Caputo-Fabrizio Derivative Research Articles

Intelligent Neural Integral Sliding-mode Controller for a space robotic manipulator mounted on a free-floating satellite

In this paper, a fractional neural integral sliding-mode controller based on the Caputo-Fabrizio derivative and Riemann–Liouville integral for a robot manipulator mounted on a free-floating satellite is proposed and developed. The mathematical model is obtained by using the Euler–Lagrange formalism and generalized through the Caputo-Fabrizio derivative. The conventional neural integral sliding-mode controller reported in the literature were also developed to make a comparative analysis between both controllers. The controllers are tuned using a swarm intelligence algorithm. External disturbances are also considered. Simulation results show the superiority and robustness of the proposed control system for trajectory tracking tasks through the root-mean-square error. Additionally, two additional reference trajectories were considered. In this sense, the numerical simulations were done by using the same orders and gains, showing the robustness of our fractional-order control strategy under different operating conditions. Further, the proposed controller uses the fractional derivative in the sense of Caputo-Fabrizio which has not been used enough in the area of robotics, so this is a start to analyse the contributions that arise when using said derivative in this robotic.

Linear regularized finite difference scheme for the quasilinear subdiffusion equation

Abstract A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with a time-fractional derivative and coefficients at the elliptic part that depend on the gradient of the solution is considered. Conditions on the coefficients ensure the monotonicity and Lipschitz property of the elliptic operator on the set of functions whose gradients in space variables are uniformly bounded. For this problem, a linear regularized mesh scheme is constructed and investigated. A sufficient condition is derived for the regularization parameter that ensures the so-called local correctness of the mesh scheme. On the basis of correctness and approximation estimates for model problems with time-fractional Caputo or Caputo–Fabrizio derivatives, accuracy estimates are given in terms of mesh and regularization parameters under the assumption of the existence of a smooth solution to the differential problem. The presented results of the numerical experiments confirm the obtained asymptotic accuracy estimates.

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.

Comparative Study of the Fractional-Order Crime System as a Social Epidemic of the USA Scenario.

Fractional derivatives are considered significant mathematical tools to design the fractional-order models of real phenomena. In this investigation, we are going to design and compare the non-integer models of the crime system by using three fractional-order operators called Atangana-Baleanu-Caputo, Caputo, and Caputo-Fabrizio derivatives for the first time. We use the real initial conditions for the subgroups of USA. To get the approximate solutions of the suggested models some numerical methods are derived. To see the performance of the numerical methods different values of the fractional orders are considered. The differences between the solutions under the used operators for each state variable are provided through some figures.

Assessing the impact of SARS-CoV-2 infection on the dynamics of dengue and HIV via fractional derivatives

A new non-integer order mathematical model for SARS-CoV-2, Dengue and HIV co-dynamics is designed and studied. The impact of SARS-CoV-2 infection on the dynamics of dengue and HIV is analyzed using the tools of fractional calculus. The existence and uniqueness of solution of the proposed model are established employing well known Banach contraction principle. The Ulam-Hyers and generalized Ulam-Hyers stability of the model is also presented. We have applied the Laplace Adomian decomposition method to investigate the model with the help of three different fractional derivatives, namely: Caputo, Caputo-Fabrizio and Atangana-Baleanu derivatives. Stability analyses of the iterative schemes are also performed. The model fitting using the three fractional derivatives was carried out using real data from Argentina. Simulations were performed with each non-integer derivative and the results thus obtained are compared. Furthermore, it was concluded that efforts to keep the spread of SARS-CoV-2 low will have a significant impact in reducing the co-infections of SARS-CoV-2 and dengue or SARS-COV-2 and HIV. We also highlighted the impact of three different fractional derivatives in analyzing complex models dealing with the co-dynamics of different diseases.

New mathematical modelings of the human liver and hearing loss systems with fractional derivatives

Fractional derivatives are significant mathematical instruments that have been practiced to model real phenomena in different areas of science. Through the current investigation study, we develop the Human Liver and Hearing Loss models by employing three fractional operators called Atangana–Baleanu–Caputo, Caputo and Caputo–Fabrizio derivatives. The numerical techniques regarding the basic theories of fractional analysis considered to obtain the approximate solutions of the offered systems. Indeed, approximate results are presented for showing the efficacy of the methods during graphs of solutions. To see the performance of the suggested models, different fractional orders are tested.

