Caputo-Fabrizio Operator Research Articles

Comparison of unsteady MHD flow of second grade fluid by two fractional derivatives

A study is conducted on the unsteady motion of a free convective flow of second grade fluid, energy transfer, and Darcy’s law over an oscillatory smooth vertical plate. The study compares two different approaches in developing a fractional model: the Caputo-Fabrizio operator with nonsingular kernel and the constant proportional Caputo fractional operator with Fourier’s and Fick’s laws. By applying the Laplace method and transforming the provided set of equations into nondimensional form, we obtained semi-analytical results and presented these results through graphical analysis. The study examines how different flow parameters, including the fractional parameters, affect the velocity, mass, and heat profiles of the physical system. The results suggest that the physical model using constant proportional Caputo derivative leads to higher temperature, stronger concentration, and increased velocity compared to the Caputo-Fabrizio model. This highlights the importance of selecting an appropriate fractional model when studying complex physical systems. It is also depicted from the whole analysis that field variables with novel hybrid fractional derivative constant proportional Caputo exhibits a more effective and declining tendency than Caputo-Fabrizio.

ON THE INITIAL VALUE PROBLEMS FOR NEUTRAL INTEGRO-DIFFERENTIAL SYSTEM WITHIN EXPONENTIAL KERNEL

This paper deals with the existence results for neutral integro-differential system (NIDS) through Caputo-Fabrizio (CF) operator. The solution to our addressing system, which can be considered innovative, is explained and proven in preliminary section. Certain novel existence findings are shown using fixed point approaches. Finally, two numerical examples are provided in the work to demonstrate the application of our theoretical findings.

Study of Hybrid Problems under Exponential Type Fractional-Order Derivatives

In this investigation, we develop a theory for the hybrid boundary value problem for fractional differential equations subject to three-point boundary conditions, including the antiperiodic hybrid boundary condition. On suggested problems, the third-order Caputo–Fabrizio derivative is the fractional operator applied. In this regard, the corresponding hybrid fractional integral equation is obtained by the Caputo–Fabrizio operator’s properties with the Green function’s aid. Then, we apply Dhage’s nonlinear alternative to the Schaefer type to prove the existence results. Finally, two examples are provided to confirm the validity of our main results.

Soret Effect on MHD Casson Fluid over an Accelerated Plate with the Help of Constant Proportional Caputo Fractional Derivative.

Non-Newtonian fluid flow is significant in engineering and biomedical applications such as thermal exchangers, electrical cooling mechanisms, nuclear reactor cooling, drug delivery, blood flow analysis, and tissue engineering. The Caputo operator has emerged as a prevalent tool in fractional calculus, garnering widespread recognition. This research aims to introduce a novel derivative by merging the proportional and Caputo operators, resulting in the fractional operator known as the constant proportional Caputo. In order to demonstrate this newly defined operator's dynamic qualities, it was employed in the analysis of the unsteady Casson flow model. In addition, the current work shows an analytical analysis to determine the Soret effect on the fractionalized MHD Casson fluid over an oscillating vertical plate. Fractional partial differential equations (PDEs) are used to formulate the problem along with IBCs. The introduction of appropriate nondimensional variables converts the PDEs into dimensionless form. The precise solutions to the fractional governing PDEs are then determined by the Laplace transform method. Velocity, concentration, and temperature profiles; the impacts of the Prandtl number; fractional parameter β and γ; and Soret and Schmidt numbers are graphically depicted. The profiles of temperature, concentration, and velocity rise with rising time and fractional parameters. Interestingly, as the Casson flow parameter is higher, fluid velocity decreases closest to the plate but increases away from the plate. Tables showing the findings for the skin-friction coefficient, Sherwood, and Nusselt numbers for a range of flow-controlling parameter values are provided. Furthermore, an investigation is undertaken to compare fractionalized and ordinary velocity fields. The results suggest that the fractional model employing a constant proportional derivative exhibits a quicker decay than the model incorporating conventional Caputo and Caputo-Fabrizio operators.

On initial value problem for diffusion equation with Caputo-Fabrizio operator on the plane

On initial value problem for diffusion equation with Caputo-Fabrizio operator on the plane

Investigation of the fractional coupled-Burgers model with the exponential kernel

The present study seeks to develop a collocation method focused on an extended cubic B-spline (ECBS) to approximate the time fractional coupled-burgers (TFCB) equation. Numerous branches of engineering and physical sciences are used to model the TFCB equation. They have been utilized to simulate the sedimentation of scale volume concentrations of two different types of particles in fluid mixtures under the influence of gravity with the non-local time. The non-local time is considered in the form of the Caputo-Fabrizio operator (CFO). The space-direction is discretized via the ECBS functions. In addition, the new theoretical study of the stability and the convergence are also provided and compared with the computational values. Three examples are tested numerically in order to check the feasibility and applicability of the methodology. The graphics are also provided to examine the order of convergence numerically.

