https://www.doczj.com/doc/ec16576302.html,
Intermediate processes, critical phenomena: theory, method,
progress of fractional operators and its applications to modern
mechanics
Xu Mingyu (徐明瑜)1 & Tan Wenchang (谭文长)2
1. Institute of Applied Mathematics, School of Math & System Science, Shandong
University, Jinan 250100, China;
2. LTCS & Department of Mechanics and Engineering Science, Peking University,
Beijing 100871, China
Correspondence should be addressed to Xu Mingyu (email: xumingyu@https://www.doczj.com/doc/ec16576302.html,) or Tan Wenchang (email:
tanwch@https://www.doczj.com/doc/ec16576302.html,)
Abstract ：From point of view of physics, especially of mechanics, we briefly introduce fractional operators (with emphasis on fractional calculus and fractional differential equations) used for describing intermediate processes and critical phenomena in physics and mechanics, and their progress in theory and methods, and applications to modern mechanics. Some authors’ researches in this area in recent years are included. Finally, prospects and evaluation for this subject are made.
Keywords: fractional operators, intermediate processes, critical phenomena, modern mechanics
1 Brief introduction to the fractional operators (FO)
When in the 17th century the integer calculus had been developed, Leibniz and L’Hospital probed into the problems on the fractional calculus (FC) and the simplest fractional differential equations (FOEs) through letters. Leibniz asked in a letter addressed to L’Hospital:
Can the meaning of derivatives of integral order d n f(x)/dx n be extended to have meaning when n is not an integer but any number (irrational, fractional or even complex-valued)?
L’Hospital responded: What if n be 1/2? d 1/2f(x)/dx 1/2=? for f(x)=x. Leibniz, in a letter dated from Sept. 30, 1695, replied: It will lead to a paradox, from which one day useful consequences will be drawn. And, 124 years later (in 1819) Lacroix gave the correct answer of this problem for the first time, i.e., π/2/2/12/1x dx x d =. After that in a long period of time, through great efforts made by many mathematicians the Riemann-Liouville (R-L) fractional operators were finally formed. In Lebesgue integrable space L 1(a, b) fractional integral operator with q order is defined as:
{}∫???Γ= t 0 10)()()
(1:)(τττd f t q t f D q q
t ，(1Re 0≤ )(?ΓAs its inversion, the fractional differential operator with ν order is defined as: - 1 - {}[{})(:)(00t f D dt d t f D n t n n t ?=νν]，（0Re The main property of FO is semigroup of operators. And it is easy to prove that it is a (C 0)-semigroup of operators [6]. The property of FO which is different from one of integer operators is that it is a global (non-local) operator and a limit in the sense of ultra-long time [7] . Therefore it is contradictory with viewpoint of classical (Newtonian) mechanics. However, from point of view of coarse graining procedure in the sense of ultra-long time both of them can unify [7]. If we define the kernel function K q (t) as causal one , then (1) can be equivalently rewritten as: ???≤>Γ=?0 t 0 0 t )(/ )(1q t t K q q {})(0t f D q t ?= (where = is generalized function and “*” denotes Laplace convolution). It is obvious that for an arbitrary non-trivial kernel function, FO has memory, i.e., it is non-Markovian process. And Markovian process recovers provided and only provided =)(*)(t f t K q )(t K q )(/1q t q Γ?+q t D ?0)(t K q )(0t K δ (K 0=const, )(t δ is Dirac delta function). It is easy to calculate that fractional derivative of an arbitrary constant C 0 is not equal to zero, and we have {})1(/000ααα?Γ=?t C C D t )10(≤<α. In a similar way, for the simplest FDE ({}0)(0=t f D q t 10< - 2 -https://www.doczj.com/doc/ec16576302.html, https://www.doczj.com/doc/ec16576302.html, 2Classical Newtonian mechanics and linear physics are subjected to challenge The classical Newtonian mechanics which is based on the absolute space, time and the Euclidean geometry thinks that space and time are without beginning or end and continuity is everywhere. Under the assumptions of above point of view of space and time, all of derived physical quantities, such as velocity, momentum, acceleration and force are continuous. Under the action of the integer operators the functions representing above physical quantities are analytical. Furthermore, classical mechanics and linear physics maintained that most of physical phenomena can be expressed by analytical functions and evolutional processes of the phenomena may be described by the motion differential equations involved the analytical functions. For instance, the solutions of Euler-Lagrange’s and Hamilton’s equations in classical mechanics are all analytical functions. Therefore, at that time most of physicists thought that the analytical functions can always describe various phenomena which take place in physics and mechanics. Indeed, the important achievements in acoustics, electromagnetics and the theory of heat transfer in 19th century and in diffusion theory and quantum mechanics in 20th century seemed to show that above mentioned conclusion of classical mechanics and linear physics is unassailable. However, some contraexamples with