Semilinear parabolic diffusion systems on the sphere with Caputo-Fabrizio operator

PDEs on spheres have many important applications in physical phenomena, oceanography and meteorology, geophysics. In this paper, we study the parabolic systems with Caputo-Fabrizio derivative. In order to establish the existence of the mild solution, we use the Banach fixed point theorem and some analysis of Fourier series associated with several evaluations of the spherical harmonics function. Some of the techniques on upper and lower bounds of the Mittag-Lefler functions are also applied. This is one of the first research results on the systems of parabolic diffusion on the sphere.

Mathematical assessment of a fractional‐order vector–host disease model with the Caputo–Fabrizio derivative

Vector–host diseases outbreak is a major public health concern, and it has greatly affected human health and economy in various regions around the globe. Different approaches have been adopted to investigate the dynamical behavior and possible control of these diseases. In this study, we present a compartmental transmission model in order to explore the dynamics of vector–host infectious diseases. The saturated incidence rate instead of bilinear (or standard) and saturated treatment function is considered in model formulation which enhance the biological suitability of the proposed model. We first formulate the model based on nonlinear classical integer‐order differential equations. Then, the proposed integer‐order model is reformulated using the fractional‐order operator in Caputo–Fabrizio sense with nonsingular kernel. We investigate the model equilibria and evaluate the expression for the most important threshold parameter known as the basic reproduction number. Furthermore, the existence and uniqueness are presented via the fixed point approach. Additionally, using an efficient numerical scheme, the iterative solution of the model is obtained. Finally, we present the model simulations to illustrate the impact of arbitrary fractional order and some of other important parameters involved in the model on the disease dynamics and minimization.

NEW RESULTS FOR PARABOLIC EQUATION ON THE SPHERE WITH CAPUTO–FABRIZIO OPERATOR

In this paper, we are interested in studying the initial value problem for parabolic problem associated with the Caputo–Fabrizio derivative. We deal the problem in two cases: linear inhomogeneous case and nonlinearity source term. For the linear case, we derive the convergence result of the mild solution when the fractional order [Formula: see text] under some various assumptions on the initial datum. For the nonlinear problem, we show the existence and uniqueness of the mild solution using Banach fixed point theory. We also prove the convergence result of the mild solution when the fractional order [Formula: see text].

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.

A NEW NUMERICAL METHOD FOR SOLVING FRACTIONAL NEW NUMERICAL METHOD FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS IN THE SENSE OF CAPUTO-FABRIZIO DERIVATIVE

In this paper, fractional differential equations in the sense of Caputo-Fabrizio derivative are transformed into integral equations. Then a high order numerical method for the integral equation is investigated by approximating the integrand with a piece-wise quadratic interpolant. The scheme is capable of handling both linear and nonlinear fractional differential equations. A detailed error analysis and stability region of the numerical scheme is rigorously established.

Existence and Uniqueness of Solutions for Mixed Immunotherapy and Chemotherapy Cancer Treatment Fractional Model with Caputo-Fabrizio Derivative

Existence and Uniqueness of Solutions for Mixed Immunotherapy and Chemotherapy Cancer Treatment Fractional Model with Caputo-Fabrizio Derivative

Analysis of the Fuzzy Fractional-Order Solitary Wave Solutions for the KdV Equation in the Sense of Caputo-Fabrizio Derivative

In this paper, we construct a system for analysis of an analytic solution of fractional fuzzy solitary wave solutions for the Korteweg–De Vries (KdV) equation. We apply the iterative method and the Laplace transform under the fractional Caputo-Fabrizio operator. The obtained series form the solution was calculated and approached the estimate values of the proposed problems. The upper and lower portions of the fuzzy result in all three problems were simulation applying two different fractional order among zero and one. The fractional operator is nonsingular and global since the exponential function is present. It provides all types of fuzzy results occurring among zero and one at any fractional order because its dynamic behaviour is globalised of the suggested problems. Because the fuzzy number provides the result in a fuzzy form, with lower and upper branches, fuzziness is also incorporated in the unknown quantity.

