Fractional Caputo-Fabrizio Derivative Research Articles

On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives

In this work, we present a design for a Newton-Leipnik system with a fractional Caputo-Fabrizio derivative to explain its chaotic characteristics. This time-varying fractional Caputo-Fabrizio derivative approach is applied to solve the model numerically, and to check the solution’s existence and uniqueness. The existence and uniqueness of results of a fractional-order model under the Caputo-Fabrizio fractional operator have been proved by fixed point theory. As well, we achieved a stable result by applying the Ulam-Hyers concept. Chaos is controlled by linear controllers. Furthermore, the Lyapunov exponent of the system indicates that the chaos control findings are accurate. Based on weighted covariant Lyapunov vectors we construct a background covariance matrix using the Kaplan-Yorke dimension. Using a numerical example, this suggested method is illustrated for its applicability and efficiency.

A numerical investigation of marriage divorce model: Fractal fractional perspective

Even if a marriage began as “till forever”, divorce is a legal way for married couples to end their relationship. Divorce, indeed, is a complex process that may be seen from several perspectives. Sociologists believe that the family gives more than simply individuals. It is the structure that allows individuals to develop mentally while also providing the necessary educational and financial assistance. Divorced spouses face economic, social, and financial exploitation. This, in turn, disintegrates families and diminishes the concept of the family as the core unit of society, causing society to become stalled. The breakup of families that will produce healthy and happy children in the future is a significant societal issue. According to Eurostat statistics, the number of crude marriages in European nations was 8 in 1964 and 4.3 in 2014. At the same period, crude divorce rates grew by more than twice as much, from 0.8 in 1964 to 1.8 in 2019. In 2022, Latvia, Lithuania, and Denmark had the most number of divorces. Considering the datas, the investigation of this phenomenon, which deeply injures and shakes society, by various branches of science and the study of its dynamics has become an important objective and mathematical analysis of divorce illuminates the motivation for this paper. Therefore, the mathematical model of divorce is considered with by the fractal fractional Caputo–Fabrizio derivative. Firstly, using Equilibrium points, the model’s linear stability is obtained. Then, the existence and uniqueness the solution of the model was proven by the Banach Fixed point theorem. Lastly, the behavior of the model is evaluated using graphics for different values of the fractal dimension and fractional derivative by developing a numerical method for the model that contains the fractal fractional derivative. Experimental results are analyzed for different instances of the key parameters that played major roles for each of the sub-population classes.

Hermite wavelets collocation method for solving a fredholm integro-differential equation with fractional Caputo-Fabrizio derivative

In this paper, we investigate the numerical study of nonlinear Fredholm integro-differential equation with the fractional Caputo-Fabrizio derivative. We use the Hermite wavelets and collocation technique to approximate the exact solution by reducing the Fredholm integro-differential equation to a nonlinear algebraic system. Furthermore, we apply this numerical method on certain examples to check its accuracy and validity.

A novel approach for numerical treatment of traveling wave solution of ion acoustic waves as a fractional nonlinear evolution equation on Shehu transform environment

Abstract In this paper, we develop and employ an efficient numerical technique for traveling wave solution of the Time Fractional Zakharov-Kuznetsov (TFZK) equation, also known as the nonlinear evolution equation, using the Modified Adomian Decomposition Approach (MADA) in collaboration with the cubic order convergence of the Newton-Raphson method (also known as the improvised Newton-Raphson method) on the Shehu Transform environment (STE). In the current study, the time fractional Caputo-Fabrizio Derivative (CFD) is used in singular and non-singular kernel derivatives to address the influence of fractional parameters. Some of the current numerical and analytical results are displayed utilizing 3D plots, while others are depicted in the form of a legend 2D plots for comparison. To validate the robustness of the current approach, the uniqueness, stability, and convergence analyses are described. The current result is compared to the analytical solution as well as previous solutions in order to demonstrate the efficiency of our suggested technique.

Dynamical analysis of a Tumor Growth model under the effect of fractal fractional Caputo-Fabrizio derivative

Abstract Fractal-fractional derivatives, which are still rather new, are frequently used to look into the complexities of an issue. Today, tumors are a prevalent and difficult-to-treat condition. The Caputo-Fabrizio-fractal-fractional derivative, which is a non-singular derivative,. has been used to explore the tumor-growth model quantitatively and numerically. By using fixed-point theorems, it has been demonstrated that the model underneath the Caputo-Fabrizio-fractal-fractional derivative exists and is unique. The Ulam-Hyres stability of the model was evaluated using non-linear analysis. Using Lagrangian-piecewise interpolation and the fundamentals of fractional calculus, we can develop an algorithm that will enable us to determine the numerical solutions for the new model. In order to show the method’s dependability and effectiveness, numerical simulations are also included. Utilizing an exponential-decay kernel, we evaluated the dynamics of the Tumor Growth model to see if the non-singular fractal fractional operator offered better dynamics for the model under consideration.