challenge which can not be explained by the classical mechanics and linear physics puzzled the physicists. Challenge One: Irregular Fluctuations in the Velocity Field of Turbulent. As early as 1926, Richardson [13] had found that the variations in wind are random in both magnitude and direction in the velocity field of turbulent wind in the atmosphere. Therefore the velocity field is not differentiable, that is to say, the velocity field can not be described by analytical functions. Furthermore, he concluded that it was impossible to model the irregular flow of the wind using the traditional differential equations representing Newton’s force law, i.e., Navier-Stokes equations in fluid mechanics. A lot of facts happened later as indicated that Richardson’s conclusions are correct. Researches on turbulent along the road of classical Navier-Stokes equations produce very little effect. At present it is proved that velocity fields of turbulent fluids at low viscosity have been shown to be multifractal [12, 14~18]. Challenge Two: Brownian Motion. Brownian motion is the base of both statistical mechanics and the theory of modern random walk and anomalous diffusion. As early as1827, Brown found that the velocity and direction of motion of small pollen grain in water are irregular and vary discontinuously through observation. However, he did not know the mechanism induced this random motion at that time. Nearly at the same time that Brown found above facts, Fourier came up with the heat conduction equation from point of view of Fick’s diffusion. In 1905, Albert Einstein unified the two approaches in his treatises on Brownian motion. An important application of Einstein’s results was the independent measurement of the Avogadro number by Perrin. And just as well that Perrin won the Nobel Prize in 1926 [19]. It should be noted that during that time some mathematicians were studying a kind of so-called functions of mathematical pathologies. For instance, in 1872, Weiestrass gave a lecture to the Berlin Academy in which he presented a function that is continuous everywhere, but nowhere differentiable. And we name it Weierstrass function (WF): - 3 - ∑∞ ==1cos )(k k k t b a t w . b a <<<10 and （3） 1>ab In classical mathematical analysis formula (3) along with Van der Waerden’s function was introduced as an example of functions that is continuous everywhere, but nowhere differentiable [20]. In researches on the theory of FO, (3) is deformed into the following generalized Weiestrass function (GWF) [21]: ∑∞ ?∞=+=n n n n t b a t G ]sin[1)(?. （4） In (4) we introduce the set of phases {}n ?. It may be selected either randomly or deterministically. Under the assumption that , the fractional dimension of this function is related to the choice of a and b. It is obvious that a b >)cos((/n n n n t b a b dt dG ?+= ∑∞?∞= () is absolutely divergence. Thus it can be seen that it is impossible to show the evolution process of this function through classical differential equations of motion. We call such a function fractal function ∞→n [22]. It can be proved that fractal function has fractional derivatives [21]. The definition of fractal function is that it has not characteristic scale, its derivative is divergence, i.e., non-differentiable. From (3) and (4) it can be seen that their scale is invariant, that is to say, the following transformation is satisfied [23]: )()(t aF bt F =. And this kind of transformation is the important property of fractal functions and fractional dynamics. Scaling invariance [24], renormalization group theory [25] and memory integral [26] are closely related to one another. They are all properties of fractals. Nearly at the same time with Weiestrass, Cantor [27], a student and later a colleague of Weiestrass, provided another famous fractal function named the middle-third Cantor set nowadays. Challenge Three: Viscoelastic Materials Have Memory. A number of experiments in early stage showed that to describe the constitutive relation the integral equation is more accurate than differential equation. Further study indicated that the constitutive relation can be represented by the convolution integral of FO and there is not characteristic time scale, i.e., the stress relaxation is non-exponential (non-Debye)[28~33]. The constitutive equation of viscoelastic materials is the earliest field of the applications of FO and is the most field obtained achievements. It is common knowledge that the stress-strain of elasticity body satisfies Hooke’s law, )(~)(t t εσ, and viscous fluid satisfies Newton’s law, . If Hooke’s law is rewritten in the form , then at least it is formally reasonable to consider that the constitutive equation of viscoelastic materials should be in the form: (11/)(~)(dt t d t εσ00/)(~)(dt t d t εσααεσdt t d t /)(~)(10≤≤α). Under the guidance of aforementioned idea, the constitutive relation of viscoelastic materials along with phase transition evolution [34], as a kind of intermediate process [35] and critical phenomenon [36] is most successful in the applications of the FO theory. The authors of this paper [37] in study on the constitutive equation of viscoelastic materials using generalized fractional