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.

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.

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.

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.

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.

Noether's theorems of a fractional Birkhoffian system within Riemann-Liouville derivatives

The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff＇s equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether＇s theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether＇s theorems are established. Finally, an example is given to illustrate the application of the results.

Variational Calculus With Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

Fractional Pfaff-Birkhoff Principle and Birkhoff′s Equations in Terms of Riesz Fractional Derivatives

The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s equations.The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics.Here,the fractional Pfaff-Birkhoff variational problem is presented and studied.The definitions of fractional derivatives,the formulae for integration by parts and some other preliminaries are firstly given.Secondly,the fractional Pfaff-Birkhoff principle and the fractional Birkhoff′s equations in terms of RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.

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.

Non-Noether symmetries of Hamiltonian systems with conformable fractional derivatives

In this paper, we present the fractional Hamilton＇s canonical equations and the fractional non-Noether symmetry of Hamilton systems by the conformable fractional derivative. First/y, the exchanging relationship between isochronous variation and fractional derivatives, and the fractional Hamilton principle of the system under this fractional derivative are proposed. Secondly, the fractional Hamilton＇s canonical equations of Hamilton systems based on the Hamilton principle are established. Thirdly, the fractional non-Noether symmetries, non-Noether theorem and non-Noether conserved quantities for the Hamilton systems with the conformable fractional derivatives are obtained. Finally, an example is given to illustrate 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.

Fractional Optical Solitons and Fractional Noether’s Theorem with Ortigueira’s Centered Derivatives

This paper shows that the centered fractional derivatives introduced by Manuel Duarte Ortigueira in 2006 are useful in the description of optical solitons. It is shown that we can construct a fractional extension of the nonlinear Schr&ouml;dinger (NLS) equation which incorporates Ortigueira’s derivatives and has soliton solutions. It is also shown that this fractional NLS equation has a Lagrangian density and can be derived from a variational principle. Finally, a fractional extension of Noether’s theorem is formulated to determine the conserved quantities associated to the invariances of the action integral under infinitesimal transformations.

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.

Fractional Noether's Theorems for El-Nabulsi's Fractional Birkhoffian Systems in Terms of Riemann-Liouville Derivatives

The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.

S-asymptotically ω-periodic Solutions of R-L Fractional Derivative-Integral Equation

The aim of this paper is to study the S-asymptotically ω-periodic solutions of R-L fractional derivative-integral equation:v′(t)=∫t0(t-s)α-2/Γ(α-1)Av(s)ds+∫+∞-∞e-|τ|f(u(t-τ))dτ,(1)v(0)=u0∈X,(2)where 1 <α <2, A:D(A)X→X is a linear densely defined operator of sectorial type on a completed Banach space X, f is a continuous function satisfying a suitable Lipschitz type condition. We will use the contraction mapping theory to prove problem(1) and(2) has a unique S-asymptoticallyω-periodic solution if the function f satisfies Lipshcitz condition.