Asymptotic Green’s functions for time-fractional diffusion equation and their application for anomalous diffusion problem

Asymptotic Green’s functions for short and long times for time-fractional diffusion equation, derived by simple and heuristic method, are provided in case if fractional derivative is presented in Caputo sense. The applicability of the asymptotic Green’s functions for solving the anomalous diffusion problem on a semi-infinite rod is demonstrated. The initial value problem for longtime solution of the time-fractional diffusion equation by Green’s function approach is resolved.

[1]  Mariusz Ciesielski,et al.  Numerical simulations of anomalous diffusion , 2003, math-ph/0309007.

[2]  T. Franosch,et al.  Anomalous transport in the crowded world of biological cells , 2013, Reports on progress in physics. Physical Society.

[3]  Fawang Liu,et al.  The space-time fractional diffusion equation with Caputo derivatives , 2005 .

[4]  L. Bécu,et al.  Evidence for three-dimensional unstable flows in shear-banding wormlike micelles. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  G. Espinosa-Paredes,et al.  Time-fractional telegraph equation for hydrogen diffusion during severe accident in BWRs , 2016 .

[6]  X. H. Zhang,et al.  Finite domain anomalous spreading consistent with first and second laws , 2009, 0911.1192.

[7]  Santos B. Yuste,et al.  Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations , 2014, Numerical Algorithms.

[8]  On the derivation of fractional diffusion equation with an absorbent term and a linear external force , 2009 .

[9]  Changpin Li,et al.  On Riemann-Liouville and Caputo Derivatives , 2011 .

[10]  Colin Atkinson,et al.  Rational Solutions for the Time-Fractional Diffusion Equation , 2011, SIAM J. Appl. Math..

[11]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[12]  Mark M. Meerschaert,et al.  A second-order accurate numerical method for the two-dimensional fractional diffusion equation , 2007, J. Comput. Phys..

[13]  R. Haydock,et al.  Vector continued fractions using a generalized inverse , 2003, math-ph/0310041.

[14]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[15]  Enrico Scalas,et al.  Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Risong Li,et al.  A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property , 2011 .

[17]  R. Sánchez,et al.  Transport equation describing fractional Lévy motion of suprathermal ions in TORPEX , 2014 .

[18]  Santos B. Yuste,et al.  Weighted average finite difference methods for fractional diffusion equations , 2004, J. Comput. Phys..