Existence of the solution for hybrid differential equation with Caputo-Fabrizio fractional derivative

In this manuscript, we are interested in the existence result of the solution of hybrid nonlinear differential equations. involving fractional Caputo Fabrizio derivatives of arbitrary order ? ?]0, 1[. By applying Dhage?s fixed point theorem and some fractional analysis techniques, we prove our main result. As an application, A non-trivial example is given to demonstrate the effectiveness of our theoretical result.

Convection heat–mass transfer of generalized Maxwell fluid with radiation effect, exponential heating, and chemical reaction using fractional Caputo–Fabrizio derivatives

Abstract This article is directed to analyze the transfer of mass and heat in a generalized Maxwell fluid flow unsteadily on a vertical flat plate oscillating in its respective plane and heated exponentially. It explains the transfer of mass and heat using a non-integer order derivative usually called a fractional derivative. It is a generalization of the classical derivatives of the famous Maxwell’s equation to fractional non-integer order derivatives used for one-dimensional flow of fluids. The definition given by Caputo–Fabrizio for the fractional derivative is used for solving the problem mathematically. The Laplace transform method is used for finding the exact analytical solution to a problem by applying it to a set of non-integer order differential equations that are dimensionless in nature. These equations contain concentration, temperature, and velocity equations with specific initial and boundary conditions. Solutions of the three equations are graphically represented to visualize the effects of various parameters, such as the radiation parameter (Nr), the thermal Grashof number, the fractional parameter (α), the mass Grashof number, Prandtl effective number, Schmidt number, Prandtl number, the chemical reaction ( η 2 ) ({\eta }_{2}) , mass, and the temperature during fluid flow.

Well-posedness and numerical simulations employing Legendre-shifted spectral approach for Caputo–Fabrizio fractional stochastic integrodifferential equations

This paper investigates the well-posedness of a class of FSIDEs utilizing the fractional Caputo–Fabrizio derivative. Herein, the well-posedness proofs are constructed by considering some applicable conditions and combining theories of Banach space, AAT, and FPST. Approximating the solutions of such equations is still challenging for many mathematicians today due to their randomness and the hardness of finding the exact one. For the numerical aim, we introduce some useful properties of the Legendre-shifted polynomials and employ them as a basis of the collocation spectral method. The idea of this scheme is to convert such stochastic equations into algebraic systems subject to [Formula: see text]-measurable independent parameters. The stochastic term is driven by one-dimensional standard Brownian motion which is the most familiar type and for simulating its trajectories we discuss an easy method. We rigorously analyze the convergence of the proposed technique and other error behavior-bound results. Finally, various tangible numerical applications are performed to verify the present scheme’s accuracy and great feasibility and support theoretical results. The acquired results reveal that the methodology used is effective and appropriate to deal with various issues in light of the fractional Caputo–Fabrizio derivative.

Heat transfer analysis in a non-Newtonian hybrid nanofluid over an exponentially oscillating plate using fractional Caputo–Fabrizio derivative

In this paper, we have been study a hybrid nanofluid over an exponentially oscillating vertical flat plate. Therefore the fractional derivatives definition of Caputo–Fabrizio approach is applied to transform the classical model for this hybrid nanofluid to fractional model. Together with an oscillating boundary motion, therefore the heat transfer is cause as a result of the buoyancy force produce due temperature differences between the plate and the fluid. The dimensionless classical model is generalized by transforming it to the time fractional model using Caputo–Fabrizio time fractional derivative. Exact analytical solutions are obtained by using Laplace transform method to the set of dimensionless fractional governing equations, containing the momentum and energy equations subjected to the boundary and initial conditions. Numerical computations and graphical illustrations are used to checked the results of the Caputo–Fabrizio time-fractional parameter, the second-grade parameter, the magnetic parameter and the Grashof numbers on the velocity field. An assessment for time spin-off is shown graphically of integer order versus fractional-order for these non-Newtonian hybrid nanofluid through Mathcad software. The fluid velocity increases for increasing the value of the fractional parameter, second-grade parameter and Grashof number. Also for increasing the values of the MHD parameter the fluid velocity decreases.

Heat-mass transfer of MHD second grade fluid flow with exponential heating, chemical reaction and porosity by using fractional Caputo-Fabrizio derivatives

This work studies the analysis of heat mass transfer of generalized second grade fluid. The problem is modelled using Caputo-Fabrizio fractional derivative approach. The effects of MHD, chemical reaction, heat source and porous media are also included. The infinite vertical plate is subjected to the phenomenon of exponential heating. The governing equations along with their corresponding initial and boundary conditions are transformed into non-dimensional form. The system of governing equations has been solved through Laplace transform. Plots are prepared for several physical parameters of interested and discussed in detail. In limiting sense, the published results are recovered analytically and a comparison is made which shows an excellent agreement.

Novel Mathematical Modelling of Platelet-Poor Plasma Arising in a Blood Coagulation System with the Fractional Caputo–Fabrizio Derivative

This study develops a fractional model using the Caputo–Fabrizio derivative with order α for platelet-poor plasma arising in a blood coagulation system. The existence of solutions ensures that there are solutions to the considered system of equations. Approximate solutions to the recommended model are presented by selecting different numbers of fractional orders and initial conditions (ICs). For each case, graphs of solutions are supplied through different dimensions.

