Efficient solution of a vibration equation involving fractional derivatives

Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980s, and they grow more and more popular nowadays. However, their efficient numerical calculation is non-trivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included.     

Relaxation and retardation functions of the Maxwell model with fractional derivatives

A four-parameter Maxwell model is formulated with fractional derivatives of different orders of the stress and strain using the Riemann-Liouville definition. This model is used to determine the relaxation and retardation functions. The relaxation function was found in the time domain with the help of a power law series; a direct solution was used in the Laplace domain. The solution can be presented as a product of a power law term and the Mittag-Leffler function. The retardation function is determined via Laplace transformation and is solely a power law type.The investigation of the relaxation function shows that it is strongly monotonic. This explains why the model with fractional derivatives is consistent with thermodynamic principles.This type of rheological constitutive equation shows fluid behavior only in the case of a fractional derivative of the stress and a first order derivative of the strain. In all other cases the viscosity does not reach a stationary value.In a comparison with other relaxation functions like the exponential function or the Kohlrausch-Williams-Watts function, the investigated model has no terminal relaxation time. The time parameter of the fractional Maxwell model is determined by the intersection point of the short- and long-rime asymptotes of the relaxation function.     

Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multi- term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ- ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.     

Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional difusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial diferential equations are converted into time-fractional ordinary diferential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-difusion problems are given to validate the proposed analytical method.     

First passage of stochastic fractional derivative systems with power-form restoring force

In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge weakly to an Itô differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.     

Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method

The smooth and discontinuous oscillator with fractional derivative damping under combined harmonic and random excitations is investigated in this paper. The short memory principle is introduced so that the evolution process of the oscillator with fractional derivative damping can be described by the Markov chain. Then the stochastic generalized cell mapping method is used to obtain the steady-state probability density functions of the response. The stochastic response and bifurcation of the oscillator with fractional derivative damping are discussed in detail. We found that both the smoothness parameter, the noise intensity, the amplitude and frequency of the harmonic force can induce the occurrence of stochastic P-bifurcation in the system. Monte Carlo simulation verifies the effectiveness of the method we adopt in the paper.     

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 rheologically unreasonable with the help of the mechanical analogues of the fractional element model. We also find that to make them more reasonable rheologically, 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.     

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann–Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.Dedicated to professor Hari M. Srivastava on the occasion of his 65th birthday     

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.     

Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances

Non-linear free damped vibrations of a rectangular plate described by three non-linear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order smaller than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales, but the fractional derivative is represented as a fractional power of the differentiation operator. It is assumed that the order of the damping coefficient depends on the character of the vibratory process and takes on the magnitude of the amplitudes’ order. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.     

A numerical method for fractional integral with applications

IntroductionThefractionalcalculushasalonghistoryandthereareamassofworkstodiscussthefractionalderivativesandfractionalintegralswitharbitrary (realorcomplex)order[1- 3 ].Thefractionalcalculushasawideapplicationbackground ,especiallyinthefieldsofchemistry ,electromagnetics,materialscienceandmechanics.Forexample,Gement[4 ]proposedthefractionalderivativeconstitutivemodelsofaviscoelasticmaterialatfirst.Themodelshavereceivedincreasingattention[5 - 7].Onlyafewparametersarecontainedinthemodelsandthemo…     

A fractional calculus interpretation of the fractional volatility model

Based on criteria of mathematical simplicity and consistency with empirical market data, a model with volatility driven by fractional noise has been constructed which provides a fairly accurate mathematical parametrization of the data. Here, the model is formulated in terms of a fractional integration of stochastic processes.     

Enhance limit cycle oscillation in a wind flow energy harvester system with fractional order derivatives

Advances in material science and mathematics in conjunction with technological needs have triggered the use of material and electric components with fractional order physical properties. This paper considers the mathematical model of a piezoelectric wind flow energy harvester system for which the capacitance of the piezoelectric material has fractional order current-voltage characteristics. Additionally the mechanical element is assumed to have fractional order damping. The analysis is focused on the effects of order of derivatives on the appearance and characteristics of limit circle oscillations (LCO). It is obtained that, the order of derivatives to enhance the amplitude of LCO and lower the threshold condition leading to LCO. The domains of efficiency of the system are illustrated in various parameters spaces.     

Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations

The stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping of order α (0<α<1) under combined harmonic and white noise excitations are studied. First, the system state is approximately represented by two-dimensional time-homogeneous diffusive Markov process of amplitude and phase difference using the stochastic averaging method. Then, the method of reduced Fokker–Plank–Kolmogorov (FPK) equation is used to predict the stationary response of the original system. The phenomenon of stochastic jump and bifurcation as the fractional orders' change is examined.     

Lattice with long-range interaction of power-law type for fractional non-local elasticity

Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case.     

Effect of the initial ramps of creep and relaxation tests on models with fractional derivatives

We analyze the influences of the initial ramps of relaxation and creep tests on the determination of the parameters of the Scott-Blair’s model, the simplest model with a fractional derivative. The true input of a test is a constant with an initial ramp. The error resulted from neglecting the initial ramp increases with the increasing of the fractional order, and the closer the data are to the beginning, the larger the errors are. Based on the error analysis, an n-times rule is generalized from the ten-times rule. For illustration, the Scott-Blair’s model is fitted with the experimental data in literature. The data are of two polymer samples, PMMA and PTFE (methylmethacrylate and polytetrafluorethylene), and self-healing and thermoreversible rubbers.     

Study on the constitutive equation with fractional derivative for the viscoelastic fluids – Modified Jeffreys model and its application

Based on the classic linear viscoelastic Jeffreys model, a modified Jeffreys model is suggested. The corresponding five-parameter equation with fractional derivatives of different orders of the stress and rate of strain is stated and the characteristic material functions of the linear viscoelasticity theory, such as the dynamic moduli, are derived. The comparison between the measured dynamic moduli of Sesbania gel and xanthan gum and the theoretical predictions of the proposed phenomenological model shows an excellent agreement. Received: 26 August 1997 Accepted: 26 May 1998     

Harmonic wavelets based response evolutionary power spectrum determination of linear and non-linear oscillators with fractional derivative elements

A harmonic wavelets based approximate analytical technique for determining the response evolutionary power spectrum of linear and non-linear (time-variant) oscillators endowed with fractional derivative elements is developed. Specifically, time- and frequency-dependent harmonic wavelets based frequency response functions are defined based on the localization properties of harmonic wavelets. This leads to a closed form harmonic wavelets based excitation-response relationship which can be viewed as a natural generalization of the celebrated Wiener–Khinchin spectral relationship of the linear stationary random vibration theory to account for fully non-stationary in time and frequency stochastic processes. Further, relying on the orthogonality properties of harmonic wavelets an extension via statistical linearization of the excitation-response relationship for the case of non-linear systems is developed. This involves the novel concept of determining optimal equivalent linear elements which are both time- and frequency-dependent. Several linear and non-linear oscillators with fractional derivative elements are studied as numerical examples. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the technique.     

Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

Many constitutive models exist to characterise the cyclic behaviour of granular soils but can only simulate deformations for very limited cycles. Fractional derivatives have been regarded as one potential instrument for modelling memory-dependent phenomena. In this paper, the physical connection between the fractional derivative order and the fractal dimension of granular soils is investigated in detail.Then a modified elasto-plastic constitutive model is proposed for evaluating the long-term deformation of granular soils under cyclic loading by incorporating the concept of factional calculus. To describe the flow direction of granular soils under cyclic loading, a cyclic flow potential considering particle breakage is used. Test results of several types of granular soils are used to validate the model performance.     

On the physical interpretation of fractional diffusion

Even if the diffusion equation has been widely used in physics and engineering, and its physical content is well understood, some variants of it escape fully physical understanding. In particular, anormal diffusion appears in the so-called fractional diffusion equation, whose main particularity is its non-local behavior, whose physical interpretation represents the main part of the present work.