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1.
Małgorzata Klimek 《Czechoslovak Journal of Physics》2001,51(12):1348-1354
The symmetric fractional derivative is introduced and its properties are studied. The Euler-Lagrange equations for models
depending on sequential derivatives of type are derived using minimal action principle. The Hamiltonian for such systems is
introduced following methods of classical generalized mechanics and the Hamilton’s equations are obtained. It is explicitly
shown that models of fractional sequential mechanics are non-conservative. The limiting procedure recovers classical generalized
mechanics of systems depending on higher order derivatives. The method is applied to fractional deformation of harmonic oscillator
and to the case of classical frictional force proportional to velocity.
Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June
2001. 相似文献
2.
This work presents a lossy partial differential acoustic wave equation including fractional derivative terms. It is derived from first principles of physics (mass and momentum conservation) and an equation of state given by the fractional Zener stress-strain constitutive relation. For a derivative order α in the fractional Zener relation, the resulting absorption α(k) obeys frequency power-laws as α(k) ∝ ω(1+α) in a low-frequency regime, α(k) ∝ ω(1-α/2) in an intermediate-frequency regime, and α(k) ∝ ω(1-α) in a high-frequency regime. The value α=1 corresponds to the case of a single relaxation process. The wave equation is causal for all frequencies. In addition the sound speed does not diverge as the frequency approaches infinity. This is an improvement over a previously published wave equation building on the fractional Kelvin-Voigt constitutive relation. 相似文献
3.
Dumitru Baleanu Ivo Petras Jihad H. Asad Maria Pilar Velasco 《International Journal of Theoretical Physics》2012,51(4):1253-1258
In this paper we study the fractional Lagrangian of Pais–Uhlenbeck oscillator. We obtained the fractional Euler–Lagrangian
equation of the system and then we studied the obtained Euler–Lagrangian equation numerically. The numerical study is based
on the so-called Grünwald–Letnikov approach, which is power series expansion of the generating function (backward and forward
difference) and it can be easy derived from the Grünwald–Letnikov definition of the fractional derivative. This approach is
based on the fact, that Riemman–Liouville fractional derivative is equivalent to the Grünwald–Letnikov derivative for a wide
class of the functions. 相似文献
4.
I. A. Lubashevskii A. A. Zemlyanov 《Journal of Experimental and Theoretical Physics》1998,87(4):700-713
Anomalous diffusion on a comb structure consisting of a one-dimensional backbone and lateral branches (teeth) of random length
is considered. A well-defined classification of the trajectories of random walks reduces the original problem to an analysis
of classical diffusion on the backbone, where, however, the time of this process is a random quantity. Its distribution is
dictated by the properties of the random walks of the diffusing particles on the teeth. The feasibility of applying mean-field
theory in such a model is demonstrated, and the equation for the Green’s function with a partial derivative of fractional
order is obtained. The characteristic features of the propagation of particles on a comb structure are analyzed. We obtain
a model of an effective homogeneous medium in which diffusion is described by an equation with a fractional derivative with
respect to time and an initial condition that is an integral of fractional order.
Zh. éksp. Teor. Fiz. 114, 1284–1312 (October 1998) 相似文献
5.
Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases
V. S. Kirchanov 《Russian Physics Journal》2009,52(4):343-353
Two model examples of the application of fractional calculus are considered. The Riemann–Liouville fractional derivative with
0 < α ≤ 1 was used. The solution of a fractional equation, which describes anomalous relaxation and diffusion in an isotropic
fractal space, has been obtained in the form of the product of a Fox function by a Mittag-Leffler function. The solution is
simpler than that given in Ref. 6 and it generalizes the result reported in Ref. 7. For the quantum case, a solution of the generalized Neumann–Kolmogorov fractional quantum-statistical equation has been
obtained for an incomplete statistical operator which describes the random walk of a quantum spin particle, retarded in traps
over a fractal space. The solution contains contributions from quantum Mittag-Leffler (nonharmonic) fractional oscillations,
anomalous relaxation, noise fractional oscillations, and exponential fractional diffusion oscillation damping. 相似文献
6.
A. T. Kosilov V. A. Mikhailov V. V. Sviridov V. A. Khonik 《Physics of the Solid State》1997,39(11):1796-1802
A generalized theoretical model is proposed for the structural relaxation of metallic glasses under load. Structural relaxation
is treated as a set of irreversible, uncorrelated, two-stage atomic displacements in some regions of the structure, the “relaxation
centers.” In loaded samples structural relaxation acquires a directional character, leading to the buildup of plastic deformation
in accordance with the magnitude and orientation of the applied mechanical stress. General equations are obtained for creep
kinetics including a continuous statistical distribution of the principal activation parameters. These equations are compared
with the results of a special experiment. The model is found to provide an adequate interpretation of the observed creep kinetics,
except for the first 101–102 seconds after loading. It is argued that the initial stage of creep is determined by reversible atomic realignments in relaxation
centers having symmetric two-well potential.
Fiz. Tverd. Tela (St. Petersburg) 39, 2008–2015 (November 1997) 相似文献
7.
E. Capelas de Oliveira F. Mainardi J. VazJr. 《The European physical journal. Special topics》2011,193(1):161-171
We revisit the Mittag-Leffler functions of a real variable t, with one, two and three order-parameters {α,β,γ}, as far as their Laplace transform pairs and complete monotonicity properties
are concerned. These functions, subjected to the requirement to be completely monotone for t > 0, are shown to be suitable models for non–Debye relaxation phenomena in dielectrics including as particular cases the
classical models referred to as Cole–Cole, Davidson–Cole and Havriliak–Negami. We show 3D plots of the relaxations functions
and of the corresponding spectral distributions, keeping fixed one of the three order-parameters. 相似文献
8.
In this paper, numerical solutions of a reaction-diffusion chemotactic model of fractional orders for bacterial growth will
be present. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. We
compare the experimental result obtained with those obtained by simulation of the chemotactic model without fractional derivatives.
The results show that the solution continuously depends on the time-fractional derivative. The resulting solutions spread
faster than the classical solutions and may exhibit asymmetry, depending on the fractional derivative used. We present results
of numerical simulations to illustrate the method, and investigate properties of numerical solutions. The Adomian’s decomposition
method (ADM) is used to find the approximate solution of fractional ‘reaction-diffusion chemotactic model. Numerical results
show that the approach is easy to implement and accurate when applied to partial differential equations of fractional order. 相似文献
9.
Meral FC Royston TJ Magin RL 《The Journal of the Acoustical Society of America》2011,129(2):1036-1045
A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed. 相似文献
10.
Sergiu I. Vacaru 《International Journal of Theoretical Physics》2012,51(5):1338-1359
We study the fractional gravity for spacetimes with non-integer fractional derivatives. Our constructions are based on a formalism
with the fractional Caputo derivative and integral calculus adapted to nonholonomic distributions. This allows us to define
a fractional spacetime geometry with fundamental geometric/physical objects and a generalized tensor calculus all being similar
to respective integer dimension constructions. Such models of fractional gravity mimic the Einstein gravity theory and various
Lagrange–Finsler and Hamilton–Cartan generalizations in nonholonomic variables. The approach suggests a number of new implications
for gravity and matter field theories with singular, stochastic, kinetic, fractal, memory etc processes. We prove that the
fractional gravitational field equations can be integrated in very general forms following the anholonomic deformation method
for constructing exact solutions. Finally, we study some examples of fractional black hole solutions, ellipsoid gravitational
configurations and imbedding of such objects in solitonic backgrounds. 相似文献
11.
Yasemin ?. ?iftci Kemal ?olako?lu Cansu ?oban Engin Delig?z 《Central European Journal of Physics》2012,10(1):197-205
The structural, elastic and thermodynamic characteristics of CeGa2 compound in the AlB2 (space group: P6/mmm) and the omega trigonal (space group: P-3m1) type structures are investigated using the methods of density
functional theory within the generalized gradient approximation (GGA). The thermodynamic properties of the considered structures
are obtained through the quasi-harmonic Debye model. The results on the basic physical parameters, such as the lattice constant,
the bulk modulus, the pressure derivative of bulk modulus, the phase-transition pressure (P
t
) from P6/mmm to P-3m1 structure, the second-order elastic constants, Zener anisotropy factor, Poisson’s ratio, Young’s modulus,
and the isotropic shear modulus are presented. In order to gain further information, the pressure and temperature-dependent
behavior of the volume, the bulk modulus, the thermal expansion coefficient, the heat capacity, the entropy, Debye temperature
and Grüneisen parameter are also evaluated over a pressure range of 0–6 GPa and a wide temperature range of 0–1800 K. The
obtained results are in agreement with the available experimental and the other theoretical values. 相似文献
12.
Eqab M. Rabei Ibrahim M. Rawashdeh Sami Muslih Dumitru Baleanu 《International Journal of Theoretical Physics》2011,50(5):1569-1576
The paper presents fractional Hamilton–Jacobi formulations for systems containing Riesz fractional derivatives (RFD’s). The
Hamilton–Jacobi equations of motion are obtained. An illustrative example for simple harmonic oscillator (SHO) has been discussed.
It was observed that the classical results are recovered for integer order derivatives. 相似文献
13.
