Caputo-Fabrizio Fractional Derivative Research Articles

Comparative study of integer-order and fractional-order artificial neural networks: Application for mathematical function generation

This paper investigates the impact of fractional derivatives on the activation functions of an artificial neural network (ANN). Based on the results and analysis, a three-layer backpropagation neural network model with fractional and integer derivatives employed in the activation function and fractional gradient descent method backpropagation learning algorithm has been proposed. Specifically, three perceptrons have been proposed based on fractional and integer derivatives applied to activation functions and learning algorithms. They are fractional derivative activation function (FDAF) perceptron, integer derivative activation function (IDAF)-fractional derivative learning algorithm (FDLA) perceptron, and fractional derivative learning algorithm (FDLA) perceptron. The Riemann-Liouville (RL) fractional derivative, Grunwald-Letnikov (GL) derivative, Caputo-Fabrizio (CF) fractional derivative, Caputo (C) fractional derivative, and Atangana-Baleanu (AB) fractional derivative have been employed. The impact of these derivatives’ fractional-order (FO) is investigated in a wide range from 0.1−0.9 on testing mean square error (MSE) and time required to train the perceptron. FO-based and IO-based perceptrons are compared with the help of performance metrics such as testing MSE and the time required to train the models. The training and testing simulation results illustrate that Caputo-Fabrizio derivative-based perceptrons outperform other fractional derivatives. Also, the performance of the FO-based perceptron is better in terms of the least MSE.

Theoretical analysis of colonic crypt and colorectal cancer model through Caputo–Fabrizio fractional derivative

This study aims to analyze the solution of a system of differential equations that describes the mathematical modeling of cell population dynamics in colonic crypt and colorectal cancer. The Caputo–Fabrizio fractional order derivatives are used to fractionalize the model. The corresponding mathematical model is solved by the Laplace transform, which helps transform differential equations into terms of algebraic equations. The Partial fraction technique is used to find the inverse Laplace of the governing equations. To assess the credibility of the results, graphical simulation has been investigated by manipulating certain parameters. There is a special scenario, namely when [Formula: see text], where the solutions obtained align with those already documented in the literature. This alignment ensures that the initial conditions are met and confirms the accuracy of our solutions.

A Fixed Point Theorem for Time-Fractional Fornberg-Whitham Equation Arising in Atmospheric Science

In this paper, we apply the Homotopy perturbation transform method of the time-fractional Fornberg-Whitham equation with Caputo-Fabrizio fractional derivative. We establish the existence and uniqueness solutions of the time-fractional Fornberg-Whitham equation using fixed point theory. The physical interpretation of the nonlinear models are also discussed through the semi analytical solutions, which demonstrate the effectiveness of the proposed method. Finally, 2D and 3D graphs of some derived solutions are depicted through suitable parameter values.

The Role of Slip Velocity in Fractional MHD Casson Fluid Flow within Porous Cylinder

Numerous academics are intrigued by exploring the fractional Casson fluid model due to its enhanced precision compared to the traditional fluid model. However, many researchers have successfully tackled the analytical solution for the fractional fluid model while overlooking the impact of boundary slip. Hence, this study aims to tackle the challenge of describing the Casson fluid behaviour with fractional properties in a cylindrical domain, considering the influence of slip velocity at the boundaries. Furthermore, magnetohydrodynamics (MHD) had been introduced into the analysis and considered a porous medium resembling a blood clot or fatty plaque. The Caputo-Fabrizio fractional derivative is employed to establish the dimensionless governing equation. Subsequently, we solve it analytically using the Laplace transform in conjunction with finite Hankel transform techniques. A thorough examination follows the graphical presentation of the analytical solution in the context of relevant parameters. The findings illustrate those higher values of the Casson parameter, slip velocity parameter, Darcy number, and fractional parameter lead to an augmentation in fluid velocity. Conversely, an escalation in magnetic parameters causes a reduction in fluid velocity. These findings can be utilized to validate the accuracy of numerical results. The findings of this study hold considerable importance in enhancing our understanding of human blood flow, especially in scenarios where a velocity gradient occurs between blood particles and the stretching motion of arteries.

