boundedness of the model system have been investigated. We have studied the local stability at all equilibrium points. Also, we have discussed global stability and Hopf bifurcation of our suggested model at interior equilibrium point. The Adam-Bashforth-Moulton approach is utilized to approximate the solution to the proposed model. With the help of MATLAB, we were able to conduct graphical demonstrations and numerical simulations.

Computational analysis of COVID-19 model outbreak with singular and nonlocal operator

<abstract><p>The SARS-CoV-2 virus pandemic remains a pressing issue with its unpredictable nature, and it spreads worldwide through human interaction. Current research focuses on the investigation and analysis of fractional epidemic models that discuss the temporal dynamics of the SARS-CoV-2 virus in the community. In this work, we choose a fractional-order mathematical model to examine the transmissibility in the community of several symptoms of COVID-19 in the sense of the Caputo operator. Sensitivity analysis of $ R_{0} $ and disease-free local stability of the system are checked. Also, with the assistance of fixed point theory, we demonstrate the existence and uniqueness of the system. In addition, numerically we solve the fractional model and presented some simulation results via actual estimation parameters. Graphically we displayed the effects of numerous model parameters and memory indexes. The numerical outcomes show the reliability, validation, and accuracy of the scheme.</p></abstract>

Mathematical assessment of the dynamics of the tobacco smoking model: An application of fractional theory

<abstract><p>In this paper we consider fractional-order mathematical model describing the spread of the smoking model in the sense of Caputo operator with tobacco in the form of snuffing. The threshold quantity $ \mathcal{R}_0 $ and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of of the solution of the considered model. The new version of numerical approximation's framework for the approximation of Caputo operator is used. Finally, the numerical results are presented to justify the significance of the arbitrary fractional order derivative. The analysis shows fractional-order model of tobacco smoking in Caputo sense gives useful information as compared to the classical integer order tobacco smoking model.</p></abstract>

On periodic solutions of fractional-order differential systems with a fixed length of sliding memory

The fractional-order derivative of a non-constant periodic function is not periodic with the same period. Consequently, any time-invariant fractional-order systems do not have a non-constant periodic solution. This property limits the applicability of fractional derivatives and makes it unfavorable to model periodic real phenomena.This article introduces a modification to the Caputo and Rieman-Liouville fractional-order operators by fixing their memory length and varying the lower terminal. It is shown that this modified definition of fractional derivative preserves the periodicity. Therefore, periodic solutions can be expected in fractional-order systems in terms of the new fractional derivative operator. To confirm this assertion, one investigates two examples, one linear system for which one gives an exact periodic solution by its analytical expression and another nonlinear system for which one provides exact periodic solutions using qualitative and numerical methods.

New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect.