Nonlinear Fractional Derivative Research Articles

On a localization-in-frequency approach for a class of elliptic problems with singular boundary data

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

Nonlinear Vibrations of an Axially Moving Beam with Fractional Viscoelastic Damping

The nonlinear vibrations of an axially moving viscoelastic beam under the transverse harmonic excitation are examined. The governing equation of motion of the viscoelastic beam is discretized into a Duffing system with nonlinear fractional derivative using Galerkin’s method. The viscoelasticity of the moving beam is described by the fractional Kelvin–Voigt model based on the Caputo definition. The primary resonance is analytically investigated by the averaging method. With the aid of response curves, a parametric study is conducted to display the influences of the fractional order and the viscosity coefficient on steady-state responses. The validations of this study are given through comparisons between the analytical solutions and numerical ones, where the stability of the solutions is determined by the Routh–Hurwitz criterion. It is found that suppression of undesirable responses can be achieved via changing the viscosity of the system.

NONLINEAR BEHAVIOR OF CLAY CREEP AND ITS FRACTIONAL DERIVATIVE CREEP MODEL

Creep test is performed on Chengdu clay to study the nonlinear property of clay creep. It is found that instantaneous elastic deformation, attenuated creep deformation, steady creep deformation, and accelerated creep deformation are included in clay deformation. The long-term elastic modulus of clay is nonlinear softening with the increase of time and stress. The viscosity coefficient is nonlinear softening with the increase of stress and nonlinear hardening with the increase of time. Based on the rheology theory, fractional calculus theory and Harris attenuation function, fractional derivative components, nonlinear elastic components and nonlinear viscous components are established, respectively. A nonlinear fractional derivative creep model with simple form, few parameters and clear concept is established. Then, the nonlinear fractional derivative creep model and Burgers creep model are compared. It is found that the fitting effect of nonlinear fractional derivative creep model is better in each stage, and can give more reasonable description of the nonlinear creep of clay, and can accurately reflect the whole process of clay creep. The scientific rationality of the nonlinear fractional derivative creep model is proved.

Identification of Nonlinear Fractional Derivative Loudspeaker Model

Identification of Nonlinear Fractional Derivative Loudspeaker Model

Anisotropy Characteristics of Stress Relaxation in Coal: An Improved Fractional Derivative Constitutive Model

To study the anisotropy characteristics of coal during stress relaxation, a series of uniaxial compressive stress relaxation experiments are conducted on samples cut perpendicular and parallel to bedding. Some conclusions are as follows: (1) the stress of the specimens undergoes rapid relaxation, transient relaxation and steady relaxation, approximately represented with a negative exponential law, before reaching the final strain. The stress decreases discontinuously at the final strain. (2) Anisotropy is observable from stress relaxation tests. The failure strengths and elastic moduli of the bedding-perpendicular samples are generally greater than those of the bedding-parallel samples due to the hardening effect. At the same strain and before the final strain, the relaxation stress of the bedding-perpendicular samples is also greater than that of the bedding-parallel samples. At the final strain, for the bedding-parallel samples, the relaxation stress increases more evidently with higher initial stress relaxation rate, and the residual stress ratio decrease more remarkably. (3) An improved nonlinear fractional derivative generalized Kelvin model is proposed by replacing the Newton dashpot by the Abel dashpot, and the parameters in the model reflect the sample anisotropy well. This new model is used to fit the stress relaxation data, and the results show an improved accuracy in depicting the nonlinear stress relaxation characteristics of raw lean coal in relation to anisotropy over that of other models.

A Numerical Method for Calculating Fractional Derivative

A numerical method is proposed for calculating the fractional order derivative and successfully resolving the integrand singularity problem based on Zhang-Shimizu algorithm. And then a method is developed to calculate the twice nonlinear fractional derivative, numerical examples demonstrate the numerical method with high precision and good stability.

Fractional Homotopy Perturbation Transform Method for Solving the Time-Fractional KdV , K ( 2 , 2 ) and Burgers Equations

In this paper, the fractional homotopy perturbation transform method (FHPTM) is employed to obtain approximate analytical solutions of the time-fractional KdV, K(2,2) and Burgers equations. The FHPTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHPTM is an appropriate method for solving nonlinear fractional derivative equation.

