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The normalized fractional derivatives (NFDs) are a normalization of the existing fractional derivatives (FDs) which admit geometrical meanings. They have finite ordinary derivatives at the initial point, which cause smoothness of solutions for related fractional differential equations (FDEs). For FDs with non-singular kernels, the NFDs do not vanish at the starting point, in general, and therefore related FDEs admit solutions without the need to impose extra conditions. In this paper, we consider FDEs with the normalized Caputo–Fabrizio derivative of Caputo type. We show in closed forms the solutions of related FDEs and show that the Cauchy problem with the NFD admits a nontrivial solution. We also define the higher-order NFDs and show that related FDEs can be solved by transforming them to integro-differential equations with integer orders. [ABSTRACT FROM AUTHOR]

Ordinary nonlinear differential equations with classical and fractional derivatives are used to simulate several real-world problems. Nonetheless, numerical approaches are used to acquire their solutions. While various have been proposed, they are susceptible to both disadvantages and advantages. In this paper, we propose a more accurate numerical system for solving nonlinear differential equations with classical and Caputo–Fabrizio derivatives by combining two concepts: the parametrized method and the predictor–corrector method. We gave theoretical analyses to demonstrate the method’s correctness, as well as several illustrated examples for both scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we apply the fractional homotopy perturbation transform method (FHPTM) to deliver an effective semi-analytical technique for determining fractional-order Kuramoto–Sivashinsky equations. The project technique combines the Laplace transform with the Caputo–Fabrizio fractional derivative of order α where α ∈ (0 , 1 ] . Fractional-order Kuramoto–Sivashinsky equation is indeed important in the field of nonlinear physics and mathematics. It is a fractional partial differential equation that describes the behaviour of waves in certain dissipative media, such as flames and chemicals. The FHPTM is described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of the presented techniques. [ABSTRACT FROM AUTHOR]

This article focuses on the investigation of the time-fractional Sharma–Tasso–Olver equation, a crucial equation with significant relevance in various scientific domains, including nonlinear optics, quantum field theory, and the physical sciences. The natural transform decomposition approach, an efficient and innovative methodology, is used in this study. The two nonsingular kernel derivatives, such as Caputo–Fabrizio and Atangana–Baleanu in the Caputo sense, are used in the suggested technique. We considered two nonlinear cases with suitable initial conditions to show that the proposed method is accurate and works well with the existing methods. Banach's fixed point theorem is used to demonstrate the uniqueness and convergence of the solution that has been achieved. The proposed solution accurately captures the behaviour of the reported findings for various fractional orders. The obtained outcomes of the suggested approach are contrasted with those of the precise solution and other numerical techniques, including the natural transform decomposition method with Caputo derivative, the natural transform iterative method, the q-homotopy analysis Elzaki transform method, and the fractional reduced differential transform method. The outcomes of the proposed method are presented in tables and figures. The findings of this study demonstrate that the investigated technique is highly effective and precise in solving nonlinear time fractional differential equations. [ABSTRACT FROM AUTHOR]

The study of the complex model associated with chaotic models is always the most complicated and fundamental in the current scientific environment. The primary goal of the current paper is to provide an illustration of the fundamental theory while analysing dynamical systems and validating the chaotic behaviour of the Lorenz–Haken (LH) equations, a system of fractional differential equations. The LH equations are used to describe the 4D chaotic laser. The Adams–Bashforth numerical method is used to extract the numerical solutions projected model. The classical model introduces the bifurcation of the parameter linked with the system. The system's uniqueness and existence are confirmed using the fixed-point hypothesis with the Caputo–Fabrizio fractional operator, followed by boundedness and dynamical analysis. Furthermore, the chaotic character of the numerical solutions with different orders obtained at different beginning circumstances is presented. [ABSTRACT FROM AUTHOR]