Małgorzata Klimek 《Czechoslovak Journal of Physics》2002,52(11):1247-1253
The models described by fractional order derivatives of Riemann-Liouville type in sequential form are discussed in Lagrangean
and Hamiltonian formalism. The Euler-Lagrange equations are derived using the minimum action principle. Then the methods of
generalized mechanics are applied to obtain the Hamilton’s equations. As an example free motion in fractional picture is studied.
The respective fractional differential equations are explicitly solved and it is shown that the limitα→1+ recovers classical model with linear trajectories and constant velocity.
Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. 相似文献
14.
采用分数阶黏弹单元替代经典模型中的黏壶, 结合非晶合金在外加载荷作用下的微观结构演化, 建立了以分数阶微积分表示的非晶合金黏弹性本构模型. 并根据Hertz弹性理论及分数阶黏弹性本构模型, 推导了块体非晶合金在纳米压痕球形压头下的位移与载荷及时间关系式. 基于推导的解析式, 对铁基块体非晶合金在表观弹性区的纳米压痕位移与载荷及时间曲线进行了非线性拟合分析. 相较于整数阶模型, 分数阶模型不仅具有较高的拟合精度, 其拟合参数能敏锐地反应加载速率对块体非晶合金黏弹性行为的影响, 且参数的变化规律与载荷作用下非晶合金微观结构演化呈现出较强的相关性. 相似文献
15.
A quantitative theory of creep in linearly heated metallic glasses is developed in terms of new ideas on the kinetics of irreversible
structural relaxation under external mechanical stress. The validity of the resulting flow equation has been confirmed by
a specially devised experiment. It is shown that the temperature dependence of Newtonian viscosity is determined by the rate
of heating and the energy spectrum of irreversible structural relaxation.
Fiz. Tverd. Tela (St. Petersburg) 39, 2186–2190 (December 1997) 相似文献
16.
The classical binomial process has been studied by Jakeman (J. Phys. A 23:2815–2825, 1990) (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular,
he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed
number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial
process. In this paper, the classical binomial process is generalized using the techniques of fractional calculus and is called
the fractional binomial process. The fractional binomial process is shown to preserve the binomial limit at large times while
expanding the class of models that include non-binomial fluctuations (non-Markovian) at regular and small times. As a direct
consequence, the generality of the fractional binomial model makes the proposed model more desirable than its classical counterpart
in describing real physical processes. More statistical properties are also derived. 相似文献
17.
Christian Cuadrado-Laborde 《Optical and Quantum Electronics》2008,40(13):983-990
In this work we present a technique for the implementation of an ultrafast all-optical temporal differentiator. A photonic
Mach–Zehnder interferometer can provide the required spectral response for fractional differentiation within certain fractional
order extent. This device shows a good accuracy calculating the fractional time derivatives of the complex field of an arbitrary
input optical waveform. Analytical expressions were found relating the photonic Mach–Zehnder parameters and the required spectral
characteristics of the differentiator for integer and fractional operation. The introduced concept is supported by numerical
simulations. 相似文献
18.
The main goal of this work is to perform a nonholonomic deformation (Fedosov type) quantization of fractional Lagrange–Finsler
geometries. The constructions are provided for a fractional almost K?hler model encoding equivalently all data for fractional
Euler–Lagrange equations with Caputo fractional derivative. 相似文献
19.
TONG Dengke & WANG Ruihe Department of Applied Mathematics Petroleum University Dongying China 《中国科学G辑(英文版)》2004,47(4):424-441
In both the oil reservoir engineering and seepage flow mechanics, heavy oil with relaxation property shows non-Newtonian rheological characteristics. The relationship between shear rate g& and shear stress t is nonlinear. Because of the relaxation phenomena of heavy oil flow in porous media, the equation of motion can be written as[1] 2,rrvpqkppqtrrtll秏骣+=-+琪抖桫 (1) where lv and lp are velocity relaxation and pressure retardation times. For most porous media, the above motion equation (1)… 相似文献
20.
A. Celletti S. Di Ruzza C. Lhotka L. Stefanelli 《The European physical journal. Special topics》2010,186(1):33-66
The influence of dissipative effects on classical dynamical models of Celestial Mechanics is of basic importance. We introduce
the reader to the subject, giving classical examples found in the literature, like the standard map, the Hénon map, the logistic
mapping. In the framework of the dissipative standard map, we investigate the existence of periodic orbits as a function of
the parameters. We also provide some techniques to compute the breakdown threshold of quasi-periodic attractors. Next, we
review a simple model of Celestial Mechanics, known as the spin-orbit problem which is closely linked to the dissipative standard
map. In this context we present the conservative and dissipative KAM theorems to prove the existence of quasi-periodic tori
and invariant attractors. We conclude by reviewing some dissipative models of Celestial Mechanics. Among the rotational dynamics
we consider the Yarkovsky and YORP effects; within the three-body problem we introduce the so-called Stokes and Poynting–Robertson
effects. 相似文献