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.

Pfaff-Birkhoff Variational Problem and Noether Symmetry in Terms of RiemannLiouville Fractional Derivatives of Variable Order

The Pfaff-Birkhoff variational problem and its Noether symmetry are studied in terms of Riemann-Liouville fractional derivatives of variable order. Based on the combination of variational principle and fractional calculus of variable order,the Pfaff-Birkhoff variational principle with Riemann-Liouville fractional derivatives of variable order is proposed, and the fractional Birkhoff's equations of variable order are derived. Then,the Noether 's theorem for the fractional Birkhoffian system of variable order is given. At last,an example is expressed to illustrate the application of the results.

Fractional Order Proportional Integral Derivative Controller Design and Simulation for Bioengineering Systems

This paper deals with the study of fractional order system tuning method based on Factional Order Proportional Integral Derivative( FOPID) controller in allusion to the nonlinear characteristics and fractional order mathematical model of bioengineering systems. The main contents include the design of FOPID controller and the simulation for bioengineering systems. The simulation results show that the tuning method of fractional order system based on the FOPID controller outperforms the fractional order system based on Fractional Order Proportional Integral( FOPI) controller. As it can enhance control character and improve the robustness of the system.

Fractal Dimension of Graph of Weierstrass Function and Its Derivative of the Fractional Order

In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.

Viscoelastic Rod Vibration Problem when Constitutive Relationship Contains a Fractional Derivative and its Limiting Case

The equation of motion of a viscoelastic rod of density rod p is considered when the Constitutive Relationships contains fractional derivatives （in the Riemann-Liouville sense） of order β, 0 ≤ β ≤ 1. The solution of this equation is given in a general case. It is shown that for the limit values of the derivative index ,β, i.e. when β = 0 or β = 1, the general solution gives rise to classical solutions of hyperbolic and parabolic equations.

Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models

Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena.In this paper,the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically.Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow.The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed.Based on the L1-approximation formula of Caputo fractional derivatives,the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented,and the numerical solutions are derived.In order to test the correctness and availability of numerical schemes,two numerical examples are established to give the exact solutions.The comparisons between the numerical solutions and the exact solutions have been made,and their high consistency indicates that the present numerical methods are effective.Then,this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions,and discusses the effects of the weight coefficient(α)in distributed order time fractional derivatives,the orderα(r,t)in variable fractional order derivatives,the relaxation timeλ,and the frequencyωof the periodic pressure gradient on the fluid flow velocity.Finally,the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

With the increasingly deep studies in physics and technology, the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research. In this paper, the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively inves- tigated. With the stability criterion of linear fractional systems, the synchronization of a fractional non-autonomous system is obtained. Specifically, an effective singly active control is proposed and used to synchronize a fractional order Duffing system. The nu- merical results demonstrate the effectiveness of the proposed methods.

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

Fractional-order visco-plastic constitutive model for uniaxial ratcheting behaviors

This paper proposes a novel unified visco-plastic constitutive model for uniaxial ratcheting behaviors. The cyclic deformation of the material presents remarkable time-dependence and history memory phenomena. The fractional(fractional-order)derivative is an efficient tool for modeling these phenomena. Therefore, we develop a cyclic fractional-order unified visco-plastic(FVP) constitutive model. Specifically, within the framework of the cyclic elasto-plastic theory, the fractional derivative is used to describe the accumulated plastic strain rate and nonlinear kinematic hardening rule based on the Ohno-Abdel-Karim model. Moreover, a new radial return method for the back stress is developed to describe the unclosed hysteresis loops of the stress-strain properly.The capacity of the FVP model used to predict the cyclic deformation of the SS304 stainless steel is verified through a comparison with the corresponding experimental data found in the literature(KANG, G. Z., KAN, Q. H., ZHANG, J., and SUN, Y. F. Timedependent ratcheting experiments of SS304 stainless steel. International Journal of Plasticity, 22(5), 858–894(2006)). The FVP model is shown to be successful in predicting the rate-dependent ratcheting behaviors of the SS304 stainless steel.

Controllability of Fractional Order Stochastic Differential Inclusions with Fractional Brownian Motion in Finite Dimensional Space

In this paper,sufficient conditions are formulated for controllability of fractional order stochastic differential inclusions with fractional Brownian motion(f Bm) via fixed point theorems,namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler fixed point theorem for the nonconvex case.The controllability Grammian matrix is defined by using Mittag-Leffler matrix function.Finally,a numerical example is presented to illustrate the efficiency of the obtained theoretical results.

Counterexample of Local Fractional Order Chain Rule and Modified Definition of Local Fractional Order

Fractional calculus is a powerful tool for modeling nonlinear systems.It is necessary to discuss the basic properties of fractional order before solving a fractional order model.Using the formula of power function defined by local fractional derivative and the chain rule to calculate a compound function,the results are inconsistent.This shows that the chain rule of local fractional derivatives similar to classical calculus is suspicious,and fractional complex transformation based on the chain rule is also suspicious and needs further discussion.In order to overcome this inconsistency,an improved definition of local fractional derivative,which can be regarded as a fractal derivative,is proposed based on the results derived from the relationship between the mass function and the Hausdorff measure.

Higher-order approximate solutions of fractional stochastic point kinetics equations in nuclear reactor dynamics

Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.

Attempt to generalize fractional-order electric elements to complex-order ones

The complex derivative D^α±jβ, with α, β ∈ R＋ is a generalization of the concept of integer derivative, where α = 1,β = 0. Fractional-order electric elements and circuits are becoming more and more attractive. In this paper, the complexorder electric elements concept is proposed for the first time, and the complex-order elements are modeled and analyzed.Some interesting phenomena are found that the real part of the order affects the phase of output signal, and the imaginary part affects the amplitude for both the complex-order capacitor and complex-order memristor. More interesting is that the complex-order capacitor can do well at the time of fitting electrochemistry impedance spectra. The complex-order memristor is also analyzed. The area inside the hysteresis loops increases with the increasing of the imaginary part of the order and decreases with the increasing of the real part. Some complex case of complex-order memristors hysteresis loops are analyzed at last, whose loop has touching points beyond the origin of the coordinate system.

A specific state variable for a class of 3D continuous fractional-order chaotic systems

A specific state variable in a class of 3D continuous fractional-order chaotic systems is presented. All state variables of fractional-order chaotic systems of this class can be obtained via a specific state variable and its （q-order and 2q-order） time derivatives. This idea is demonstrated by using several well-known fractional-order chaotic systems. Finally, a synchronization scheme is investigated for this fractional-order chaotic system via a specific state variable and its （q-order and 2q-order） time derivatives. Some examples are used to illustrate the effectiveness of the proposed synchronization method.

Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

Fractional order distance regularized level set method with bias correction

The existing level set segmentation methods have drawbacks such as poor convergence,poor noise resistance,and long iteration times.In this paper,a fractional order distance regularized level set segmentation method with bias correction is proposed.This method firstly introduces fractional order distance regularized term to punish the deviation between the level set function(LSF)and the signed distance function.Secondly a series of covering template is constructed to calculate fractional derivative and its conjugate of image pixel.Thirdly introducing the offset correction term and fully using the local clustering property of image intensity,the local clustering criterion of image intensity is defined and integrated with the neighborhood center to obtain the global criterion of image segmentation.Finally,the fractional distance regularization,offset correction,and external energy constraints are combined,and the energy optimization segmentation method for noisy image is established by level set.Experimental results show that the proposed method can accurately segment the image,and effectively improve the efficiency and robustness of exiting state of the art level set related algorithms.

CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.

Existence of positive solutions for integral boundary value problem of fractional differential equations

In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.