Existence and Stability Results for Piecewise Caputo–Fabrizio Fractional Differential Equations with Mixed Delays

In this article, by using the differential Caputo–Fabrizio operator, we suggest a novel family of piecewise differential equations (DEs). The issue under study contains a mixed delay period under the criteria of anti-periodic boundaries. It is possible to utilize the piecewise derivative to describe a variety of complex, multi-step, real-world situations that arise from nature. Using fixed point (FP) techniques, like Banach’s FP theorem, Schauder’s FP theorem, and Arzelá Ascoli’s FP theorem, the Hyer–Ulam (HU) stability and the existence theorem conclusions are investigated for the considered problem. Eventually, a supportive example is given to demonstrate the applicability and efficacy of the applied concept.

Asymptotically almost periodic synchronization in fuzzy competitive neural networks with Caputo-Fabrizio operator

This paper researches a class of Caputo-Fabrizio fractional-order fuzzy competitive neural networks (CF-FCNNs) with different time scales and time delays. Firstly, according to Banach contracting mapping principle and some appropriate assumptions, it derives CF-FCNNs possess a unique weak solution. Then, exponential Euler difference scheme of discrete-time CF-FCNNs (DCF-FCNNs) is presented by the constant variation methods and the difference format is implicit Euler difference and is solved by fractional PECE algorithm. Next, the existence of a unique bounded asymptotically almost periodic sequence solution, global exponential stability and synchronization of DCF-FCNNs are considered. In the end, numerical examples are given to describe the effectiveness of the acquired conclusions.

A numerical study of the Van der Pol model derived by the Caputo–Fabrizio operator

The study of nonlinear oscillators is an important topic in the development of the theory of dynamical systems. In this research, a nonlinear fractional model is introduced, which is called the fractional Van der Pol model. This modified model is derived using the Caputo–Fabrizio operator. Achieving the solution of this model is not easy. Therefore, in this research, an efficient algorithm for solving this fractional model is evaluated. This algorithm is supported by the three-step Adams–Bashforth process. A significant feature of this research is the graphical presentation of achieving limit cycles for various parameters.

Quadratic and cubic logistic models involving Caputo–Fabrizio operator

Quadratic and cubic logistic models involving Caputo–Fabrizio operator

On mathematical modeling of fractional-order stochastic for tuberculosis transmission dynamics

Tuberculosis remains one of the most dangerous diseases globally and has affected many people in Sub-Saharan Africa. In this paper, a fractional stochastic model of tuberculosis disease was formulated and analyzed. The existence and uniqueness of solutions are presented in the new approach far from the deterministic fractional operators. We carry out numerical simulations of the model using three fractional operators. Our numerical results suggest that the Caputo–Fabrizio operator has a more random effect among the three different operators than the Atangana–Baleanu and the Caputo operators. Further, it is envisaged that the fractional-order derivatives significantly impact the dynamics of the disease.

On dynamical behavior for approximate solutions sustained by nonlinear fractional damped Burger and Sharma–Tasso–Olver equation

In this paper, we study a nonlinear fractional Damped Burger and Sharma–Tasso–Olver equation using a new novel technique, called homotopy perturbation transform method (FHPTM). There are three examples used to demonstrate and validate the proposed algorithm’s efficiency. This nonlinear model depicts nonlinear wave processes in fluid dynamics, ecology, solid-state physics, shallow-water wave propagation, optical fibers, fluid mechanics, plasma physics, and other applied science, engineering, and mathematical physics disciplines, as well as other phenomena. Numerous algebraic properties of the fractional derivative Caputo–Fabrizio operator are illustrated concerning the Laplace transformation to demonstrate their utility. Different graphs and tables compare the results obtained by R. Nawaz et al. [Alex. Eng. J. 60, 3205 (2021)] and M. S. Rawashdeh [Appl. Math. Inform. Sci. 9, 1239 (2015)]. The proposed scheme accelerates the convergence of the series solution and guarantees the convergence associated with the homotopy parameter. Furthermore, the physical nature of various fractional orders has been captured in plots. The obtained results demonstrate that the employed solution procedure is dependable and methodical in investigating the behaviors of nonlinear models of both integer and fractional orders.

Mathematical epidemiological modeling and analysis of monkeypox dynamism with non-pharmaceutical intervention using real data from United Kingdom.