element networks in generalized function space gave out the unified representations of all results including piecewise self-similar properties. - 4 - 3 Applications of FO in various complex systems At present, one of the important subjects of non-linear science in the world is complex system (or named complex phenomena). As a new research field, although a precise definition of complex system is unavailable [38] nowadays, one can summarize its intension in terms of two simple ideas, capable of explaining the behavior of many complex systems. One is self-organized criticality [39], the other is the principle of active walks. The former asserts that large dynamical systems tend to drive themselves to a critical state with no characteristic spatial and temporal scales. The latter describes how the elements in a complex system communicate with their environment and with each other through the interaction with the landscape they share. The principle of active walks has been applied successfully to many different problems, such as the formation of dielectric breakdown patterns, ion transport in glasses, and the cooperation of ants in food collection. However, a few of research workers have briefly defined the complex system as such physical systems that have long-time memory and/or long-range spatial interactions [22]. In the following, we outline the applications of FO to every kind of complex systems. (1) The applications of FO to dynamical problems of linear and nonlinear hereditary mechanics of solids. It is worth notice that recently Rossikhin and Shitikova [40,41] systematically and exhaustively summarized and reviewed the works of this respect. From history of FO to the recent development of it, damped vibrations of every kind of viscoelastic oscillators, forced vibrations of the hereditarily elastic oscillator with Rabotnov kernel, nonstationary waves in a hereditarily elastic rod with weakly singular kernel of heredity (shock waves, stress waves, the impulse load propagation in a hereditarily elastic rod), harmonic waves in a 3D hereditarily elastic medium as well as nonlinear waves in 1D hereditarily elastic media with FO, were all included in the review Ref.[41] included 174 references concerned. (2) The applications of FO to non-Newtonian fluid mechanics. Viscoelastic materials may be divided into two groups: viscoelastic solids and viscoelastic fluids. At present, applications of FO have been extended to the constitutive equation of viscoelastic fluids in 1D scalar form. Replacing one order derivative with respect to time by R-L FO, under certain special geometric boundary conditions one can get well-posed Cauchy’s problems. Using certain special functions concerned with FO, for instance, H-Fox function [42,43], generalized Mittag-Leffler (M-L) function [44,45], as well as Wright function [46,47] the analytical solutions to the problems can be often obtained. The obtained analytical solutions reveal the characteristics of flows of viscoelastic fluids. And when fractional derivative 1→α, the obtained solutions approach to ones of integer order Newtonian fluid. Recently, Chinese scholars have done a lot of works in this research field and have gotten some meaningful results [48~60]. It should be noted that for the above analytical solutions, the establishment of steady state of flows satisfies the scaling law, which has a typical property of FO. (3) The applications of FO to biophysics and biomechanics In fact, in early researches of biomechanics on morphology and anatomy of organs, tissues and capillaries the allometric law [61] came into use which is also called allometric determination equation [62], namely , where Y is anatomic variable (volume of the organ), M is the mass of animal, and a , b are constants varying with different organs. Mammals have many similarities in anatomy and physiology. This is the earliest scaling law found in the body of animal. It is a kind of the form of expression of fractals. And the power law is one of the most important allometric law. It describes a kind of macroscopic phenomenon of biological tissues and organs which show lack of characteristic scale and/or characteristic time in structure and function. In fact, as early as 1949, Adolph has pointed out b aM Y = - 5 - that many physiological processes vary with seven tenths power ~ eight tenths power of weight of animal and anatomic variable varies almost with first power of weight (linear relation). Recently, some of the research workers found that even human gait is related to the allomietric law [63]. Combining this law with fractional dimension and using the model of diffusion limited aggregation (DLA) from shear stress Ref.