Caputo-Fabrizio Fractional Derivative Research Articles

On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel

In this paper, we give analytical solutions of a fractional-time wave equation with memory effect and frictional memory kernel of Mittag–Leffler type via the Atangana–Baleanu fractional order derivative. The method of separation of variables and the Laplace transform has been used to obtain the exact solutions for the fractional order wave equations. Additionally, we present analytical solutions considering the Caputo–Fabrizio fractional derivative with exponential kernel. We showed that the solutions obtained via Caputo–Fabrizio fractional order derivative were a particular case of the solutions obtained with the new fractional derivative based in the Mittag–Leffler law.

An analysis of the semi-analytic solutions of a viscous fluid with old and new definitions of fractional derivatives

In this paper we present the natural convection flow of an incompressible viscous fluid subject to Newtonian heating and constant mass diffusion using a recently developed definition of the Caputo–Fabrizio fractional derivative. Boundary layer equations in dimensionless form are obtained by means of dimensionless variables. The expressions for the temperature, concentration and velocity fields are obtained in the Laplace transformed domain. The inverse Laplace transform for the temperature, concentration and velocity field are found numerically by means of Stehfest's and Tzou's algorithms. A comparative analysis has been carried between the Caputo–Fabrizio and the Caputo fractional model obtained by Vieru (2015) through graphical illustration. At the end, we can see the impact of the flow parameters, including the new fractional parameter, on the flow which is presented graphically. As a result, the fractional viscous fluid model with the Caputo–Fabrizio fractional derivative has a higher velocity than with the Caputo.

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.

On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative

The uniqueness of thermal radiation effects on materials has an adhesive role in engineering. This article emphasizes the effects of thermal radiation of Jeffery fluid with magnetic field by employing Caputo-Fabrizio fractional derivative. Analytical solutions for the velocity field and temperature distribution are based on integral transforms (Fourier Sine and Laplace transform) with their inversion. The general solutions have been established in terms of elementary functions and product of convolution theorem satisfying initial and boundary conditions. This analysis is one of the infrequent contributions to elucidate the rheology of Jeffery fluid for free convective problem on oscillating plate. In order to bring physical insights, four models have been prepared for comparison i-e (i) Jeffery fluid with magnetic field (ii) Jeffery fluid without magnetic field (iii) Second grade fluid with magnetic field and (iv) Second grade fluid without magnetic field for several rheological parameters for instance, viscosity ν, thermal radiation Rd, thermal conductivity k, Jeffrey fluid parameter λ, Hartmann number Ha, magnetic field M, Prandtl number Pr, Grashof number on fluid flow. At the end, our results suggested that Ordinary Jeffery fluid with magnetic field oscillates more rapidly to remaining models of fluid but fractionalized Jeffery fluid with magnetic field has reciprocal behavior of fluid flow.

On the analysis of fractional diabetes model with exponential law

In this work, we study the diabetes model and its complications with the Caputo–Fabrizio fractional derivative. A deterministic mathematical model pertaining to the fractional derivative of the diabetes mellitus is discussed. The analytical solution of the diabetes model is derived by exerting the homotopy analysis method, the Laplace transform and the Padé approximation. Moreover, existence and uniqueness of the solution are examined by making use of fixed point theory and the Picard–Lindelof approach. Ultimately, for illustrating the obtained results some numerical simulations are performed.

A new fractional model for the dynamics of the hepatitis B virus using the Caputo-Fabrizio derivative

Hepatitis B is the major public health issue of the entire world. In mathematical epidemiology, mathematical models play a vital role in understanding the dynamics of infectious diseases. Therefore, in the present paper, we formulate a mathematical model for the hepatitis B virus with the Caputo-Fabrizio fractional derivative with non-singular kernel. Initially, we discuss some basic results involved in the model and then apply the fractional calculus to the proposed model to describe the hepatitis B virus with an arbitrary-order derivative having non-singular kernel. An iterative method is proposed for the solution of the model. For the model variables existence, we use the fixed-point theorem. Further, the uniqueness of the solution is verified. Graphical results are obtained for different values of the fractional parameter.

New fractional derivatives with non-singular kernel applied to the Burgers equation.

