Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.

A Hybrid ESA-CCD Method for Variable-Order Time-Fractional Diffusion Equations

In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.

Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

In this paper,we propose a fast second-order approximation to the variable-order(VO)Caputo fractional derivative,which is developed based on L2-1σformula and the exponential-sum-approximation technique.The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative.This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme(F L2-1σscheme)for the multi-dimensional VO time-fractional sub-diffusion equations.Theoretically,F L2-1σscheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1σscheme.Therefore,F L2-1σscheme is strictly proved to be unconditionally stable and convergent.A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

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 themean of an ordinary and first-order differential equation.More precisely,it is solved through an initial value problem.We proposed a newmodeling 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 theUlam-Hyers(UH)technique was also investigated.The application of the proposedmodel on two real data sets revealed that the Caputo derivative wasmore 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.

Three Kinds of Discrete Formulae for the Caputo Fractional Derivative

In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The truncation error estimates and the properties of the coeffcients of all these discretisations are analysed in more detail.Finally,the theoretical analyses areverifiedby thenumerical examples.

Fractional cyclic integrals and Routh equations of fractional Lagrange system with combined Caputo derivatives

In this paper, we develop a fractional cyclic integral and a Routh equation for fractional Lagrange system defined in terms of fractional Caputo derivatives. The fractional Hamilton principle and the fractional Lagrange equations of the system are obtained under a combined Caputo derivative. Furthermore, the fractional cyclic integrals based on the Lagrange equations are studied and the associated Routh equations of the system are presented. Finally, two examples are given to show the applications of the results.

A High Order Formula to Approximate the Caputo Fractional Derivative

We present here a high-order numerical formula for approximating the Caputo fractional derivative of order𝛼for 0<α<1.This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diff erential equations.In comparison with the previous formulae,the main superiority of the new formula is its order of accuracy which is 4−α,while the order of accuracy of the previous ones is less than 3.It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost.The eff ectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples.Moreover,an application of the new formula in solving some fractional partial diff erential equations is presented by constructing a fi nite diff erence scheme.A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

Asymptotic Analysis of Linear and Interval Linear Fractional-Order Neutral Delay Differential Systems Described by the Caputo-Fabrizio Derivative

Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel characteristic equation is derived. Stability criteria are established based on an algebraic approach and norm-based criteria are also presented. It is shown that asymptotic stability is ensured for linear fractional-order neutral delay differential systems provided that the underlying stability criterion holds for any delay parameter. In addition, sufficient conditions are derived to ensure the asymptotic stability of interval linear fractional order neutral delay differential systems. Examples are provided to illustrate the effectiveness and applicability of the theoretical results.

Local Discontinuous Galerkin Method for the Time-Fractional KdV Equation with the Caputo-Fabrizio Fractional Derivative

This paper studies the time-fractional Korteweg-de Vries (KdV) equations with Caputo-Fabrizio fractional derivatives. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. Stability and convergence are demonstrated by a specific choice of numerical fluxes. Finally, the efficiency and accuracy of the scheme are verified by numerical experiments.

A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients

The goal of this research is to develop a new,simplified analytical method known as the ARA-residue power series method for obtaining exact-approximate solutions employing Caputo type fractional partial differential equations(PDEs)with variable coefficient.ARA-transform is a robust and highly flexible generalization that unifies several existing transforms.The key concept behind this method is to create approximate series outcomes by implementing the ARA-transform and Taylor’s expansion.The process of finding approximations for dynamical fractional-order PDEs is challenging,but the ARA-residual power series technique magnifies this challenge by articulating the solution in a series pattern and then determining the series coefficients by employing the residual component and the limit at infinity concepts.This approach is effective and useful for solving a massive class of fractional-order PDEs.Five appealing implementations are taken into consideration to demonstrate the effectiveness of the projected technique in creating solitary series findings for the governing equations with variable coefficients.Additionally,several visualizations are drawn for different fractional-order values.Besides that,the estimated findings by the proposed technique are in close agreement with the exact outcomes.Finally,statistical analyses further validate the efficacy,dependability and steady interconnectivity of the suggested ARA-residue power series approach.

On the partial stability of nonlinear impulsive Caputo fractional systems

In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.

Conformable and Caputo’s Derivatives in Generalized Viscoelastic Models

We study two generalized versions of a system of equations which describe the time evolution of the hydrodynamic fluctuations of density and velocity in a linear viscoelastic fluid. In the first of these versions, the time derivatives are replaced by conformable derivatives, and in the second version left-handed Caputo’s derivatives are used. We show that the solutions obtained with these two types of derivatives exhibit significant similarities, which is an interesting (and somewhat surprising) result, taking into account that the conformable derivatives are local operators, while Caputo’s derivatives are nonlocal operators. We also show that the solutions of the generalized systems are similar to the solutions of the original system, if the order α of the new derivatives (conformable or Caputo) is less than one. On the other hand, when α is greater than one, the solutions of the generalized systems are qualitatively different from the solutions of the original system.

