Analytical solutions of the space–time fractional Telegraph and advection–diffusion equations

Abstract The aim of this paper is to develop a fractional derivative model of energetic particle transport for both uniform and non-uniform large-scale magnetic field by studying the fractional Telegraph equation and the fractional advection–diffusion equation. Analytical solutions of the space–time fractional Telegraph equation and space–time fractional advection–diffusion equation are obtained by use of the Caputo fractional derivative and the Laplace–Fourier technique. The solutions are given in terms of Fox’s H function. As an illustration they are applied to the case of solar energetic particles.

[1]  S. Momani.  An algorithm for solving the fractional convection–diffusion equation with nonlinear source term , 2007 .

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

[3]  F. Effenberger,et al.  THE DIFFUSION APPROXIMATION VERSUS THE TELEGRAPH EQUATION FOR MODELING SOLAR ENERGETIC PARTICLE TRANSPORT WITH ADIABATIC FOCUSING. I. ISOTROPIC PITCH-ANGLE SCATTERING , 2014, 1410.1225.

[4]  Brian Berkowitz,et al.  Anomalous transport in laboratory‐scale, heterogeneous porous media , 2000 .

[5]  Zakari,et al.  Equations of state and transport equations in viscous cosmological models. , 1993, Physical review. D, Particles and fields.

[6]  T. Atanacković.  A modified Zener model of a viscoelastic body , 2002 .

[7]  R. Schlickeiser,et al.  THE TELEGRAPH APPROXIMATION FOR FOCUSED COSMIC-RAY TRANSPORT IN THE PRESENCE OF BOUNDARIES , 2015, 1505.05134.

[8]  F. Effenberger,et al.  ANALYTICAL SOLUTIONS OF A FRACTIONAL DIFFUSION-ADVECTION EQUATION FOR SOLAR COSMIC-RAY TRANSPORT , 2014, 1410.1223.

[9]  H. Qi,et al.  Solutions of the space-time fractional Cattaneo diffusion equation , 2011 .

[10]  I. M. Sokolov,et al.  Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities , 2016, 1611.06202.

[11]  J. Klafter,et al.  Subdiffusive transport close to thermal equilibrium: from the langevin equation to fractional diffusion , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  A. Plastino,et al.  General Solution of a Fractional Diffusion-Advection Equation for Solar Cosmic-Ray Transport , 2014, 1412.0255.

[13]  J. Earl.  Analytical description of charged particle transport along arbitrary guiding-field configurations , 1981 .

[14]  A. Godec,et al.  Quantifying non-ergodicity of anomalous diffusion with higher order moments , 2017, Scientific Reports.

[15]  R. Schlickeiser,et al.  Cosmic-Ray Diffusion Approximation with Weak Adiabatic Focusing , 2008, 2008 IEEE 35th International Conference on Plasma Science.

[16]  S. Saha Ray,et al.  Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques (G′/G)-expansion method and improved (G′/G)-expansion method , 2016 .

[17]  Vickie E. Lynch,et al.  Fractional diffusion in plasma turbulence , 2004 .

[18]  On the fractional derivative model of the transport of cosmic rays in the Galaxy , 2010 .

[19]  I. Goychuk.  Fractional-time random walk subdiffusion and anomalous transport with finite mean residence times: faster, not slower. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  R. Schlickeiser,et al.  A NEW COSMIC RAY TRANSPORT THEORY IN PARTIALLY TURBULENT SPACE PLASMAS: EXTENDING THE QUASILINEAR APPROACH , 2011 .

[21]  D. Shiota,et al.  SIGN FOR SUPER-DIFFUSIVE TRANSPORT OF ENERGETIC IONS ASSOCIATED WITH A CORONAL-MASS-EJECTION-DRIVEN INTERPLANETARY SHOCK , 2011 .

[22]  G. Lebon,et al.  Extended irreversible thermodynamics , 1993 .

[23]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[24]  Ruggeri,et al.  Shock waves and second sound in a rigid heat conductor: A critical temperature for NaF and Bi. , 1990, Physical review letters.

[25]  Arak M. Mathai,et al.  The H-function with applications in statistics and other disciplines , 1978 .

[26]  G. Zimbardo,et al.  Quasi-ballistic and superdiffusive transport for impulsive solar particle events , 2011 .

[27]  A. Fasoli,et al.  Superdiffusive transport in laboratory and astrophysical plasmas , 2015 .

[28]  Andrey G. Cherstvy,et al.  Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. , 2014, Physical chemistry chemical physics : PCCP.

[29]  B. Braaksma,et al.  Asymptotic expansions and analytic continuations for a class of Barnes-integrals , 1964 .

[30]  R. Schlickeiser,et al.  The telegraph equation for cosmic-ray transport with weak adiabatic focusing , 2013 .

[31]  R. Kimmich,et al.  Sub- and superdiffusive molecular displacement laws in disordered porous media probed by nuclear magnetic resonance. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  O. P. Singh,et al.  An analytic algorithm for the space-time fractional advection-dispersion equation , 2011, Comput. Phys. Commun..

[33]  A. M. Kowalski,et al.  Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy , 2008 .

[34]  J. Klafter,et al.  Some fundamental aspects of Lévy flights , 2007 .

[35]  S. Krucker,et al.  A diffusive description of the focused transport of solar energetic particles - Intensity- and anisotropy-time profiles as a powerful diagnostic tool for interplanetary particle transport conditions , 2011 .