A spectral collocation-based approximate algorithm is adapted for the numerical evaluation of a class of fractional-order delay differential equations. The involved fractional operators are defined as the Caputo-Fabrizio and Liouville-Caputo derivatives. A novel family of polynomials called Narayana polynomials and their generalization forms are utilized in our collocation procedure. We study the convergent analysis of the Narayana polynomials in a weighted L2 norm and obtain an upper bound for their series expansion form. The performance of the present matrix collocation is justified by solving three test examples using both fractional operators and various fractional orders. The outcomes are compared with two existing numerical approaches, i.e., the modified operational matrix method (MOMM) and the operational matrix of integration relied on Taylor bases (OMTB). A comparison study reveals that our results are more accurate than these two methods and thus the presented matrix algorithm is superior in terms of efficacy and applicability. [ABSTRACT FROM AUTHOR]

This paper investigates the necessary conditions relating to the existence and uniqueness of solution to impulsive system fractional differential equation with a nonlinear p-Laplacian operator. Our problem is based on two kinds of fractional order derivatives. That is, Atangana–Baleanu–Caputo (ABC) fractional derivative and the Caputo–Fabrizio derivative. To achieve our main aims, we will first convert the proposed impulse system into an integral equation form. Next, we prove the existence and uniqueness of solutions with the help of Leray–Schauder's theory and the Banach contraction principle. We analyze the operator for continuity, boundedness, and equicontinuity. Further, we investigate the stability solution to the proposed impulsive system by using stability techniques. In the last part, we demonstrate the results via an illustrative example for the application of the results. [ABSTRACT FROM AUTHOR]

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India. [ABSTRACT FROM AUTHOR]

In this article, we studied the time‐fractional Burgers–Huxley equation using the natural transform decomposition method. The fractional operator is treated in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu senses. We employed the natural transform with the Adomian decomposition process on time‐fractional Burgers–Huxley equation to obtain the solution. To establish the uniqueness and convergence of the accomplished solution, the Banach's fixed point theorem is used. The obtained findings are visually shown in two‐ and three‐dimensional graphs for various fractional orders. To illustrate the efficacy of the method under discussion, numerical simulations are provided. The proposed solution captures the behavior of the reported findings for various fractional orders. A comparative study was conducted to ascertain the proposed method's correctness. The findings of this study establish that the technique investigated is both efficient and accurate for solving nonlinear fractional differential equations that arise in science and technology. [ABSTRACT FROM AUTHOR]

The objective of this manuscript is to present a novel approach to modeling influenza A disease dynamics by incorporating the Caputo-Fabrizio (CF) fractional derivative operator into the model. Particularly distinct contact rates between exposed and infected individuals are taken into account in the model under study, and the fractional derivative concept is explored with respect to this component. We demonstrate the existence and uniqueness of the solution and obtain the series solution for all compartments using the Laplace transform method. The reproduction number of the Influenza A model, which was created to show the effectiveness of different contact rates, was obtained and examined in detail in this sense. To validate our approach, we applied the predictor-corrector method in the sense of the Caputo-Fabrizio fractional derivative and demonstrate the effectiveness of the fractional derivative in accurately predicting disease dynamics. Our findings suggest that the use of the Caputo-Fabrizio fractional derivative can provide valuable insights into the mechanisms underlying influenza A disease and enhance the accuracy of disease models. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system Dαt X(t) = AX(t) described by fractional differential equations, where D is the fractional derivative in the Caputo-Fabrizio sense and A = (ai j)2x2 is nonsingular coefficient matrix with aij ∈ R. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system. [ABSTRACT FROM AUTHOR]

In this paper, the solutions of some typical nonlinear fractional differential equations are discussed, and the implicit analytical solutions are obtained. The fractional derivative concerned here is the Caputo-Fabrizio form, which has a nonsingular kernel. The calculation results of different fractional orders are compared through images. In addition, by comparing the results obtained in this paper with those under Caputo fractional derivative, it is found that the solutions change relatively gently under Caputo-Fabrizio fractional derivative. It can be concluded that the selection of appropriate fractional derivatives and appropriate fractional order is very important in the modeling process. [ABSTRACT FROM AUTHOR]

Smoking has caused the illness and death of many people around the world for a long time. Therefore, many researchers have investigated many methods to quit smoking and reduce its use. In this paper, a smoking model with determination is first examined broadly with Caputo-Fabrizio derivative. We give the equilibrium points and their stability analysis for our fractional model. Taking advantage of fixed point theory, we give some solution properties for the fractional smoking model with determination. Our findings are highlighted by presenting some explanatory graphics. [ABSTRACT FROM AUTHOR]

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India.

