A numerical method for the distributed order time-fractional diffusion equation

This paper is devoted to the numerical approximation of the diffusion equation with distributed order in time. A numerical method is proposed in the case where the order of the time derivative is distributed over the interval [0, 1], and results concerning the stability and convergence of that scheme are provided. Two numerical examples are presented illustrating the theoretical numerical results.

[1]  R. Gorenflo,et al.  Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density , 2013 .

[2]  Francesco Mainardi,et al.  The Two Forms of Fractional Relaxation of Distributed Order , 2007 .

[3]  Fawang Liu,et al.  Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation , 2008, Appl. Math. Comput..

[4]  Fawang Liu,et al.  Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term , 2009, J. Comput. Appl. Math..

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

[6]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[7]  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.

[8]  Zhi-Zhong Sun,et al.  A compact finite difference scheme for the fractional sub-diffusion equations , 2011, J. Comput. Phys..

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

[10]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[11]  Santos B. Yuste,et al.  On an explicit finite difference method for fractional diffusion equations , 2003, ArXiv.

[12]  Fenghui Huang.  A Time-Space Collocation Spectral Approximation for a Class of Time Fractional Differential Equations , 2012 .

[13]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[14]  Yingjun Jiang,et al.  Moving finite element methods for time fractional partial differential equations , 2013 .

[15]  Kai Diethelm,et al.  Numerical analysis for distributed-order differential equations , 2009 .

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

[17]  Mingrong Cui,et al.  Compact finite difference method for the fractional diffusion equation , 2009, J. Comput. Phys..

[18]  M. Naber.  DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION , 2003, math-ph/0311047.

[19]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[20]  Francesco Mainardi,et al.  Some aspects of fractional diffusion equations of single and distributed order , 2007, Appl. Math. Comput..

[21]  YuanTong Gu,et al.  Anomalous sub-diffusion equations by the meshless collocation method , 2012 .

[22]  Time-fractional Diffusion of Distributed Order , 2007, cond-mat/0701132.

[23]  Fawang Liu,et al.  A Fourier method for the fractional diffusion equation describing sub-diffusion , 2007, J. Comput. Phys..

[24]  Xuan Zhao,et al.  A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions , 2011, J. Comput. Phys..

[25]  HongGuang Sun,et al.  A semi-discrete finite element method for a class of time-fractional diffusion equations , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[27]  Santos B. Yuste,et al.  An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations , 2004, SIAM J. Numer. Anal..

[28]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .