Caputo-Fabrizio Fractional Derivative Research Articles

Analytical Treatment for Accelerated Riga Plate on Fractional Caputo-Fabrizio Casson Fluid

The Riga plate is a substantial alteration in the world of engineering. Mainly used in submarines to regulate water flow, studying the behaviour of fluid flowing over a Riga plate is very advantageous. Although there are ample studies on fluid flowing over a Riga plate, the introduction of fractional derivatives, coupled with a non-Newtonian fluid, has yet to be done. Within the field of fluid mechanics, specifically boundary layer flow, fractional derivatives do not have a proven geometrical representation. However, analytical solutions would be useful in aiding experimental researches in the future. Thus, this study aims to present an analytical function for a Caputo-Fabrizio fractional derivative on an unsteady Casson fluid flowing over an accelerating vertical Riga plate by using the Laplace transform method. The parametric effects considered in this study is elucidated. Through observation of obtained graphical results generated via the obtained analytical solutions, it is found that amplification of the fractional parameter and modified Hartmann number increases the fluid velocity with an average increment of 42.05% and 1.56%, respectively. While amplification of the Casson parameter and Prandtl number dampens the fluid velocity by an average of 45.09% and 43.56%, respectively

The existence of positive solutions for the Caputo-Fabrizio fractional boundary value problems at resonance

This paper deals with a class of nonlinear fractional boundary value problems at resonance with Caputo-Fabrizio fractional derivative. We establish some necessary conditions for the existence of positive solutions by using the Leggett-Williams norm-type theorem for coincidences. Some examples are constructed to support our results.

Solving Fractional Optimal Control Problems Involving Caputo–Fabrizio Derivative Using Hermite Spline Functions

In the current research, we develop a collocation method based on the biorthogonal Hermite cubic spline functions to solve a class of fractional optimal control problems using Caputo–Fabrizio derivative operator. We design dual bases for Hermite cubic spline functions for the first time in this work. So, we present two direct and efficient algorithms to solve a class of fractional optimal control problems. Two new operational matrices of Caputo–Fabrizio fractional derivative are derived using Hermite cubic spline functions. By employing these matrices, we reduce these problems to systems of algebraic equations. Illustrative examples are examined to demonstrate the important features of the new algorithm.

Stability, existence, and uniqueness for solving fractional glioblastoma multiforme using a Caputo–Fabrizio derivative

In this study, fractional order is applied to the glioblastoma multiforme (GBM) and IS interaction models. The monoclonal brain tumor GBM gives rise to other tumors with varying growth rates and treatment susceptibilities once it reaches a certain density. The two populations of macrophages and activated macrophages make up the IS cells as well. Because of this, this model depicts two conversions: the transformation of a sensitive tumor cell into a resistant tumor cell and the transformation of passive macrophages into active macrophages, as well as an interaction among the tumor cell and the macrophages. The memory and nonlocalization properties of fractional derivatives make them an excellent tool for simulating the spread of epidemics. Numerous characteristics exist in the Caputo–Fabrizio fractional derivative kernel, including nonsingularity, nonlocality, and an exponential establish. For simulating disease spreading systems, it is preferred. We suggest defining the Caputo–Fabrizio fractional derivation procedure as the paradigm for the GBM disease in this work. The hypothesized fractional‐order GBM disease paradigm's stability is investigated. The hypothesized fractional‐order GBM disease paradigm's existence and originality are established. A suggested novel signal flow graph introduces the system. We have established the existence of GBM and its uniqueness. Additionally, the numerical integrations necessary for the fractional‐order glioma immune model are carried out using the predictor–corrector Adams–Bashforth–Moulton (ABM) approach for fractional‐order differential equations. These simulations demonstrate that the behavior of the model can be affected by changing the parameters for the fractional order, contact evaluation, or quarantine evaluation. The findings demonstrate how epidemiologically sound the proposed fractional model is.

On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator

Abstract In this article, we considered the pseudo-parabolic equation with Caputo-Fabrizio fractional derivative. This equation has many applications in different fields, such as science, technology, and so on. In this article, we gave the formula of mild solution, which is represented in the form of Fourier series by some operators . In the linear case, we investigated the continuity of the mild solution with respect to the fractional order. For the nonlinear case, we investigated the existence and uniqueness of a global solution. The main proof technique is based on the Banach fixed point theorem combined with some Sobolev embeddings. For more detailed, we obtained two other interesting results: the continuity of mild solution with respect to the derivative order and the convergence of solution as the coefficient k approaches to zero.