In this paper, we investigate the solution of the Riccati differential equations of fractional order (FRDEs) involving Caputo derivative (CD), Caputo-Fabrizio derivative (CFD) or Atangana-Baleanu derivative (ABD) using a semi-analytical iterative approach. Temimi and Ansari introduced this method and called it TAM. The linearization of the method involves splitting the problem into a linear and nonlinear part. While this approach is a semi-analytical iterative technique, there is a significant amount of analytical work where the computational times of integrals are needed to be carried out numerically. The novelty and the motivation behind this paper is the comparison of the time used in minutes which is given for three derivatives: CD, CFD and ABD. Meanwhile, the comparison of the approximate solutions with CD, CFD and ABD are presented. With the help of the software Mathematica, the computational results have been obtained. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the behaviour of solution for the system exemplifying model of tumour invasion and metastasis by the help of q-homotopy analysis transform method (q-HATM) with the fractional operator. The analyzed model consists of a system of three nonlinear differential equations elucidating the activation and the migratory response of the degradation of the matrix, tumour cells and production of degradative enzymes by the tumour cells. The considered method is graceful amalgamations of q-homotopy analysis techniquewith Laplace transform(LT), and Caputo-Fabrizio (CF) fractional operator is hired in the present study. By using the fixed point theory, existence and uniqueness are demonstrated. To validate and present the effectiveness of the considered algorithm, we analyzed the considered system in terms of fractional order with time and space. The error analysis of the considered scheme is illustrated. The variations with small change time with respect to achieved results are effectively captured in plots. The obtained results confirm that the considered method is very efficient and highly methodical to analyze the behaviors of the system of fractional order differential equations. [ABSTRACT FROM AUTHOR]

In this research work, we study the model of nonlinear reaction–diffusion equation, diffusion‐wave equation, and Cattaneo equation with the help of a numerical method in which time‐fractional derivative is of Caputo–Fabrizio type (C‐F). First, we derive a formula of the fractional‐order C‐F derivative of xk. Using this formula and some properties of shifted Chebyshev polynomials, we find out the operational matrix of the C‐F derivative. We solve a new class of fractional partial differential equations (FPDEs) using the collocation method and C‐F operational matrix. We depict the feasibility and validity of this numerical method by giving the solution of some numerical examples. We take the diffusion equation, nonlinear reaction–diffusion equation, diffusion‐wave equation, and Cattaneo equation as a particular case of our model. By comparing the obtained numerical results with the given exact solution, we draw an error table depicting that our results are so accurate and desirable accuracy. In the authors' knowledge, the C‐F shifted Chebyshev operational matrix is derived for the first time in this work. [ABSTRACT FROM AUTHOR]

The aim of this study, is to present the fractional model of energy supply-demand system (ES-DS) based on the Caputo-Fabrizio derivative. For the first time, the existence and uniqueness of solution of the fractional model of ES-DS are proved and t is the main novelty of this paper. Also, we know that the obtained results from mathematical models with fractional order are more accurate than usual models. This model is based on four important functions, energy resources demand (ERD) ei, energy resource supply (ERS) S2, energy resource import (ESI) S3 and renewable energy resources (RER) e4. Also, applying the obtained numerical results, we can forecast the rate of these functions for spacial interval of time. [ABSTRACT FROM AUTHOR]

