Fractal scaling of fractional diffusion processes

The fractal growth of fractional diffusion is analyzed from the viewpoint of the influence of fractional derivative order on scaling exponents. Fractional diffusion is considered here as deterministic with stochastic forcing, and with time and space fractional derivatives defined as the Caputo and Riesz-Feller forms, respectively. Fractal growth, as characterized by a dynamic scaling exponent, z, roughness, α and growth exponent, β, is model-dependent, and we show that the exponents are distinguished by the order of fractional derivatives and the form of stochastic process. Finally, scaling exponents for nonconservative (or uncorrelated) and correlated stochastic processes are found in the lowest-order linear fractional differential equations, and the prospect for construction of stochastic nonlinear fractional diffusion evolution equations is explored here. In application to signal processing, this amounts to modeling dynamic processes (signal evolution, image growth, surface and interface growth, structural growth, etc.) and their equivalence (or approximation) in fractional differencing models of discrete data.

[1]  Tsallis,et al.  Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  J. Lévy-Véhel.  FRACTAL APPROACHES IN SIGNAL PROCESSING , 1995 .

[3]  Pietro Cornetti,et al.  A fractional calculus approach to the description of stress and strain localization in fractal media , 2002 .

[4]  A. Compte,et al.  The generalized Cattaneo equation for the description of anomalous transport processes , 1997 .

[5]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[6]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[7]  A. M. Mathai,et al.  On fractional kinetic equations , 2002 .

[8]  Bruce J. West,et al.  Fractional Calculus and the Evolution of Fractal Phenomena , 1999 .

[9]  Evolution of Burgers' turbulence in the presence of external forces , 1997 .

[10]  A. Stanislavsky,et al.  Memory effects and macroscopic manifestation of randomness. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Ralf Metzler,et al.  Boundary value problems for fractional diffusion equations , 2000 .

[12]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[13]  Massimiliano Giona,et al.  Fractional diffusion equation for transport phenomena in random media , 1992 .

[14]  W. Wyss.  The fractional diffusion equation , 1986 .

[15]  S. Edwards,et al.  The surface statistics of a granular aggregate , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[16]  Damián H. Zanette,et al.  Macroscopic current in fractional anomalous diffusion , 1998 .

[17]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[18]  R. Hilfer.  Fractional Diffusion Based on Riemann-Liouville Fractional Derivatives † , 2000 .

[19]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[20]  Alexander I. Saichev,et al.  Fractional kinetic equations: solutions and applications. , 1997, Chaos.

[21]  I. Podlubny.  Fractional differential equations , 1998 .

[22]  V. Uchaikin.  Multidimensional symmetric anomalous diffusion , 2002 .

[23]  Bruce J. West,et al.  \FRACTIONAL TUNING" OF THE RICCATI EQUATION , 1997 .

[24]  A. Compte,et al.  Anomalous diffusion in linear shear flows , 1997 .

[25]  W. Woyczynski,et al.  Fractal Burgers Equations , 1998 .

[26]  G. Zaslavsky,et al.  Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics. , 1997, Chaos.

[27]  P. Meakin.  The growth of rough surfaces and interfaces , 1993 .

[28]  William R. Young,et al.  Strongly and weakly self-similar diffusion , 2001 .

[29]  Jacques Lévy Véhel,et al.  Signal representation and segmentation based on multifractal stationarity , 2002, Signal Process..

[30]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[31]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[32]  R. Gorenflo,et al.  Discrete random walk models for space-time fractional diffusion , 2002, cond-mat/0702072.

[33]  Paul Meakin,et al.  Fractals, scaling, and growth far from equilibrium , 1998 .

[34]  A. Barabasi,et al.  Fractal concepts in surface growth , 1995 .

[35]  Francesco Mainardi,et al.  The fractional Fick's law for non-local transport processes , 2001 .

[36]  I. Procaccia,et al.  Diffusion on fractals. , 1985, Physical review. A, General physics.

[37]  R. Metzler,et al.  Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation , 2002 .

[38]  Stéphane Canu,et al.  The long-term memory prediction by multiscale decomposition , 2000, Signal Process..

[39]  Jan Beran,et al.  Testing for a change of the long-memory parameter , 1996 .

[40]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[41]  J. L. Véhel.  Fractal Approaches in Signal Processing , 1995 .

[42]  W. A. Woyczyński.  Burgers-KPZ Turbulence , 1998 .

[43]  P. Grigolini,et al.  Fractional calculus as a macroscopic manifestation of randomness , 1999 .

[44]  Ralf Metzler,et al.  Fractional diffusion: exact representations of spectral functions , 1997 .

[45]  West,et al.  Dynamical approach to Lévy processes. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[46]  E. Lutz.  Fractional Langevin equation. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  R. Gorenflo,et al.  Time Fractional Diffusion: A Discrete Random Walk Approach , 2002 .

[48]  Ralf Metzler,et al.  FRACTIONAL DIFFUSION, WAITING-TIME DISTRIBUTIONS, AND CATTANEO-TYPE EQUATIONS , 1998 .

[49]  F. Tatom.  THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS , 1995 .

[50]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[51]  T. Vicsek,et al.  Dynamics of fractal surfaces , 1991 .

[52]  Fred J. Molz,et al.  A physical interpretation for the fractional derivative in Levy diffusion , 2002, Appl. Math. Lett..

[53]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[54]  A. Olemskoi,et al.  The theory of spatiotemporal pattern in nonequilibrium systems , 2000 .

[55]  Compte,et al.  Stochastic foundations of fractional dynamics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[56]  Evolution, its Fractional Extension and Generalization , 1999, math-ph/9912023.

[57]  Karl Heinz Hoffmann,et al.  The similarity group and anomalous diffusion equations , 2000 .

[58]  S. A. El-Wakil,et al.  Fractional Fokker–Planck equation , 2000 .

[59]  A. El-Sayed,et al.  Fractional-order diffusion-wave equation , 1996 .

[60]  V. Privalko,et al.  Temporal fractal model for the anomalous dielectric relaxation of inhomogeneous media with chaotic structure. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[61]  Structure of random fractals and the probability distribution of random walks. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[62]  H. Roman,et al.  Continuous-time random walks and the fractional diffusion equation , 1994 .

[63]  Karl Heinz Hoffmann,et al.  Fractional Diffusion and Entropy Production , 1998 .

[64]  Ke Zhao,et al.  Fractal analysis with applications to seismological pattern recognition of underground nuclear explosions , 2000, Signal Process..