Caputo-Fabrizio Derivative Research Articles

Modeling the dynamics of coronavirus with super-spreader class: A fractal-fractional approach

Super-spreaders of the novel coronavirus disease (or COVID-19) are those with greater potential for disease transmission to infect other people. Understanding and isolating the super-spreaders are important for controlling the COVID-19 incidence as well as future infectious disease outbreaks. Many scientific evidences can be found in the literature on reporting and impact of super-spreaders and super-spreading events on the COVID-19 dynamics. This paper deals with the formulation and simulation of a new epidemic model addressing the dynamics of COVID-19 with the presence of super-spreader individuals. In the first step, we formulate the model using classical integer order nonlinear differential system composed of six equations. The individuals responsible for the disease transmission are further categorized into three sub-classes, i.e., the symptomatic, super-spreader and asymptomatic. The model is parameterized using the actual infected cases reported in the kingdom of Saudi Arabia in order to enhance the biological suitability of the study. Moreover, to analyze the impact of memory index, we extend the model to fractional case using the well-known Caputo–Fabrizio derivative. By making use of the Picard-Lindelöf theorem and fixed point approach, we establish the existence and uniqueness criteria for the fractional-order model. Furthermore, we applied the novel fractal-fractional operator in Caputo–Fabrizio sense to obtain a more generalized model. Finally, to simulate the models in both fractional and fractal-fractional cases, efficient iterative schemes are utilized in order to present the impact of the fractional and fractal orders coupled with the key parameters (including transmission rate due to super-spreaders) on the pandemic peaks.

The Ulam Stability of Fractional Differential Equation with the Caputo-Fabrizio Derivative

The aim of this paper is to establish the Ulam stability of the Caputo-Fabrizio fractional differential equation with integral boundary condition. We also present the existence and uniqueness results of the solution for the Caputo-Fabrizio fractional differential equation by Krasnoselskii’s fixed point theorem and Banach fixed point theorem. Some examples are provided to illustrate our theorems.

Well-posed results for nonlocal fractional parabolic equation involving Caputo-Fabrizio operator

In this paper, we study the parabolic problem associated with non-local conditions, with the Caputo-Fabrizio derivative. Equations on the sphere have many important applications in physics, phenomena, and oceanography. The main motivation for us to study non-local boundary value problems comes from two main reasons: the first reason is that current major interest in several application areas. The second reason is to study approximation for the terminal value problem. With some given data, we prove that the problem has only the solution for two cases. In case \(\epsilon = 0,\) we prove the problem has a local solution. In case \(\epsilon > 0,\) then the problem has a global solution. The main tools and techniques in our demonstration are of using Banach's fixed point theorem in conjunction with some Fourier series analysis involved some evaluation of spherical harmonic function. Several upper and lower upper limit techniques for the Mittag-Lefler functions are also applied.

A new result for fractional differential equation with nonlocal initial value using Caputo-Fabrizio derivative

In this paper, using Caputo-Fabrizio fractional derivative, we obtain some new results for the existence and uniqueness of solutions of differential equation with nonlocal initial value about fractional order 0 < ? < 1 . These results are applied with the help of Arzela-Ascoli theorem and Schauder fixed point. Also based on some ?-contractive maps for such problems, some new unique theorem has been introduced and proved. Finally, some illustrative example is considered to show the effectiveness of the results.

Utilizing fixed point approach to investigate piecewise equations with non-singular type derivative

<abstract><p>In this research work, we establish some new results about piecewise equation involving Caputo Fabrizio derivative (CFD). The concerned class has been recently introduced and these results are fundamental for investigation of qualitative theory and numerical interpretation. We derive some necessary results for the existence, uniqueness and various form of Hyers-Ulam (H-U) type stability for the considered problem. For the required results, we need to utilize usual classical fixed point theorems due to Banach and Krasnoselskii's. Moreover, results devoted to H-U stability are derived by using classical tools of nonlinear functional analysis. Some pertinent test problems are given to demonstrate our results.</p></abstract>

Differential equations of arbitrary order under Caputo-Fabrizio derivative: some existence results and study of stability.

In this work, we consider the problem of the existence and uniqueness of solution, and also the simple existence of solution, for implicit differential equations of arbitrary order involving Caputo-Fabrizio derivative. The main tools for this study are contraction mapping principle and Schaefer's fixed point result. We also study the stability of the equations in the sense of Ulam-Hyers and also from the perspective of Ulam-Hyers-Rassias.