[64] proposed a model of vasculogenesis. This research extended classical optimization principle of grow and distribution of biological tissues and capillary networks (e.g. Roux’s Law and the law of one third power of flux). Recently, Ref [65] studied the fractal dynamics in the growth of root and demonstrated that fractals enable to increase significantly the surface of the system while its volume changes slightly. Therefore the authors of the paper proposed a new fractal optimization principle of biological growth. This principle is quite different from that of increasing surface under the condition of integer dimension in regular Euclidean space to increase the uptake [66]. To study the structure of network of blood vessels, T. Sun et al [67] presented a model of minimum energy dissipation and fractal structure of vascular systems. This network structure is self-similar. Precisely based on this self-similarity of biological tissues and organs， Best B.J. [68] proposed and summarized “error-tolerant” of human body organs in 1993. He thought that the self-similarity of organs makes life have a great deal of survival potential and makes organism have more tolerance to the random fluctuation of environment and to certain genetic errors. Obviously, it has important significance to the theory of biological evolution and optimization and optimization of tissues and organs. The authors of this paper using the theory of FO expressed the viscoelasticity of human organs and tissues [69, 70]. The obtained results showed that for certain tissues by using FO to denote the constitutive relation may be fit in with the experiment data better. Making use of the theory of FO to the ultra-slow relaxation of polymeric materials, one of the authors of this paper and co-worker [71] gave out an exact particular solution to the moving boundary problem with fractional anomalous diffusion in drug release devices and theoretically proved the well-known semi-empirical Ritger-Peppas’ formula in controlled drug release system to be right. In Refs. [72, 73] the theory of FO was used to illustrate the problems of anomalous diffusion in self-similar protein dynamics and in yeast, respectively. Applying the theory of FO to the study of non-linear dynamics of DNA is one of the branches of new and rapidly developing fields of nonlinear science created at the dawn of the 21st century. It is named non-linear biomolecular dynamics nowadays [74]. At present, researches on the helical structure of DNA sequence have mode considerable success. Pen C. K. et al [75] and Stanley H.E. et al [76] based on random walk ideas they conclude that protein-coding DNA sequences ( cDNA, exons) are remarkably like Brownian motion, having Gaussian statistics and zero correlation length. On the other hand, non-coding DNA sequences (introns) have long-range correlations. Voss [77] thought that all DNA sequences can be described by anomalous diffusive processes. And Buldyrev et al [78] argue that fBm can account for the second moment properties of DNA sequences. In 1988, Allegrini et al [79] by taking into account the influence of folding on the statistics of random walkers resolved the tertiary structure. Nowadays there are many references and monographs to consult in this field [80~82]. (4)The applications of FO to the theory of random walk and anomalous diffusion. Classical - 6 - (integer order) partial differential equation of diffusion and wave has been extended to the equation with fractional time and/or space by means of FO [83]. Furthermore, it has been extended to the problems of every kind of nonlinear fractional differential equation. And to present the solutions to the problems of initial and boundary values subject to above equations is another rapidly developing field of FO applications. In general, all of these equations have important background of practice applications, such as dispersion in fractals and porous media, semiconductor physics, turbulence and condensed matter physics [84~86]. Historically, foundation and development of diffusion equation have come from two different points of view. The one, starting with the 1st and 2nd Fick’s law and setting up the constitutive equation between flow and flux, is called determinacy point and the other, starting with the random walk is called random one. In the early stage the diffusion equation named Einstein-Kolmogorov is the typical example of random point. After the fractional constitutive relation and generalized concept of random walk have been established, two different points of view gave out the unified form of fractional diffusion equation [87, 88]. In general, the nature of fractional diffusion equation is characterarized by the temporal scaling of the mean-square displacement λt t r ∝)(2, 1=λ denotes standard (integer order) Gaussian diffusion, where as 1<λ and 1>λ denote the sub-diffusion and super-diffusion, respectively. Replacing time derivative of integer order and space Laplace operator by fractional time derivative and fractional Laplace operator, respectively, the fractional diffusion equation in fractals media can be obtained. Under the condition of spherical symmetry the non-dimensional form of it is as follows [89, 90]: w w d d dt t r P d /2/2),(＝),(11t r P r r r r s s d d ??????. （5） Where is probability density, is anomalous diffusion exponent and is spectral dimension, is the Hausdoff dimension. At present, most of studies on the anomalous diffusion equations, such as Refs. [91, 92] published in recent years, made progress on the basis of Eq. (5). In general, the measurements of physical quantities (e.g. concentration of tracer), especially in