Existence and continuity results for Kirchhoff parabolic equation with Caputo–Fabrizio operator

This paper deals with the solution of the Kirchhoff parabolic equation involving the Caputo–Fabrizio fractional derivative with non-singular kernel. We represent the mild solution by equation operators which are defined via Fourier series. The main results of this work are about the existence, uniqueness and the continuity property with respect to the derivative order of the mild solution. The result of the existence of a solution consists of two parts: The existence of a local solution with a nonlinear source function. The existence of a global solution in the case of a linear source function. To achieve global results, we had to introduce a new space with norm contains weighted function. The key analysis is based on the Banach fixed point theorem and some Sobolev embeddings. The order-continuity problem has two main results: The first result proves the continuity related to order α∈(0,1). The remaining results examine the continuity when the fractional order is approaching 1−. Finally, we illustrate the achieved theoretical results by some numerical examples.

The existence and uniqueness of a fractional q-integro differential equation involving the Caputo-Fabrizio fractional derivative and the fractional q -integral of the Riemann-Liouville type with q -nonlocal condition

The existence and uniqueness of a fractional q-integro differential equation involving the Caputo-Fabrizio fractional derivative and the fractional q -integral of the Riemann-Liouville type with q -nonlocal condition

Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

This study focuses on the dynamics of drug concentration in the blood. In general, the concentration level of a drug in the blood is evaluated by the mean of an ordinary and first-order differential equation. More precisely, it is solved through an initial value problem. We proposed a new modeling technique for studying drug concentration in blood dynamics. This technique is based on two fractional derivatives, namely, Caputo and Caputo-Fabrizio derivatives. We first provided comprehensive and detailed proof of the existence of at least one solution to the problem; we later proved the uniqueness of the existing solution. The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques. Stability via the Ulam-Hyers (UH) technique was also investigated. The application of the proposed model on two real data sets revealed that the Caputo derivative was more suitable in this study. Indeed, for the first data set, the model based on the Caputo derivative yielded a Mean Squared Error (MSE) of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360. Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem, with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532. For the second experiment, Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066, whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299. Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.

Analytical and numerical study of a nonlinear Volterra integro-differential equation with the Caputo-Fabrizio fractional derivative

In this paper we interest concerns an integro-differential equation of Volterra when the derivative of the solution appears under the sign of integration in a non-linear way. The derivation is the Caputo-Fabrizio Fractional Derivative (CF-FD), which increases the interest of this study. we built Lipchitz conditions to obtain the existence and uniqueness of the solution in a Banach space specific to the derivative of Caputo. The analytical study is followed by a complete numerical study.

On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

<abstract><p>The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.</p></abstract>

Computational modeling of financial crime population dynamics under different fractional operators

<abstract><p>This paper presents an analysis and numerical simulation of financial crime population dynamics using fractional order calculus and Newton's polynomial. The dynamics of financial crimes are modeled as a fractional-order system, which is then solved using numerical methods based on Newton's polynomial. The results of the simulation provide insights into the behavior of financial crime populations over time, including the stability and convergence of the systems. The study provides a new approach to understanding financial crime populations and has potential applications in developing effective strategies for combating financial crimes. Fractional derivatives are commonly applied in many interdisciplinary fields of science because of its effectiveness in understanding and analyzing complicated phenomena. In this work, a mathematical model for the population dynamics of financial crime with fractional derivatives is reformulated and analyzed. A fractional-order financial crime model using the new Atangana-Baleanu-Caputo (ABC) derivative is introduced. The reproduction number for financial crime is calculated. In addition, the relative significance of model parameters is also determined by sensitivity analysis. The existence and uniqueness of the solution in consideration of the ABC derivative are discussed. A number of conditions are established for the existence and Ulam-Hyers stability of financial crime equilibria. A numerical scheme is presented for the proposed model, starting with the Caputo-Fabrizio fractional derivative, followed by the Caputo and Atangana-Baleanu fractional derivatives. Finally, we solve the models with fractal-fractional derivatives.</p></abstract>

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.

Existence of Approximate Solutions to Nonlinear Lorenz System under Caputo-Fabrizio Derivative

In this article, we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative (CFFD). The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii. Also, we enriched our work by establishing a stable result based on the Ulam-Hyers (U-H) concept. Also, the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method. We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders. The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier. Also, the suggested method results for the proposed system under the mentioned derivative are new. Furthermore, the adopted technique has some useful features, such as the lack of prior discrimination required by wavelet methods. our proposed method does not depend on auxiliary parameters like the homotopy method, which controls the method. Our proposed method is rapidly convergent and, in most cases, it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.