Fractional dynamics of coronavirus with comorbidity via Caputo-Fabrizio derivative

Coronavirus and its associated comorbidities have been the drivers of many deaths across the globe in recent times. Individuals with underlying medical conditions are at higher risk of becoming critically ill and developing complications if they are infected with the Coronavirus. In this paper, a Caputo-Fabrizio fractional-order model of coronavirus disease with comorbidity is formulated to access the impact of comorbidity diseases on COVID-19 transmission using both a fractional-order as well as a stochastic approach. Exponential law is utilized to present the existence and uniqueness of solutions using the fixed-point theory. The fractional stochastic approach is adopted to examine the same model to explore the random effect. Numerical simulations are used to support the theoretical results and the simulation results suggest that the increase of comorbidity development and the fractional-order derivative factor simultaneously increases the prevalence of the infection and the spread of the disease. The fractional stochastic numerical results suggest that the prediction of infection rate is more stochastic than deterministic.

Treatment of Nonlinear Electrical Circuits by the Caputo-Fabrizio Derivative

Treatment of Nonlinear Electrical Circuits by the Caputo-Fabrizio Derivative

Qualitative theory and approximate solution to a dynamical system under modified type Caputo-Fabrizio derivative

<abstract><p>Qualitative theory, together with approximate solutions to a dynamic system, are investigated. The proposed mathematical model is composed of protected, susceptible, infected and treated classes. The adopted model expresses the mechanism of disease due to Typhoid fever. A modified type Caputo-Fabrizio fractional derivative (CFFD) is considered for the intended results. With the help of fixed point theory, some sufficient conditions for the existence of approximate solutions are developed. Also, to compute an approximate solution with respect to each compartment, we utilize the Laplace Transform and the Adomian decomposition method (ADM). A graphical presentation corresponding to some fundamental data is given.</p></abstract>

Magnetic drug targeting during Caputo-Fabrizio fractionalized blood flow through a permeable vessel.

Nanoparticle-based drug targeting is an important platform for the treatment of cardiovascular disorders. Magnetic drug targeting is more significant as it is a noninvasive procedure and biocompatible. The present problem aims to understand magnetic drug delivery to a specific location in a permeable blood vessel under the vibration and magnetic environment. Caputo-Fabrizio fractional-order time derivatives are used in the governing equations. The momentum equations are solved analytically and presented in the form of Lorenzo-Hartley and Robotonov-Hartley functions and convolution of the Laplace transform. Convolution integrations are solved by using the numerical integration technique. The Fourth order Runge-Kutta method (RK4) is used to solve the force balance equation. The influence of pertinent parameters such as Reynolds number, pulsatile frequency, magnetic field strength, Darcy number and fractional-order parameters are presented through graphs. It is observed that increasing Reynolds number results in decreasing the tendency of the drug to capture near the tumor site, whereas the pulsatile frequency presents an opposite phenomenon. Increasing the magnetic field strength and Darcy number boosts the capture efficiency of drug particles near the tumor site. The short memory effect efficiently captures the magnetic drug carriers to a specific location under the action of suitable magnetic field strength.

Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

<abstract><p>In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 < \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.</p></abstract>

Existence and uniqueness results for coupled system of fractional differential equations with exponential kernel derivatives

<abstract><p>In the framework of Caputo-Fabrizio derivatives, we study a new coupled system of fractional differential equations of higher orders supplemented with coupled nonlocal boundary conditions. The existence and uniqueness results of the solutions are proved. We consider the classical fixed-point theories due to Banach and Krasnoselskii for the main results. An example illustrating the main results is introduced.</p></abstract>

Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives

<abstract><p>The approximate solution of the Kersten-Krasil'shchik coupled Korteweg-de Vries-modified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in traffic flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method's effectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems.</p></abstract>

Numerical Approaches of the Generalized Time-Fractional Burgers’ Equation with Time-Variable Coefficients

The generalized time-fractional, one-dimensional, nonlinear Burgers equation with time-variable coefficients is numerically investigated. The classical Burgers equation is generalized by considering the generalized Atangana-Baleanu time-fractional derivative. The studied model contains as particular cases the Burgers equation with Atangana-Baleanu, Caputo-Fabrizio, and Caputo time-fractional derivatives. A numerical scheme, based on the finite-difference approximations and some integral representations of the two-parameter Mittag-Leffler functions, has been developed. Numerical solutions of a particular problem with initial and boundary values are determined by employing the proposed method. The numerical results are plotted to compare solutions corresponding to the problems with time-fractional derivatives with different kernels.