The Effects of Magnetic Casson Blood Flow in an Inclined Multi-stenosed Artery by using Caputo-Fabrizio Fractional Derivatives

This paper investigates the magnetic blood flow in an inclined multi-stenosed artery under the influence of a uniformly distributed magnetic field and an oscillating pressure gradient. The blood is modelled using the non-Newtonian Casson fluid model. The governing fractional differential equations are expressed by using the fractional Caputo-Fabrizio derivative without singular kernel. Exact analytical solutions are obtained by using the Laplace and finite Hankel transforms for both velocities. The velocities of blood flow and magnetic particles are graphically presented. It shows that the velocity increases with respect to the Reynolds number and the Casson parameter. Meanwhile, the velocity decreases as the Hartmann number increases. These results are useful for the diagnosis and treatment of certain medical problems.

MHD flow of generalized second grade fluid with modified Darcy’s law and exponential heating using fractional Caputo-Fabrizio derivatives

The aim of the present work is to apply the Caputo-Fabrizio fractional derivative to the heat transformation of a second grade fluid. The flow is analyzed under the effect of magneto hydrodynamic together with heat transfer as well as Darcy’s law. The infinite vertical plate is subjected to a more general time dependent motion. Moreover, on the thermal aspects of the infinite plate, we are taking into account the exponential heat phenomena. Exact analytical solutions are obtained by using Laplace transform method to the set of dimensionless fractional governing equations, containing the momentum and energy equations subjected to the boundary and initial conditions. Solutions are plotted graphically to analyze the effect of different parameters i.e. fractional parameter α, Grashof number and Prandtl number on the physical behavior of the fluid flow and temperature.

On the mathematical model of Rabies by using the fractional Caputo\u2013Fabrizio derivative

Using the fractional Caputo–Fabrizio derivative, we investigate a new version of the mathematical model of Rabies disease. Using fixed point results, we prove the existence of a unique solution. We calculate the equilibrium points and check the stability of solutions. We solve the equation by combining the Laplace transform and Adomian decomposition method. In numerical results, we investigate the effect of coefficients on the number of infected groups. We also examine the effect of derivation orders on the behavior of functions and make a comparison between the results of the integer-order derivative and the Caputo and Caputo–Fabrizio fractional-order derivatives.

On Lyapunov-type inequalities for boundary value problems of fractional Caputo-Fabrizio derivative

In this study, Lyapunov-type inequalities for fractional boundary value problems involving the fractional Caputo Fabrizio differential equation with mixed boundary conditions when the fractional order of $\beta \in (1,2]$ and Dirichlet-type boundary condition when the fractional order of $\sigma \in (2,3]$ have been derived. Some consequences of the results related to the fractional Sturm?Liouville eigenvalue problems have also been given. Additionally, we examine the nonexistence of the solution of the fractional boundary value problem.

A new numerical solution of the competition model among bank data in Caputo-Fabrizio derivative

A new numerical scheme for the Caputo-Fabrizio operator is proposed. We initially present a bank model with real data and then present the model in the fractional Caputo-Fabrizio derivative. We estimate and fit the model parameter using the least square curve fitting. The Caputo-Fabrizio model is solved numerically by using three steps Adams-Bashforth method. The proposed scheme is used to obtain graphical results for bank data of rural and commercial. The real data of rural and commercial banks are used to fit with Caputo-Fabrizio model. We show that the Caputo-Fabrizio model show good fitting for the fractional order parameters versus the real data of rural and commercial bank. Further, we show graphical illustration for some values of the fractional order in order to show the effectiveness of the proposed new numerical scheme.

Analysis of the model of HIV-1 infection of CD4^{+} T-cell with a new approach of fractional derivative

By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

Erratum to: Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo-Fabrizio derivatives

Erratum to: Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo-Fabrizio derivatives

Heat transfer analysis of Walters’-B fluid with Newtonian heating through an oscillating vertical plate by using fractional Caputo–Fabrizio derivatives

This paper presents a study of Walters’-B fluid with Caputo–Fabrizio fractional derivatives through an infinitely long oscillating vertical plate by Newtonian heating under the action of transverse magnetic field. The fractional calculus approach is employed to obtain a system of fractional partial differential equations. The governing equations of momentum and energy are converted first into dimensionless form and then solved by employing Laplace transformation. The Laplace inverse transform has been evaluated both analytically and numerically. The graphical illustrations represent the behavior of material parameters on the solutions. A comparison between exact and numerical solutions is presented in tabular and graphical form. The variation in Nusselt number with the change in fractional and physical parameters is also presented. The velocity and temperature of the fluid decreases with the enhancement in the fractional parameter for small values of time, and it has the opposite behavior for greater values of time.

Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo-Fabrizio derivatives

In this work, we study the flow of both blood and magnetic particles using Caputo-Fabrizio fractional derivative model approach. The fluid flow through a circular cylinder is influenced by an external magnetic field which is perpendicular to the circular tube and an oscillating pressure gradient. Integral transforms are used to find solutions for the blood and magnetic particle velocities. Comparison of profiles of velocities for different values of α, the impact of physical variables on the dynamics of fluid and magnetic parameters are highlighted graphically.