A note on the definition of fractional derivatives applied in rheology

It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives-Riemann-Liouville （R-L） definition and Caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R-L definition and Caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model （Scott-Blair model）. We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R-L definition with lower terminal a →∞ and the Caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal a →∞ （or, equivalently, the Caputo derivatives with lower terminal a →∞） not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott-Blair model and give an exact solution of this equation under given initial conditions.

Fractional derivative multivariable grey model for nonstationary sequence and its application

Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC(q, N) is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization(PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC(q, N) model is used to predict the municipal solid waste(MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model.

On the definition of fractional derivatives in rheology

During the last two decades fractional calculus has been increasingly applied to physics, especially to rheology.It is well known that there are obivious differences between Riemann-Liouville (R-L) definition and Caputo definition,which are the two most commonly used definitions of fractional derivatives.The multiple definitions of fractional derivatives have hindered the application of fractional calculus in rheology.In this paper,we clarify that the R-L definition and Caputo definition are both Theologically unreasonable with the help of the mechanical analogues of the fractional element model.We also find that to make them more reasonable Theologically,the lower terminals of both definitions should be put to—∞.We further prove that the R-L definition with lower terminal—∞and the Caputo definition with lower terminal—∞are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points.Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal—∞(or,equivalently,the Caputo derivatives with lower terminal—∞) not only for steady-state processes,but also for transient processes.

ON A COUPLED INTEGRO-DIFFERENTIAL SYSTEM INVOLVING MIXED FRACTIONAL DERIVATIVES AND INTEGRALS OF DIFFERENT ORDERS

By applying the standard fixed point theorems,we prove the existence and uniqueness results for a system of coupled differential equations involving both left Caputo and right Riemann-Liouville fractional derivatives and mixed fractional integrals,supplemented with nonlocal coupled fractional integral boundary conditions.An example is also constructed for the illustration of the obtained results.

Fuzzy computational study on the generalized fractional smoking model with caputo gH-type derivatives

This work considers a generalized fuzzy fractional smoking model with Caputo gHtypes fractional derivatives upon considering the case of uncertainty quantification.The disease-free equilibrium point and stability of the equilibrium point have been discussed for the fuzzy nonlinear fractional smoking model.The analytical proofs for the existence and uniqueness of the proposed model are concerned with the help of the fixed-point theorem,Banach contraction,and Schauder theorem.A robust double parametric approach with a generalized transform is used to study the behavior of the fuzzy fractional model in an uncertain context and obtain the convergence analysis of the study in a crisp context.Finally,the obtained results of the proposed model have been validated with the Runge-Kutta method of fourth order in crisp case(s=1,l=O).

A NONLOCAL HYBRID BOUNDARY VALUE PROBLEM OF CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, we discuss the existence of solutions for a nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations. Our main result is based on a hybrid fixed point theorem for a sum of three operators due to Dhage, and is well illustrated with the aid of an example.

Analysis of anomalous transport with temporal fractional transport equations in a bounded domain

Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations.The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α(α ∈(0, 1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result,if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.

Fractional Order Modeling of Predicting COVID-19 with Isolation and Vaccination Strategies in Morocco

In this work,we present a model that uses the fractional order Caputo derivative for the novel Coronavirus disease 2019(COVID-19)with different hospitalization strategies for severe and mild cases and incorporate an awareness program.We generalize the SEIR model of the spread of COVID-19 with a private focus on the transmissibility of people who are aware of the disease and follow preventative health measures and people who are ignorant of the disease and do not follow preventive health measures.Moreover,individuals with severe,mild symptoms and asymptomatically infected are also considered.The basic reproduction number(R0)and local stability of the disease-free equilibrium(DFE)in terms of R0 are investigated.Also,the uniqueness and existence of the solution are studied.Numerical simulations are performed by using some real values of parameters.Furthermore,the immunization of a sample of aware susceptible individuals in the proposed model to forecast the effect of the vaccination is also considered.Also,an investigation of the effect of public awareness on transmission dynamics is one of our aim in this work.Finally,a prediction about the evolution of COVID-19 in 1000 days is given.For the qualitative theory of the existence of a solution,we use some tools of nonlinear analysis,including Lipschitz criteria.Also,for the numerical interpretation,we use the Adams-Moulton-Bashforth procedure.All the numerical results are presented graphically.