Random-order fractional differential equation models

This paper proposes a new concept of random-order fractional differential equation model, in which a noise term is included in the fractional order. We investigate both a random-order anomalous relaxation model and a random-order time fractional anomalous diffusion model to demonstrate the advantages and the distinguishing features of the proposed models. From numerical simulation results, it is observed that the scale parameter and the frequency of the noise play a crucial role in the evolution behaviors of these systems. In addition, some potential applications of the new models are presented.

[1]  Ya. L. Kobelev,et al.  Anomalous diffusion with time-and coordinate-dependent memory , 2003 .

[2]  Jun Sugiyama,et al.  Li diffusion in LixCoO2 probed by muon-spin spectroscopy. , 2009, Physical review letters.

[3]  R. Riera,et al.  Truncated Lévy walks and an emerging market economic index , 2001 .

[4]  C. Mallows,et al.  A Method for Simulating Stable Random Variables , 1976 .

[5]  Michele Caputo,et al.  Diffusion with space memory modelled with distributed order space fractional differential equations , 2003 .

[6]  Wen Chen.  A speculative study of fractional Laplacian modeling of turbulence , 2006 .

[7]  Hongguang Sun,et al.  Author's Personal Copy Computers and Mathematics with Applications Anomalous Diffusion Modeling by Fractal and Fractional Derivatives , 2022 .

[8]  E. Lutz,et al.  Fractional transport equations for Lévy stable processes. , 2001, Physical review letters.

[9]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

[10]  E. Schutter,et al.  Anomalous Diffusion in Purkinje Cell Dendrites Caused by Spines , 2006, Neuron.

[11]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

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

[13]  I M Sokolov,et al.  Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  K. Weron,et al.  On the Cole-Cole relaxation function and related Mittag-Leffler distribution , 1996 .

[15]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[16]  Frank Jenko,et al.  Intermediate non-Gaussian transport in plasma core turbulence , 2007 .

[17]  Guido Germano,et al.  Efficient Monte Carlo simulation of uncoupled continuous-time random walks and approximate solution of the fractional diffusion equation , 2007 .

[18]  G. Voth,et al.  Fluid particle accelerations in fully developed turbulence , 2000, Nature.

[19]  T. Mexia,et al.  Author ' s personal copy , 2009 .

[20]  Mark M. Meerschaert,et al.  Inhomogeneous Fractional Diffusion Equations , 2005 .

[21]  Igor M. Sokolov,et al.  Fractional diffusion in inhomogeneous media , 2005 .

[22]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

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

[24]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[25]  Stefan Samko,et al.  Fractional integration and differentiation of variable order , 1995 .

[26]  Joo-Young Go,et al.  A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion-controlled transfer processes , 2006 .

[27]  Carlos F.M. Coimbra,et al.  Mechanics with variable‐order differential operators , 2003 .

[28]  I. M. Sokolov,et al.  Distributed-Order Fractional Kinetics , 2004 .

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

[30]  Bruce J. West,et al.  FRACTIONAL DIFFUSION AND LEVY STABLE PROCESSES , 1997 .

[31]  E. Scalas,et al.  Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Fawang Liu,et al.  Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation , 2007 .

[33]  Barbara Atamaniuk,et al.  Application of Fractional Derivative Operators to Anomalous Diffusion and Propagation Problems , 2007 .