Finite difference Methods for fractional differential equations

In this review paper, the finite difference methods (FDMs) for the fractional differential equations are displayed. The considered equations mainly include the fractional kinetic equations of diffusion or dispersion with time, space and time-space derivatives. In some way, these numerical methods have similar form as the case for classical equations, some of which can be seen as the generalizations of the FDMs for the typical differential equations. And the classical tools, such as the von Neumann analysis method, the energy method and the Fourier method are extended to numerical methods for fractional differential equations accordingly. At the same time, the techniques for improving the accuracy and reducing the computation and storage are also introduced.

[1]  Zaid M. Odibat,et al.  Computational algorithms for computing the fractional derivatives of functions , 2009, Math. Comput. Simul..

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

[3]  Enrico Scalas,et al.  Waiting-times and returns in high-frequency financial data: an empirical study , 2002, cond-mat/0203596.

[4]  I. Podlubny.  Matrix Approach to Discrete Fractional Calculus , 2000 .

[5]  Weihua Deng,et al.  Numerical algorithm for the time fractional Fokker-Planck equation , 2007, J. Comput. Phys..

[6]  Fawang Liu,et al.  Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation , 2011, Numerical Algorithms.

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

[8]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

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

[10]  Fawang Liu,et al.  Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method , 2007, J. Comput. Phys..

[11]  Changpin Li,et al.  On the fractional Adams method , 2009, Comput. Math. Appl..

[12]  K. Diethelm,et al.  The Fracpece Subroutine for the Numerical Solution of Differential Equations of Fractional Order , 2002 .

[13]  Yangquan Chen,et al.  Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion , 2011, Comput. Math. Appl..

[14]  Changpin Li,et al.  Chaos in Chen's system with a fractional order , 2004 .

[15]  Dumitru Baleanu,et al.  A Central Difference Numerical Scheme for Fractional Optimal Control Problems , 2008, 0811.4368.

[16]  Fathalla A. Rihan.  Computational methods for delay parabolic and time‐fractional partial differential equations , 2010 .

[17]  Christopher T. H. Baker,et al.  A perspective on the numerical treatment of Volterra equations , 2000 .

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

[19]  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..

[20]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[21]  I. Turner,et al.  Two New Implicit Numerical Methods for the Fractional Cable Equation , 2011 .

[22]  M. A. Akanbi,et al.  Numerical solution of initial value problems in differential - algebraic equations , 2005 .

[23]  B. West.  Fractional Calculus in Bioengineering , 2007 .

[24]  Zhi‐zhong Sun,et al.  A compact difference scheme for the fractional diffusion-wave equation , 2010 .

[25]  Fawang Liu,et al.  Numerical simulation for the 3D seepage flow with fractional derivatives in porous media , 2008 .

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

[27]  Fawang Liu,et al.  Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation , 2010, SIAM J. Sci. Comput..

[28]  Fawang Liu,et al.  Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations , 2009 .

[29]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

[30]  Fawang Liu,et al.  Finite difference approximations for the fractional Fokker–Planck equation , 2009 .

[31]  Neville J. Ford,et al.  The numerical solution of fractional differential equations: Speed versus accuracy , 2001, Numerical Algorithms.

[32]  Aiguo Xiao,et al.  Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients , 2010, J. Comput. Appl. Math..

[33]  Kai Diethelm,et al.  Numerical solution of fractional order differential equations by extrapolation , 1997, Numerical Algorithms.

[34]  John Paul Roop.  Numerical approximation of a one-dimensional space fractional advection-dispersion equation with boundary layer , 2008, Comput. Math. Appl..

[35]  Mehdi Maerefat,et al.  Explicit and implicit finite difference schemes for fractional Cattaneo equation , 2010, J. Comput. Phys..

[36]  C. Lubich.  Discretized fractional calculus , 1986 .

[37]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[38]  Fawang Liu,et al.  New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation , 2008, SIAM J. Numer. Anal..

[39]  Changpin Li,et al.  Synchronization in fractional-order differential systems , 2005 .

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

[41]  N. Ford,et al.  Pitfalls in fast numerical solvers for fractional differential equations , 2006 .

[42]  Fawang Liu,et al.  Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term , 2011, Appl. Math. Comput..

[43]  Changpin Li,et al.  Numerical approaches to fractional calculus and fractional ordinary differential equation , 2011, J. Comput. Phys..

[44]  Fawang Liu,et al.  Implicit difference approximation for the time fractional diffusion equation , 2006 .

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

[46]  V. Anh,et al.  Numerical Simulation of the Nonlinear Fractional Dynamical Systems with Fractional Damping for the Extensible and Inextensible Pendulum , 2007 .

[47]  Xianjuan Li,et al.  A Space-Time Spectral Method for the Time Fractional Diffusion Equation , 2009, SIAM J. Numer. Anal..

[48]  Fawang Liu,et al.  The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation , 2008 .