fractal media, mostly reveal not probability density but some scattering functions, e.g. dynamical structure factor ),(t r P 2>w d w f s d d d /2=f d ),(~?k p P , namely, the image function in Fourier and Laplace space [93, 94]. Recently, the authors and co-workers solved distribution of probability density and gave out the analytical expression of scattering functions caused by instantaneous point source in Euclidean space and in disordered fractal media, respectively [95,96]. When diffusion coefficient is a function of radius or is a power function of the concentration [97, 98] although it is non-linear fractional equation, the analytical solution with physical meaning sometimes can be found by means of the technique of transformation group [99,100]. In this case, most of solutions have such a sort of property which is akin to a propagating wave [101]. It is quite different from the solution of classical diffusion equation. The former has a finite velocity and the latter’s velocity is infinite. (5) The applications of FO to the research of DLA theory. As a critical phenomenon, study on DLA has begun since 1981 [102,103]. It came from study on every kind of patterns which extensively exist in natural science, such as fractal crystals, patterns of breakdown of dielectric, viscous fingering, lightning sparks, branching aggregations of bacteria, and so forth. The aggregation of fluid particles was its dynamic model proposed in the early stage [104]. There are two kinds of problems to be considered: a. kinetics. It involves the quantitative description of the time evolution of aggregations and their size distribution. b. geometry. It - 7 - studies quantitative description of the structure of the aggregation. And DLA is the simplest and most universal kinetical model. Their common feature is that a suitably defined potential governed by the Laplace equation exists. It denotes a kind of conservation property in nature. For instance, in seepage flow of porous media due to Darcy’s law and incompressibility, the equation satisfied by the pressure: 0)(=???p k . Therefore, some literatures also named DLA Laplacian growth mechanism of hydrodynamical origin [102, 103]. At present, it is proved that DLA aggregation is fractal structure with a dimension of about 1.7 in two dimensions and 2.5 in three dimension [105, 106] and the growth of the percolation cluster is governed by the so-called harmonic measure [107]. (The harmonic measure is a solution of Laplace’s equation for the electrostatic potential.) 4 Some notes and outlook A. Notes 1. In the world, the nonlinear science usually may be divided into six parts: fractals, chaos, pattern formation, solitons, cellular automata and complex system. Because in a dissipative system the signature of chaos is the existence of strange attractor(s) in the phase space, which is a fractal, only solitons have not been found related to FO yet. Thus it can be seen that the theory of FO occupied very important place in nonlinear science. 2. The applications of FO are extensive. In addition to aforementioned fields, there are some following domains of its application: (1) Generalized classical mechanics with fractional order. F. Riewe [108] proposed that Lagrangian and Hamiltonian mechanics can be formulated to include derivatives of fractional order and arrived at the motion equation of non-conservative system including friction force. Soon after that he developed fractional derivative mechanics by deriving a modified Hamilton’s principle, introducing two types of canonical transformations, and deriving the Hamilton-Jacobi equation [109]. And using the theory of FO, Ref. [110] solved the problem of the tautochrone under arbitrary potentials. Agrawal O. P. [111] presented the formulation of Euler-Lagrange equations for fractional variational problems. Recently, Achar et al [112] using the generalized M-L function presented the solution to the dynamical equation of fractional oscillator. (2) Fractional micro-system in quantum mechanics. In fact, in quantum mechanics the first attempt to apply the fractality concept was the Feynman path integral approach. Recently, through a lot of research works done by Laskin [113~116] the concepts and methods of FO were extended to the classical quantum mechanics. He gave out the following fractional Schr?dinger equation satisfied by the wave function ψ: ψψψααα)]()([x V D H t i +??==??h h , where denotes Hamiltonian operator with fractional order, and αH )()(x V D H +??=αααh ∫?=?ααπ||21)(/p dpe ipx h h h , is Riesz fractional derivative. Laskin presented the density matrix of a free particle and fractional plane wave function in terms of H-functions, and proved that the Hermiticity of the fractional Hamilton operator and established the parity conservation law for fractional quantum mechanics and the fractional “Bohr atom”. Naber αH [117] set up time fractional Schr?dinger equation in terms of Caputo fractional derivative, obtained time dependent Hamiltonian operator , and found the probability current was not conserved. αH - 8 - Moreover, he gave out the solutions for a free particle and for an infinite potential well. Using the theory of scaling, Sidharth [118] studied micro and macro (cosmical) systems and got the formulas which are analogous to the Heisenberg’s Uncertainity Principle. And he named it Scaled Quantum Mechanics. Refs. [119] and [120] presented a family of boson coherent sates by means of M-L functions and quantization of the anomalous Brownian motion, respectively. (3) The integer (standard) exterior derivative was extended to the fractional one by means of the theory of FO. Ref. [121] founded new vector spaces of finite and infinite dimension (fractional differential form spaces), and gave out the coordinate transformation rules. And many (standard) special functions maybe generalized to the form of fractional order [122, 123]. Recently, the authors of this paper presented a series of functional forms of generalized M-L functions 1), e.g. =)(2/1,2z E 2/1)(4/14/1z e z erf z ))sin(2([22/14/14/1iz z i S z i π?))]cos(2(2/14/1iz z i C π +2/1?+π and so on (where. , and )(?erf )(?S )(?C denote Error function, Fresnel sine function and Fresnel cosine function, respectively). B. Outlook. 1. At present, to solve the non-linear equation with fractional order and to study the properties of the solution have not been entirely developed yet. 2. The studies on the behaviors of statics and kinetics of some fractal media (e.g. the middle third Cantor set, Sierpinski gasket) have begun just now. Carpinteri and Cornetti [124] applying the local FO and the virtual work principle solved the problem of description of stress and strain localization in fractal media. There are going to be important meaning of theory and good prospects of applications in this aspect. 3. In the foregoing we briefly introduced the theory of FO under the Euclidean measure. Because in probability the Euclidean measure is applied, the theory of FO is extensively applied in random analysis. The research under other measure has not been developed yet. 4. In 1994, S. Westerland 2) in a report of Kalmar university, Sweden, proposed a generalization of Newton’s second law and considered that it can be written in the form: ()(0t x D k F t α=21≤<α,is displacement). It should be noted that historically, this is an important problem at issue and to be studied further. In 2001, the authors starting from Navier-Stokes equation arrived at a conclusion which supports Westerland’s argument )(t x [48]. 5. Over the past twenty years, significant changes in our knowledge of non-linear phenomena have occurred. The methodology has moved away from complete reliance on the tools of linear, analytical quantitative physics towards a combination of nonlinear, numerical and qualitative techniques [125,126]. Many non-linear phenomena satisfy fractal statistics [127]. For example, the frequency-area statistics of earthquakes, the time series of the earth’s magnetic field and so on. Solutions to classical differential equations cannot give this type of behavior. In order to reproduce 1) Xu MY, Tan WC. Some functional forms of generalized Mittag-Leffler function in fractional calculus. Submitted. 2) Westerland S., Causality, report No. 940426, University of Kalmar, 1994 - 9 - the observed phenomena and data various models of cellular automata were proposed nowadays, e.g., the slider-block model for earthquakes. From macrocosmos to microcosmos (mesocosmos) the fractal phenomena can be seen everywhere. However, it is very difficult to explain the formation of fractals from point of view of dynamics. The difficulty and reason of this are that the irreversible or “historical” processes of fractals take place on very different time scale than typical observations. For instance, earthquakes take place in a much shorter time scale than landscape formation, and it is a kind of irreversible process. The authors of Ref [128] in terms of the model of avalanche dynamics unified the origin of fractals. And it indicated that the fractal dynamics has gotten into a rapid development period. In the researches on DLA, the problems of fingering concerning with viscoelastic fluids have been studied[129]. Its governing equation is not Laplacian one. This is a kind of highly nonlinear free-boundary value problem. The Hele-Shaw and Saffman-Taylor instabilities are still the simplest model of fluid dynamics. C.Evaluation Recently, it is claimed that cellular automata will lead to a new scientific revolution and it includes general relativity, computability, turbulence, genetic coding, and many others [130]. It gives rise to extensive issues in the international scientific circles [131]. It should be noted that as a useful tool, FO has some limitations. For example, in the researches on the constitutive equation of viscoelastic materials at present, it is proved that high dimension is instable [132]. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No.10272067, No.10372007, No.10572006), the Doctoral Program Foundation of Education Ministry of China (Grant No.20030422046) and New Century Training Programme Foundation for the Talents by Education Ministry of China. References [1] Oldham K B, Spanier J.， The fractional calculus，New York -London : Academic Press, 1974. 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