Bifurcation of a Duffing Oscillator Having Nonlinear Fractional Derivative Feedback

Active feedback control is commonly used to attenuate undesired vibrations in vibrating machineries and structures, such as bridges, highways and aircrafts. In this paper, we investigate the primary resonance and 1/3 subharmonic resonance of a harmonically forced Duffing oscillator under fractional nonlinear feedback control. By means of the first order averaging method, slow flow equations governing the modulations of amplitude and phase of the oscillator are obtained. An approximate solution for the steady state periodic response is derived and its stability is determined by the Routh–Hurwitz criterion. We demonstrate that appropriate choices on the control strategies and feedback gains can delay or eliminate the undesired bifurcations and reduce the amplitude peak both of the primary and subharmonic resonances. Analytical results are verified by comparisons with the numerical integration results.

Individualization of a pharmacokinetic model by fractional and nonlinear fit improvement

This study presents application of a new linear and nonlinear fractional derivative two compartmental model to the evaluation of individual pharmacokinetics. In the model, the integer order derivatives are replaced by derivatives of real order. A specific nonlinear function is used for the fit improvement of a fractional derivative two compartmental model with the mass balance conservation. The agreement of the values predicted by the proposed model with the values obtained through experiments with bumetanide tablets in human volunteers is shown to be good. Thus, pharmacokinetics of bumetanide can be described well by a linear or a nonlinear two compartmental model with fractional derivatives of the same order proposed here. Parameters in the model are determined by the least squares method and the particle swarm optimization numerical procedure is used. The results show that the linear fractional order two compartmental model for bumetanide is useful improvement of the classical (integer order) two compartmental model and that the nonlinear fractional order model is useful improvement of the linear model in a great number of volunteers.

Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law

Two types of models are proposed for describing nonlinear fractional derivative dynamical behavior of viscoelastic materials subject to impulse forces. The models are derived based on the thermodynamic elasticity in terms of entropy and on the “scale-free response of the material” under the basic assumption that the viscoelastic materials consist of stable coils of polymers, which we refer to as blobs. The blobs, which may be connected to each other by chemical bonds or physical bonds, are considered here as the elementary constituent of viscoelastic materials from which the nonlinear fractional derivative models are derived. Responses of individual blobs can determine the net collective response of the viscoelastic material to impulse forces. From the above consideration, two types of models are proposed in which the force elements or the stress elements are connected by the generalized Maxwell law.

Modeling of elastomeric materials using nonlinear fractional derivative and continuously yielding friction elements

Modeling of elastomeric materials using nonlinear fractional derivative and continuously yielding friction elements

Modeling of elastomeric materials using nonlinear fractional derivative and continuously yielding friction elements

A model of the one-dimensional dynamic behavior of elastomeric materials is developed based on a previously existing model. An initial model consisted of nonlinear multiple anelastic displacement fields in parallel with discrete friction elements. The motivation for the development of a new model is reduction of the number of parameters needed to accurately capture material behavior. A new element, a “continuously yielding element,” is developed, which conceptually represents a distribution of parallel friction elements. This element replaces the entire collection of discrete friction elements used in the initial model. In addition, a linear fractional derivative anelastic displacement field element replaces the multiple linear anelastic displacement field elements used in the older model. Finally, nonlinearity is introduced into the fractional derivative anelastic displacement field element, in an attempt to capture observed amplitude dependence of storage and loss moduli at higher amplitudes. The different parts of the new model are first compared individually to those of the initial model then combined and compared as an integrated whole. The new model captures the frequency and amplitude variation of the storage and loss moduli of the material better than the initial model, while the total number of parameters is reduced to seven from sixteen.

Anomalous diffusion associated with nonlinear fractional derivative fokker-planck-like equation: exact time-dependent solutions

We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives ( partial differential/ partial differentialt)P(x,t)=D( partial differential(gamma)/ partial differentialx(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2-gamma)/(1+gamma)(-infinity<gamma</=2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q=(gamma+3)/(gamma+1)(0<gamma</=2), with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the one already known for Levy-like superdiffusion (i.e., nu=1 and 0<gamma</=2). Finally, for (gamma,nu)=(2,0) we obtain q=5/3, which differs from the value q=2 corresponding to the gamma=2 solutions available in the literature (nu<1 porous medium equation), thus exhibiting nonuniform convergence.