A robust stability criterion in the one-dimensional subdiffusion equation with Caputo–Fabrizio fractional derivative

A robust stability criterion in the one-dimensional subdiffusion equation with Caputo–Fabrizio fractional derivative

Analysis and numerical simulation of fractional-order blood alcohol model with singular and non-singular kernels

Abstract In this manuscript, we examine the blood alcohol model to investigate the dynamics of alcohol concentration in the human body. The classical model of blood alcohol concentration is converted into the fractional model by using Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo derivatives. The existence and uniqueness theory for the model’s solution is constructed using the Banach fixed point theory. Also, the stability of the solution is established by Ulam-Hyers conditions. For the numerical simulation of the considered model, the Adams-Bashforth method with a two-step Lagrange polynomial is used and the numerical solution of the model with three different derivatives is presented in the tabular and graphical form. The comparison between the exact solution and observed solution is made by root mean square technique which is found to be in good agreement. Finally, the results from the three fractional derivatives are also compared with the exact data, which revealed that the CF fractional derivative performs better than the other two fractional derivatives.

Magnetohydrodynamic influence on fractional nanofluid convection in infinite vertical porous media with constant heat flux

This study investigates the complex dynamics governing free convection flow in nanofluids under Magnetohydrodynamics (MHD) within a porous medium along an infinite vertical surface. We specifically focus on scenarios involving constant and uniform heat flux as well as heat sources. The governing equations, incorporating the Boussinesq approximation, continuity, and momentum equations, are formulated and solved using Caputo-Fabrizio derivatives and Laplace transforms. This comprehensive approach integrates the Boussinesq approximation and Caputo-Fabrizio fractional derivatives, while the use of Laplace transforms enables precise analytical solutions. Additionally, our investigation highlights the significant impact of nanometer-sized copper particles on fluid properties, transforming the behavior from Newtonian (water-like) to non-Newtonian. This transformation profoundly affects thermal conductivity and velocity behavior within the system. Overall, this research establishes a robust foundation for future studies and provides a framework for exploring non-Newtonian nanofluid systems in the context of MHD-driven free convection flow.

Efficient techniques for nonlinear dynamics: a study of fractional generalized quintic Ginzburg-Landau equation

In this research, we explore the solution of the fractional Generalized Quintic Complex Ginzburg-Landau equation (GQCGLE) using the Controlled Picard Transform Method. We solve the equation by incorporating the Laplace transform (LT) and the Caputo-Fabrizio fractional derivative (CF). The fractional GQCGLE is a complex-valued equation that exhibits intricate dynamics and plays a significant role in various physical phenomena, particularly in the realm of nonlinear optics. The suggested method can efficiently address a wide variety of differential equations, encompassing both integer and fractional-order equations. This is achieved by integrating an extra parameter that enhances convergence, making it especially suitable for non-linear differential equations. To confirm the precision and convergence of our approach, we validate it by comparing it with other methods, presenting graphical representations to clarify the effects of different parameters on the solution behaviour, and confirming its precision and convergence. The existence and uniqueness of the solutions are also examined.

Analysis of the human liver model through semi-analytical and numerical techniques with non-singular kernel

This work consists of the study of the time-fractional human liver model with the Caputo–Fabrizio fractional derivative. The existence and uniqueness of the proposed model are shown using fixed point theory. Also, the stability of the considered model is shown using the Ulam Hyres theorem and the Lyapunov function. The solution of the proposed model is obtained using a semi-analytical and numerical scheme. The series solution obtained from the semi-analytical method gives the proper result at any time, similarly, the numerical scheme gives the solution for a long time. The obtained numerical results are compared with real clinical data and earlier published work and found to be very close to real data than earlier published work. Results in the graphs and tables show that the proposed fractional-order model is superior to the traditional model.

The impact of Caputo-Fabrizio fractional derivative and the dynamics of noise on worm propagation in wireless IoT networks

The main objective of this article on Wireless sensor network of the Internet of Things (IoT). The wireless network, B bluetooth network, infrared network, and other networks are the key components of the Internet of Things (IoT). The major emphasis of this work was on the impact of Caputo-Fabrizio fractional derivative on worm propagation in heterogeneous susceptible-exposed-infected-recovered wireless IoT devices. We first determined the equilibrium points and fundamental reproduction number for the Caputo-Fabrizio HSEIR system, and then we discussed the stability of the system at the worm propagation equilibrium point. Using the Picard-Lindeof method, we determine the existence and unique solution for the fractional CF system of the heterogeneous SEIR model. Next, we use fixed point theory to judge the stability of the iterative method. We investigate the impact of the derivative order on the behaviour of the resultant functions and acquired numerical values by computing the model's findings for various fractional orders. In addition, we compute the integer-order model's results and contrast them with the results of the fractional-order model. We develop a periodically intermittent controller driven by white noise with the amazing benefits of reduced cost and more adaptable control technique to restrict the spread of worms in wireless IoT networks. To clearly define the conditions for stability in probability one, we employ the stochastic analysis technique. Our results show that the nonlinear worm propagation system may be stabilised by intermittent stochastic perturbation under the parameters of intermittent time linked to stochastic perturbation strength. Our theoretical conclusions may be used to analyse the observable processes of the worm, design countermeasures to prevent its spread, and evaluate the consequences of various system parameters.