We present a fractional-order model for the COVID-19 transmission with Caputo–Fabrizio derivative. Using the homotopy analysis transform method (HATM), which combines the method of homotopy analysis and Laplace transform, we solve the problem and give approximate solution in convergent series. We prove the existence of a unique solution and the stability of the iteration approach by using fixed point theory. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative. [ABSTRACT FROM AUTHOR]

In this work, we consider a new fractional derivative with nonsingular kernel introduced by Caputo-Fabrizio (CF) and propose a finite difference method for computing the CF fractional derivatives. Based on an iterative technique, we can reduce the computational complexity from O(J²N) to O(JN), and the corresponding storage will be cut down from O(JN) to O(N), which makes the computation much more efficient. Besides, by adopting piece-wise Lagrange polynomials of degrees 1, 2, and 3, we derive the second, third, and fourth order discretization formulas respectively. The error analysis and numerical experiments are carefully provided for the validation of the accuracy and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

In this work we discuss higher order multi-term partial differential equation (PDE) with the Caputo-Fabrizio fractional derivative in time. Using method of separation of variables, we reduce fractional order partial differential equation to the integer order. We represent explicit solution of formulated problem in particular case by Fourier series.

In this article, Laplace homotopy analysis method in order to solve fractional heatlike equation with variable coefficients, are introduced. Laplace homotopy analysis method, founded on combination of homotopy methods and Laplace transform is used to supply a new analytical approximated solutions of the fractional partial differential equations in case of the Caputo-Fabrizio. The solutions obtained are compared with exact solutions of these equations. Reliability of the method is given with graphical consequens and series solutions. The results show that the method is a powerfull and efficient for solving the fractional heat-like equations with variable coefficients. [ABSTRACT FROM AUTHOR]

In the computer security and for any defensive strategy, computer viruses are very significant aspect. To understand their expansion and extension is very important component. In order to deal with this issue, we consider a fractional epidemiological model. In this article, we analyze moderate epidemiological model to describe computer viruses with an arbitrary order derivative having non-singular kernel. We obtain the solution of the problem by using an iterative method. By using the fixed-point theorem the existence of the solution is discussed. The uniqueness of the solution is verified. We perform some numerical simulations and show graphically to observe the impact of the arbitrary order derivative. [ABSTRACT FROM AUTHOR]

In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario. The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional power α = 1. Numerical results are finally given to justify the effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

A collocation method based on multiquadric radial basis functions is proposed for numerical solution of fractional differential equations. The fractional derivative is sense of Caputo-Fabrizio derivative. An efficient error bound of the method is also introduced in the $$L_2$$ norm, using properties of native spaces. We test this approach for two examples. The obtained numerical results confirm theoretical prediction of the convergence of this method. [ABSTRACT FROM AUTHOR]

The new fractional derivative, which has no singular kernel, was recently introduced by Caputo and Fabrizio. In this paper, we consider the nonlinear KdV equation with the new fractional derivative and we study the existence of a solution by using H-stability of an iteration procedure. Numerical simulations are also reported to show H-stability of presented algorithm. [ABSTRACT FROM AUTHOR]

The key purpose of the present work is to examine a fractional giving up smoking model pertaining to a new fractional derivative with non-singular kernel. The numerical simulations are conducted with the aid of an iterative technique. The existence of the solution is discussed by employing the fixed point postulate, and the uniqueness of the solution is also proved. The effect of various parameters is shown graphically. The numerical results for the smoking model associated with the new fractional derivative are compared with numerical results for a smoking model pertaining to the standard derivative and Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

We present a new method to investigate some fractional integro-differential equations involving the Caputo-Fabrizio derivation and we prove the existence of approximate solutions for these problems. We provide three examples to illustrate our main results. By checking those, one gets the possibility of using some discontinuous mappings as coefficients in the fractional integro-differential equations. [ABSTRACT FROM AUTHOR]

By using the fractional Caputo-Fabrizio derivative, we introduce two types new high order derivations called CFD and DCF. Also, we study the existence of solutions for two such type high order fractional integro-differential equations. We illustrate our results by providing two examples. [ABSTRACT FROM AUTHOR]