In this study, a mathematical model for studying the dynamics of monkeypox virus transmission with non-pharmaceutical intervention is created, examined, and simulated using real-time data. Positiveness, invariance, and boundedness of the solutions are thus examined as fundamental features of mathematical models. The equilibrium points and the prerequisites for their stability are achieved. The basic reproduction number and thus the virus transmission coefficient ℜ0 were determined and quantitatively used to study the global stability of the model's steady state. Furthermore, this study considered the sensitivity analysis of the parameters according to ℜ0. The most sensitive variables that are important for infection control are determined using the normalized forward sensitivity index. Data from the United Kingdom collected between May and August 2022, which also aid in demonstrating the usefulness and practical application of the model to the spread of the disease in the United Kingdom, were used. In addition, using the Caputo-Fabrizio operator, Krasnoselskii's fixed point theorem has been used to analyze the existence and uniqueness of the solutions to the suggested model. The numerical simulations are presented to assess the system dynamic behavior. More vulnerability was observed when monkeypox virus cases first appeared recently as a result of numerical calculations. We advise the policymakers to consider these elements to control monkeypox transmission. Based on these findings, we hypothesized that another control parameter could be the memory index or fractional order.

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.

Analytical Study of the Complexities in a Three Species Food Web Model with Modified Caputo–Fabrizio Operator

This article presents the analytical study of the three species fractional food web model in the framework of the Modified Caputo–Fabrizio operator. With the help of fixed point theory, the existence and uniqueness results are investigated for the fractional order model. To obtain the approximate solution for the suggested model, the well-known Laplace–Adomian decomposition method is used. The solutions are validated through simulations with a variety of fractional orders and initial values, where the complex nature of the system can be observed. The technique used here can be easily used to study a range of complex problems in different branches of science. From the figures, it can be observed that, at integer higher fractional order, there are a number of oscillations in the system and the system behaves chaotically, while, at lower fractional orders, the oscillation amplitudes decrease, resulting in the faster converging towards the equilibrium point. According to the results, the Modified Caputo–Fabrizio fractional-order derivative may be used in a variety of future fractional dynamics scenarios.

Transmission dynamics of fractional order yellow virus in red chili plants with the Caputo–Fabrizio operator

The yellow virus in the red chili fractional order model is investigated in this scientific study. The Caputo–Fabrizio and Fractal fractional derivative operator, which incorporates an antiretroviral treatment compartment, are used to investigates this pandemic occurrence. It is essential to figure out how to develop methods to halt the spread of the yellow virus in red chili. While measures are being taken to curb the pandemic of the yellow virus, the more contagious yellow virus found in red chilies is emerging in several areas. It is essential to develop methods for preventing the spread of the yellow virus. To maintain a certain level of protection while simulating the yellow virus’s spread in red chili plants. We investigated the potential for an epidemic in red chili plants as a case study.In this study, we use fixed point theory to create existence, uniqueness, and stability conditions for the fractional order yellow virus in the red chilli model using the Caputo Fabrizio operator. For these kinds of biological models, the fractional derivative technique is quite new. By using the iterative Laplace transform approach, we have also discovered the initial approximate solutions for a suggested model that are readily available. This method combines the Laplace transform method with one of the most dependable methods, the new iterative method. Finally, we analysed parameters that represent the progression of sickness and used charts to display the numerical simulations.

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 fuzzy fractional acoustic waves model in terms of the Caputo-Fabrizio operator

<abstract><p>This paper proposes an analytical solution for a fractional fuzzy acoustic wave equation. Under the fractional Caputo-Fabrizio operator, we use the Laplace transformation and the iterative technique. In the present study, the achieved series type result was determined, and we approximated the estimated values of the suggested models. All three problems used two various fractional-order simulations between 0 and 1 to obtain the upper and lower portions of the fuzzy results. Since the exponential function is present, the fractional operator is non-singular and global. Due to its dynamic behaviors, it provides all fuzzy form solutions that happen between 0 and 1 at any level of fractional order. Because the fuzzy numbers return the solution in a fuzzy shape with upper and lower branches, the unknown quantity likewise incorporates fuzziness.</p></abstract>

Numerical analysis of fractional heat transfer and porous media equations within Caputo-Fabrizio operator

<abstract><p>This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering.</p></abstract>

A fractional numerical study on a plant disease model with replanting and preventive treatment

Food security has become a significant issue due to the growing human population. In this case, a significant role is played by agriculture. The essential foods are obtained mainly from plants. Plant diseases can, however, decrease both food production and its quality. Therefore, it is substantial to comprehend the dynamics of plant diseases as they can provide insightful information about the dispersal of plant diseases. In order to investigate the dynamics of plant disease and analyze the effects of strategies of disease control, a mathematical model can be applied. We show that this model provides the non-negative solutions that population dynamics requires. The model was investigated by using the Atangana-Baleanu in Caputo sense (ABC) operator which is symmetrical to the Caputo-Fabrizio (CF) operator with a different function. Whereas the ABC operator uses the generalized Mittag-Leffler function while the CF operator employs the exponential kernel. For the proposed model, we have displayed the local and global stability of a nonendemic and an endemic equilibrium, existence and uniqueness theorems. By applying the fractional Adams-Bashforth-Moulton method, we have implemented numerical solutions to illustrate the theoretical analysis.