New fractional modelling and control analysis of the circumscribed self-excited spherical strange attractor

The purpose of this study is to present and examine a novel non-integer model of the circumscribed self-excited spherical strange attractor, which has not been worked yet. To design the fractional-order model, we use the Caputo-Fabrizio derivative. In order to ensure the existence of the solution Picard-Lindel and fixed-point theories are provided. Moreover, the stability of the considered fractional-order model is shown using the Picard iteration and fixed point theory approach. To get the approximate solutions of the proposed fractional model an efficient numerical scheme called the fractional Euler method(FEM) is used. To see the performance of the used method, the behavior of the numerical solutions of the model is examined under various initial conditions(ICs) and fractional orders. Considerable chaotic behaviors of the solutions are obtained which prove the accuracy and reliability of FEM.

New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator

In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard (H-H) type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed.

Unsteady MHD Flow for Fractional Casson Channel Fluid in a Porous Medium: An Application of the Caputo-Fabrizio Time-Fractional Derivative

Theoretically, this work describes the exact solutions of fractional Casson fluid through a channel under the effect of MHD and porous medium. The unsteady fluid motion of the bottom plate, which is confined by parallel but perpendicular sidewalls, supports the flow. By introducing the dimensionless parameters and variables, the momentum equation, as well as the initial and boundary conditions, has been transformed to a dimensionless form. A mix of Laplace and Fourier transformations is used to get the exact solution for the momentum equation. The constitutive equations for Caputo-Fabrizio’s time-fractional derivative are also incorporated for recovering the exact solutions of the flow problem under consideration. After recovering the exact solutions for flow characteristics, three different cases at the surface of the bottom plate are discussed, by addressing the limiting cases under the influence of the side walls. Moreover, these solutions are captured graphically, and the effects of the Reynolds number Re , fractional parameter α , effective permeability K eff , and dimensionless parameter for Casson fluid β on the fluid’s motion are observed.

A heat transfer problem with exponential memory and the associated thermal stresses

AbstractIn this study, a heat transfer problem defined by the Caputo–Fabrizio derivative, which is known to behave by the exponential decaying law, is addressed in an axially symmetric cylindrical region. Thus, the fundamental solutions of the heat diffusion process and the associated thermal stresses are aimed to find. For this purpose, Laplace and finite Hankel integral transforms are applied according to the geometry of the region. To obtain the thermal stresses, constitutive relations of the classical thermoelasticity theory are used. The effects of fractional orders on the diffusion process are illustrated graphically using MATLAB.

A numerical scheme for solving a class of time fractional integro-partial differential equations with Caputo–Fabrizio derivative

In this paper, we find the numerical solution of the time-fractional partial integro-differential equation with Caputo–Fabrizio fractional derivative. The problem is discretized by some finite difference schemes in the time direction, and then the Sinc collocation method is applied to the resulting problems in the spatial direction. The convergence and stability of the method are proved and illustrated by three test examples.

Comparative Numerical Study of Spline-Based Numerical Techniques for Time Fractional Cattaneo Equation in the Sense of Caputo–Fabrizio

This study focuses on numerically addressing the time fractional Cattaneo equation involving Caputo–Fabrizio derivative using spline-based numerical techniques. The splines used are the cubic B-splines, trigonometric cubic B-splines and extended cubic B-splines. The space derivative is approximated using B-splines basis functions, Caputo–Fabrizio derivative is discretized, using a finite difference approach. The techniques are also put through a stability analysis to verify that the errors do not pile up. The proposed scheme’s convergence analysis is also explored. The key advantage of the schemes is that the approximation solution is produced as a smooth piecewise continuous function, allowing us to approximate a solution at any place in the domain of interest. A numerical study is performed using various splines, and the outcomes are compared to demonstrate the efficiency of the proposed schemes.

Fractional Quantum Problem Based on Caputo-Fabrizio Derivative

Fractional Quantum Problem Based on Caputo-Fabrizio Derivative