In this paper, we extend the model of the Burgers (B) to the new model of time fractional Burgers (TFB) based on Liouville-Caputo (LC), Caputo-Fabrizio (CF), and Mittag-Leffler (ML) fractional time derivatives, respectively. We utilize the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of TFB using LC, CF, and ML in the Liouville-Caputo sense. We study the convergence analysis of HATM by computing the interval of the convergence, the residual error function (REF), and the average residual error (ARE), respectively. The results are very effective and accurate.

MHD fractional Jeffrey’s fluid flow in the presence of thermo diffusion, thermal radiation effects with first order chemical reaction and uniform heat flux

Exact analysis is about the natural convection flow of non-Newtonian Jeffrey fluid with the Caputo-Fabrizio fractional derivative of non-singular kernel has been discussed in this work. The Laplace transform method is used to obtain the solutions of dimensionless temperature, concentration and velocity fields with non-integer order derivative. Moreover, in the mathematical modelling of the problem, the additional effects like Soret effect, MHD, heat sink, radiations, chemical reaction, porous medium and uniform heat flux are also considered. Our results are reduced to the known solutions in the existing literature for viscous fluid. Finally, we have plotted some graphical illustration to see the physical insight of the studied problem for different flow parameters and found that fluid velocity can be enhanced with the Caputo-Fabrizo approach by increasing the value of non-integer order parameter while skin friction coefficient decreases.

Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties

We presented an analysis of evolutions equations generated by three fractional derivatives namely the Riemann–Liouville, Caputo–Fabrizio and the Atangana–Baleanu fractional derivatives. For each evolution equation, we presented the exact solution for time variable and studied the semigroup principle. The Riemann–Liouville fractional operator verifies the semigroup principle but the associate evolution equation does not. The Caputo–Fabrizio fractional derivative does not satisfy the semigroup principle but surprisingly, the exact solution satisfies very well all the principle of semigroup. However, the Atangana–Baleanu for small time is the stretched exponential derivative, which does not satisfy the semigroup as operators. For a large time the Atangana–Baleanu derivative is the same with Riemann–Liouville fractional derivative, thus satisfies semigroup principle as an operator. The solution of the associated evolution equation does not satisfy the semigroup principle as Riemann–Liouville. With the connection between semigroup theory and the Markovian processes, we found out that the Atangana–Baleanu fractional derivative has at the same time Markovian and non-Markovian processes. We concluded that, the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects, which are not always Markovian. We presented the exact solutions of some evolutions equation using the Laplace transform. In addition to this, we presented the numerical solution of a nonlinear equation and show that, the model with the Atangana–Baleanu fractional derivative has random walk for small time. We also observed that, the Mittag-Leffler function is a good filter than the exponential and power law functions, which makes the Atangana–Baleanu fractional derivatives powerful mathematical tools to model complex real world problems.

Blood flow problem in the presence of magnetic particles through a circular cylinder using Caputo-Fabrizio fractional derivative

In this paper, the flow of blood mixed with magnetic particles subjected to uniform transverse magnetic field and pressure gradient in an axisymmetric circular cylinder is studied by using a new trend of fractional derivative without singular kernel. The governing equations are fractional partial differential equations derived based on the Caputo-Fabrizio time-fractional derivatives NFDt. The current result agrees considerably well with that of the previous Caputo fractional derivatives UFDt.

Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gomez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.

A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives

This research article is analyzed for the comparative study of RL and RC electrical circuits by employing newly presented Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. The governing ordinary differential equations of RL and RC electrical circuits have been fractionalized in terms of fractional operators in the range of $0 \leq \xi \leq 1$ and $0 \leq \eta \leq 1$ . The analytic solutions of fractional differential equations for RL and RC electrical circuits have been solved by using the Laplace transform with its inversions. General solutions have been investigated for periodic and exponential sources by implementing the Atangana-Baleanu and Caputo-Fabrizio fractional operators separately. The investigated solutions have been expressed in terms of simple elementary functions with convolution product. On the basis of newly fractional derivatives with and without singular kernel, the voltage and current have interesting behavior with several similarities and differences for the periodic and exponential sources.