Hybrid cubic and hyperbolic b-spline collocation methods for solving fractional Painlevé and Bagley-Torvik equations in the Conformable, Caputo and Caputo-Fabrizio fractional derivatives

In this paper, we approximate the solution of fractional Painlevé and Bagley-Torvik equations in the Conformable (Co), Caputo (C), and Caputo-Fabrizio (CF) fractional derivatives using hybrid hyperbolic and cubic B-spline collocation methods, which is an extension of the third-degree B-spline function with more smoothness. The hybrid B-spline function is flexible and produces a system of band matrices that can be solved with little computational effort. In this method, three parameters m, η, and λ play an important role in producing accurate results. The proposed methods reduce to the system of linear or nonlinear algebraic equations. The stability and convergence analysis of the methods have been discussed. The numerical examples are presented to illustrate the applications of the methods and compare the computed results with those obtained using other methods.

Non-Standard Finite Difference and Vieta-Lucas Orthogonal Polynomials for the Multi-Space Fractional-Order Coupled Korteweg-de Vries Equation

This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For this purpose, the nonstandard finite difference method and spectral collocation method with the properties of the Shifted Vieta-Lucas orthogonal polynomials are employed for converting these models into a system of algebraic equations. The Newton-Raphson technique is then applied to solve these algebraic equations. Since there is no exact solution for non-integer order, we use the absolute two-step error to verify the accuracy of the proposed numerical results.

Stability analysis of fractional order breast cancer model in chemotherapy patients with cardiotoxicity by applying LADM

AbstractBreast cancer is the most common type of cancer in women. Chemotherapy is primarily used for patients with stage 2 to 4 breast cancer. Most chemotherapy drugs are effective at destroying rapidly growing and proliferating cancer cells. However, drugs also damage normal, rapidly growing cells, which can lead to serious side effects. Breast cancer treatment with chemotherapy can affect heart health. Side effects of chemotherapy on the heart are called cardiotoxicity. Therefore, we have constructed a mathematical model from the breast cancer patient population. In this article, we utilize the Caputo–Fabrizio fractional order derivative for mathematical modeling of the breast cancer stages in chemotherapy patients. The use of Caputo–Fabrizio fractional derivative provides a more valuable insight into the complexity of the breast cancer model. The stability of the fractional order model is also proven by the $\mathscr{P}$ P -stable approach of the fixed point theorem. Also, the numerical simulations are performed via Laplace Adomian decomposition method to establish the dependence of the breast cancer dynamics on the order of the fractional derivatives. Based on the geometric results in the figures, we can conclude that the magnitude of the fractional order has a considerable impact on the days, which the maximum or minimum of the system solutions are reached, with a shift in the time at which this happens as the fractional order decreases from 1. However, it is obvious that the solutions of Caputo–Fabrizio fractional model approach the relevant results of the classical integer order system, when the fractional order approaches to 1.

Solving Fractional Riccati Differential Equation with Caputo-Fabrizio Fractional Derivative

This article offers an analytical solution for the fractional Riccati differential equation in three distinct cases. These cases are determined by the discriminant and the analytical solution based on the properties of the Caputo-Fabrizio fractional derivative and integral. Several examples were tested using this analytical solution. It is noteworthy that various methods have yielded related results as indicated in the literature.

Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative

Abstract This study demonstrates the use of fractional calculus in the field of epidemiology, specifically in relation to dengue illness. Using noninteger order integrals and derivatives, a novel model is created to examine the impact of temperature on the transmission of the vector–host disease, dengue. A comprehensive strategy is proposed and illustrated, drawing inspiration from the first dengue epidemic recorded in 2009 in Cape Verde. The model utilizes a fractional-order derivative, which has recently acquired popularity for its adaptability in addressing a wide variety of applicable problems and exponential kernel. A fixed point method of Krasnoselskii and Banach is used to determine the main findings. The semi-analytical results are then investigated using iterative techniques such as Laplace-Adomian decomposition method. Computational models are utilized to support analytical experiments and enhance the credibility of the results. These models are useful for simulating and validating the effect of temperature on the complex dynamics of the vector–host interaction during dengue outbreaks. It is essential to note that the research draws on dengue outbreak studies conducted in various geographic regions, thereby providing a broader perspective and validating the findings generally. This study not only demonstrates a novel application of fractional calculus in epidemiology but also casts light on the complex relationship between temperature and the dynamics of dengue transmission. The obtained results serve as a foundation for enhancing our understanding of the complex interaction between environmental factors and infectious diseases, leading the way for enhanced prevention and control strategies to combat global dengue outbreaks.