A new fractional viscoelastic model for an infinitely thermoelastic body with a spherical cavity including Caputo-Fabrizio operator without singular kernel

Many applications and contexts including materials science, metallurgy, and solid-state physics, are concerned with the study of the behavior of viscoelastic materials. In addition, any composite or complex construction containing embedded polymers exhibits viscoelastic behavior under static and dynamic stress conditions. Wide types of linear/nonlinear constitutive models have been proposed to define the viscoelastic deformation process of viscoelastic materials in order to explore their mechanical behavior. However, it has been shown that the constitutive relationship in the integer-order of stress-strain available in conventional viscoelastic models may fail in some types of situations and do not match well with empirical evidence. In this paper, a novel mathematical model is provided that uses Caputo-Fabrizio fractional-order derivatives to describe the viscoelastic phenomena and is consistent with thermodynamic principles. The Caputo-Fabrizio kernel has many features, such as nonlocality and non-singularity in addition to the exponential form. The suggested model is used to study the dynamic reactions of an unbounded body with a spherical cavity made of viscoelastic material subjected to time-varying heat. Using the Laplace transform technique, numerical calculations of many physical fields are obtained and explored in depth.

Analysis of Multiterm Initial Value Problems with Caputo–Fabrizio Derivative

In this paper, we discuss the solvability of a class of multiterm initial value problems involving the Caputo–Fabrizio fractional derivative via the Laplace transform. We derive necessary and sufficient conditions to guarantee the existence of solutions to the problem. We also obtain the solutions in closed forms. We present two examples to illustrate the validity of the obtained results.

A study of fractional TB model due to mycobacterium tuberculosis bacteria

The main purpose of the proposed work is to study a TB infection model under incomplete treatment numerically and identify potential infection dynamics. We use the arbitrary-order Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) derivatives to extend the classical TB infection model into a non-classical model. In addition, through fixed-point results, the existence and uniqueness of solutions for the arbitrary order TB model are discussed. We illustrate computational techniques from existing literature for the numerical solutions of the mentioned fractional models. The numerical simulations are presented using MATLAB software with various choices of the fractional-order parameter.

Fractional dynamics and stability analysis of COVID-19 pandemic model under the harmonic mean type incidence rate

In this research, COVID-19 model is formulated by incorporating harmonic mean type incidence rate which is more realistic in average speed. Basic reproduction number, equilibrium points, and stability of the proposed model is established under certain conditions. Runge-Kutta fourth order approximation is used to solve the deterministic model. The model is then fractionalized by using Caputo-Fabrizio derivative and the existence and uniqueness of the solution are proved by using Banach and Leray-Schauder alternative type theorems. For the fractional numerical simulations, we use the Adam-Moulton scheme. Sensitivity analysis of the proposed deterministic model is studied to identify those parameters which are highly influential on basic reproduction number.

Prediction studies of the epidemic peak of coronavirus disease in Japan: From Caputo derivatives to Atangana–Baleanu derivatives

New atypical pneumonia caused by a virus called Coronavirus (COVID-19) appeared in Wuhan, China in December 2019. Unlike previous epidemics due to the severe acute respiratory syndrome (SARS) and the Middle East respiratory syndrome coronavirus (MERS-CoV), COVID-19 has the particularity that it is more contagious than the other previous ones. In this paper, we try to predict the COVID-19 epidemic peak in Japan with the help of real-time data from January 15 to February 29, 2020 with the uses of fractional derivatives, namely, Caputo derivatives, the Caputo–Fabrizio derivatives, and Atangana–Baleanu derivatives in the Caputo sense. The fixed point theory and Picard–Lindel of approach used in this study provide the proof for the existence and uniqueness analysis of the solutions to the noninteger-order models under the investigations. For each fractional model, we propose a numerical scheme as well as prove its stability. Using parameter values estimated from the Japan COVID-19 epidemic real data, we perform numerical simulations to confirm the effectiveness of used approximation methods by numerical simulations for different values of the fractional-order [Formula: see text], and to give the predictions of COVID-19 epidemic peaks in Japan in a specific range of time intervals.

Note on a Allen-Cahn equation with Caputo-Fabrizio derivative

In this short note, we investigate the Allen-Cahn equation with the appearance of the Caputo-Fabizzio derivative. We obtain a local solution when the initial value is small enough. This is an equation that has many practical applications. The power term in the nonlinear component of the source function and the Caputo-Fabizzio operator combine to make finding the solution space more difficult than the classical problem. We discovered a new technique, connecting Hilbert scale and $L^p$ spaces, to overcome these difficulties. Evaluation of the smoothness of the solution was also performed. The research ideas in this paper can be used for many other models.