Cancer treatment model with the Caputo-Fabrizio fractional derivative

In this article, a model for cancer treatment is examined. The model is integrated into the Caputo-Fabrizio fractional derivative first, to examine the existence of the solution. Then, the uniqueness of the solution is investigated and we identified under which conditions the model provides a unique solution.

Numerical Simulation of Fractional-Differential Filtration-Consolidation Dynamics Within the Framework of Models with Non-Singular Kernel

Non-classical mathematical models to describe the fractional-differential filtration-consolidation dynamics of soil media saturated with salt solutions are constructed based on the concept of the Caputo–Fabrizio fractional derivative. The corresponding boundary-value problems for the system of fractional-differential equations of filtration and salt transfer are posed, the technique for their numerical solution is developed, an approach to the parallelization of computations is presented, and the results of numerical experiments on modeling the dynamics of the process are given.

Calculus of variations involving Caputo-Fabrizio fractional differentiation

This paper is devoted to study some variational problems with functionals containing the Caputo-Fabrizio fractional derivative, that is a fractional derivative with a non-singular kernel.

Reduction of order formula and fundamental set of solutions for linear fractional differential equations

In this paper, we consider a class of linear fractional differential equations involving the Caputo–Fabrizio fractional derivative of order 1 < α < 2 . We derive, in closed form, a reduction of order formula to obtain a second linearly independent solution. We then establish a fundamental set of solutions result to the equation. One example is presented to illustrate the validity of the obtained results.

No nonlocality. No fractional derivative

The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small neighborhood of the considered point. Therefore differential equation, which is considered for this point and contains a finite number of integer-order derivatives, cannot describe nonlocality in space and time. This allows us to propose a principle of nonlocality for fractional derivatives. We state that if the differential equation with fractional derivative can be presented as a differential equation with a finite number of integer-order derivatives, then this fractional derivative cannot be considered as a derivative of non-integer order. This means that all results obtained for this type of fractional derivatives can be derived by using differential operators with integer orders. To illustrate the application of the nonlocality principle, we prove that the conformable fractional derivative, the M-fractional derivative, the alternative fractional derivative, the local fractional derivative and the Caputo–Fabrizio fractional derivatives with exponential kernels cannot be considered as fractional derivatives of non-integer orders.

A closed form expression for the Gaussian–based Caputo–Fabrizio fractional derivative for signal processing applications

The Gaussian function has been employed in a vast number of practical and theoretical applications since it was proposed. Likewise, Gaussian function and its ordinary derivatives are considered as powerful tools for signal processing and control applications, e.g., smoothing, sampling, change detection, blob detection, and transforms based on the Hermite polynomials. Nonetheless, it has impressive characteristics hidden amongst its fractional derivatives eager to be explored and studied in-depth. This work proposes a closed formula for the (n+ν)–order fractional derivative of the Gaussian function, based on the Caputo–Fabrizio definition, as an approach for analysing those attributes. The obtained expression was numerically tested with several fractional orders, and their resulting behaviours were eventually analysed. Finally, three practical applications on signal processing via this closed formula were discussed, i.e., customisable wavelets, image processing filters, and Rayleigh distributions.

On fractional modelling of dye removal using fractional derivative with non-singular kernel

Several studies showed that the adsorption process for the dye removal has its unique fractional property. In line with this, we focused on the fractional model to study the transport of direct textile industry wastewater. We used the recently introduced fractional derivative without singular kernel, the Caputo-Fabrizio fractional derivative to obtain the analytic solution for the adsorption transport equation and plot the concentration that corresponds to the dye with a given initial conditions.

Properties of the Caputo-Fabrizio fractional derivative and its distributional settings

Abstract The Caputo-Fabrizio fractional derivative is analyzed in classical and distributional settings. The integral inequalities needed for application in linear viscoelasticity are presented. They are obtained from the entropy inequality in a weak form. Moreover, integration by parts, an expansion formula, approximation formula and generalized variational principles of Hamilton’s type are given. Hamilton’s action integral in the first principle, do not coincide with the lower bound in the fractional integral, while in the second principle the minimization is performed with respect to a function from a specified space and the order of fractional derivative.