On the existence and numerical simulation of Cholera epidemic model

Abstract A model describing the transmission dynamics of cholera is considered in this article. The concerned model is investigated under the Caputo-Fabrizio fractal fractional derivative. The objective of this article is to study theoretical and numerical results for the model under our consideration. Classical fixed point approach is used to obtain sufficient conditions for the existence of solution to the proposed model. Adam’s Bashforth numerical method is utilized for the numerical interpretation of the suggested model. The considered technique is a powerful mathematical tool, that provides a numerical solution for the concerned problem. To discuss the transmission dynamics of the considered model, several graphical presentations are given.

Solution of the fractional diffusion equation by using Caputo-Fabrizio derivative: application to intrinsic arsenic diffusion in germanium

In this work, we focused on solving the space-time fractional diffusion equation with an application on the intrinsic arsenic diffusion in germanium. At first we have treated the differential equation in a semi-infinite medium by using Caputo-Fabrizio fractional derivative. We have introduced the Laplace transform to solve this type of equations. Secondly, Based on the obtained solution, we have simulated an profile of arsenic diffusion in germanium under intrinsic conditions. Accurate simulations have been achieved showing that the fractional derivative orders affect on the estimation of the diffusion coefficient, where increasing the time fractional derivative order α reduces the value of the diffusion coefficient, while increasing the space fractional derivative order β increases the value of the diffusion coefficient.

Study of fractional forced KdV equation with Caputo–Fabrizio and Atangana–Baleanu–Caputo differential operators

It is essential for mathematicians, physicists, and engineers to construct fractional mathematical models for specific phenomena and develop numerical or analytical solutions for these models. In this work, we implement the natural decomposition approach with nonsingular kernel derivatives to investigate the solution of nonlinear fractional forced Korteweg–de Vries (FF-KdV) equation. We first investigate the FF-KdV equation under the Caputo–Fabrizio fractional derivative. The similar equations are then examined using the Atangana–Baleanu derivative. This approach combines the decomposition method with the Natural transform method. The series solution of the suggested equations is thus obtained using the natural transform. The key benefit of this novel approximate-analytical approach is that it may provide an analytical solution for the FF-KdV problem in the form of convergent series with simple computations. For each equation, three unique situations are chosen to demonstrate and test the viability of the proposed method. To guarantee the competence and dependability of the proposed method, the nature for various values of the Froude number Fr have been provided. The present approach is also used to calculate solutions at various fractional orders. The approximate series solution’s behavior for various fractional orders has been graphically displayed. The outcomes demonstrate that the methodology is simple to use and reliable when applied to numerous fractional differential equations.

A fractional dual-phase-lag thermoelastic model for a solid half-space with changing thermophysical properties involving two-temperature and non-singular kernels

<abstract> <p>The thermal and mechanical properties of materials show differences depending on the temperature change, which necessitates consideration of the dependence of the properties of these materials on this change in the analysis of thermal stress and deformation of the material. As a result, in the present work, a mathematical framework for thermal conductivity was formulated to describe the behavior of non-simple elastic materials whose properties depend on temperature changes. This derived model includes generalized fractional differential operators with non-singular kernels and two-stage delay operators. The fractional derivative operators under consideration include both the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative, in addition to the traditional fractional operator. Not only that, but the system of governing equations includes the concept of two temperatures. Based on the proposed model, the thermodynamic response of an unlimited, constrained thermoelastic medium subjected to laser pulses was considered. It was taken into account that the thermal elastic properties of the medium, such as the conductivity coefficient and specific heat, depend on the temperature. The governing equations of the problem were formulated and then solved using the Laplace transform method, followed by the numerical inverse. By presenting the numerical results in graphical form, a detailed analysis and discussion of the effects of fractional factors and the dependence of properties on temperature are presented. The results indicate that the fractional order coefficient, discrepancy index, and temperature-dependent properties significantly affect the behavior fluctuations of all physical domains under consideration.</p> </abstract>

Analysis of a nonlinear problem involving discrete and proportional delay with application to Houseflies model

<abstract><p>This manuscript established a comprehensive analysis of a general class of fractional order delay differential equations with Caputo-Fabrizio fractional derivative (CFFD). Functional analysis was used to examine the existence and uniqueness of the suggested class and to generate sufficient requirements for Ulam-Hyers (UH) type stability. Further, a numerical method based on Lagrange interpolation is used to compute approximate solution. Then, some applications in physical dynamics including a houseflies model and a Cauchy type problem were discussed to illustrate the established analysis with graphical illustrations.</p></abstract>