Study of a coupled system with anti-periodic boundary conditions under piecewise Caputo-Fabrizio derivative

A coupled system under Caputo-Fabrizio fractional order derivative (CFFOD) with antiperiodic boundary condition is considered. We use piecewise version of CFFOD. Sufficient conditions for the existence and uniqueness of solution by ap?plying the Banach, Krasnoselskii?s fixed point theorems. Also some appropriate results for Hyers-Ulam (H-U) stability analysis is established. Proper example is given to verify the results.

Numerical study of a nonlinear fractional chaotic Chua's circuit

<abstract> <p>As an exponentially growing sensitivity to modest perturbations, chaos is pervasive in nature. Chaos is expected to provide a variety of functional purposes in both technological and biological systems. This work applies the time-fractional Caputo and Caputo-Fabrizio fractional derivatives to the Chua type nonlinear chaotic systems. A numerical analysis of the mathematical models is used to compare the chaotic behavior of systems with differential operators of integer order versus systems with fractional differential operators. Even though the chaotic behavior of the classical Chua's circuit has been extensively investigated, our generalization can highlight new aspects of system behavior and the effects of memory on the evolution of the chaotic generalized circuit.</p> </abstract>

Analytical and numerical study of a linear coupled system involving Caputo-Fabrizio fractional derivative with boundary conditions

Analytical and numerical study of a linear coupled system involving Caputo-Fabrizio fractional derivative with boundary conditions

SEVERAL VARIATIONAL INCLUSIONS FOR A FRACTIONAL DIFFERENTIAL INCLUSION OF CAPUTO-FABRIZIO TYPE

We establish several fractional variational inclusions for solutions of a nonconvex fractional differential inclusion involving Caputo-Fabrizio fractional derivative.

Addressing impulsive fractional integro-differential equations with Caputo-Fabrizio via monotone iterative technique in Banach spaces

In this paper, we investigate the existence of solutions for initial value problem of the impulsive integro-differential for fractional differential equations involving a Caputo-Fabrizio fractional derivative of order r ? (0, 1) in Banach spaces. Under some monotonicity conditions and the noncompactness measure condition of nonlinearity functions. We obtain the existence of extremal solutions between lower and upper solutions.

Analytical Solutions for Fractional Caputo-Fabrizio Casson Nanofluid on Riga Plate with Newtonian Heating

Introduction of fractional derivatives to the mechanics of fluid flow is relatively new. Even though the exact geometrical representations of fractional derivatives on fluid mechanics have not been discovered, recent literatures have proven that it is a paradox that will be useful in the future. Meanwhile, Riga plates are actuators that is convenient for controlling the velocity of fluid flows. Widely used in the field of marine engineering, the properties of fluid flowing over Riga plates are worth investigating. Thus, the aim of this study is to investigate the analytical solutions of an unsteady incompressible Casson nanofluid flowing over a Riga plate with presence of Newtonian heating. Carboxymethyl Cellulose (CMC) water was used as a prime example of Casson fluid with Copper-Oxide (CuO) nanoparticles. Coupled with a non-Newtonian fluid, the Casson fluid, and the Caputo-Fabrizio fractional derivative, the analytical solutions obtained will be beneficial in the engineering world as a tool for validating experimental and numerical studies. Through this study, analytical solutions were obtained and the profiles of both velocity and temperature of fluid with variations in parameters were investigated. It is observed that variations in the fractional derivative parameter produces a spectrum of solutions that abides the initial and boundary conditions set. An amplification of the modified Hartmann number increases both the velocity and temperature profiles, while an amplification of the nanoparticle volume fraction decreases the velocity profile but increase the temperature profile.

An Efficient Analytical Method for Analyzing the Nonlinear Fractional Klein–Fock–Gordon Equations

The purpose of this article is to solve a nonlinear fractional Klein–Fock–Gordon equation that involves a recently created non-singular kernel fractional derivative by Caputo–Fabrizio. Motivated by some physical applications related to the fractional Klein–Fock–Gordon equation, we focus our study on this equation and some phenomena rated to it. The findings are crucial and essential for explaining a variety of physical processes. In order to find satisfactory approximations to the offered problems, this work takes into account a modern methodology and fractional operator in this context. We first take the Yang transform of the Caputo–Fabrizio fractional derivative and then implement it to solve fractional Klein–Fock–Gordon equations. We will consider three cases of the nonlinear fractional Klein–Fock–Gordon equation to ensure the applicability and effectiveness of the suggested technique. In order to determine an approximate solution to the fractional Klein–Fock–Gordon equation in the fast convergent series form, we can use the fractional homotopy perturbation transform approach. The numerical simulation is provided to demonstrate the effectiveness and dependability of the suggested method. Furthermore, several fractional orders will be used to describe the behavior of the given solutions. The results achieved demonstrate the high efficiency, ease of use, and applicability of this strategy for resolving other nonlinear issues.