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

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

[51]  Fawang Liu,et al.  A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation , 2009 .

[52]  Ercília Sousa,et al.  Finite difference approximations for a fractional advection diffusion problem , 2009, J. Comput. Phys..

[53]  Lijuan Su,et al.  Finite difference methods for fractional dispersion equations , 2010, Appl. Math. Comput..

[54]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[55]  Daolin Xu,et al.  Chaos synchronization of the Chua system with a fractional order , 2006 .

[56]  Dumitru Baleanu,et al.  Fractional Bloch equation with delay , 2011, Comput. Math. Appl..

[57]  Linzhang Lu,et al.  Implicit numerical approximation scheme for the fractional Fokker-Planck equation , 2010, Appl. Math. Comput..

[58]  Fawang Liu,et al.  Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives , 2008, Appl. Math. Comput..

[59]  Alan D. Freed,et al.  On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity , 1999 .

[60]  H. R. Hicks,et al.  Numerical methods for the solution of partial difierential equations of fractional order , 2003 .

[61]  Roberto Garrappa,et al.  On some explicit Adams multistep methods for fractional differential equations , 2009 .

[62]  Barkai,et al.  From continuous time random walks to the fractional fokker-planck equation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[63]  Changpin Li,et al.  Fractional differential models for anomalous diffusion , 2010 .

[64]  Fawang Liu,et al.  Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term , 2010 .

[65]  Shyam L. Kalla,et al.  Numerical treatment of fractional heat equations , 2008 .

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

[67]  K. Diethelm.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

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

[69]  Fawang Liu,et al.  Detailed analysis of a conservative difference approximation for the time fractional diffusion equation , 2006 .

[70]  Duarte Valério,et al.  Variable-order fractional derivatives and their numerical approximations , 2011, Signal Process..

[71]  Fawang Liu,et al.  Implicit difference approximation for the two-dimensional space-time fractional diffusion equation , 2007 .

[72]  Fawang Liu,et al.  A Computationally Effective Predictor-Corrector Method for Simulating Fractional Order Dynamical Control System , 2006 .

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

[74]  Hong Wang,et al.  A direct O(N log2 N) finite difference method for fractional diffusion equations , 2010, J. Comput. Phys..

[75]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[76]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[77]  Chunhong Wu.  Numerical solution for Stokes' first problem for a heated generalized second grade fluid with fractional derivative , 2009 .

[78]  Mark M. Meerschaert,et al.  A second-order accurate numerical approximation for the fractional diffusion equation , 2006, J. Comput. Phys..

[79]  Fawang Liu,et al.  Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends , 2005 .

[80]  Chang-pin Li,et al.  Fractional derivatives in complex planes , 2009 .

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

[82]  Weihua Deng,et al.  Remarks on fractional derivatives , 2007, Appl. Math. Comput..

[83]  Zhi‐zhong Sun,et al.  A fully discrete difference scheme for a diffusion-wave system , 2006 .

[84]  I. Turner,et al.  Numerical methods for fractional partial differential equations with Riesz space fractional derivatives , 2010 .

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

[86]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[87]  Fawang Liu,et al.  Finite Difference Approximation for Two-Dimensional Time Fractional Diffusion Equation , 2007 .

[88]  CHANG-MING CHEN,et al.  Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation , 2012, Math. Comput..

[89]  Fawang Liu,et al.  A Fourier method and an extrapolation technique for Stokes' first problem for a heated generalized second grade fluid with fractional derivative , 2009 .

[90]  Fawang Liu,et al.  Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation , 2010, Numerical Algorithms.

[91]  Fawang Liu,et al.  ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation , 2008 .

[92]  Fawang Liu,et al.  Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation , 2009, Appl. Math. Comput..

[93]  Vijay P. Singh,et al.  Numerical Solution of Fractional Advection-Dispersion Equation , 2004 .

[94]  Diego A. Murio,et al.  On the stable numerical evaluation of caputo fractional derivatives , 2006, Comput. Math. Appl..

[95]  Roberto Garrappa,et al.  On Multistep Methods for Differential Equations of Fractional Order , 2006 .

[96]  Xiaohong Joe Zhou,et al.  Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. , 2008, Journal of magnetic resonance.

[97]  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..

[98]  Ercília Sousa,et al.  How to Approximate the fractional derivative of Order 1 < α ≤ 2 , 2012, Int. J. Bifurc. Chaos.

[99]  Zhaoxia Yang,et al.  Finite difference approximations for the fractional advection-diffusion equation , 2009 .

[100]  Changpin Li,et al.  Introduction to fractional integrability and differentiability , 2011 .

[101]  Yangquan Chen,et al.  Matrix approach to discrete fractional calculus II: Partial fractional differential equations , 2008, J. Comput. Phys..

[102]  Fawang Liu,et al.  Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation , 2009 .

[103]  Weihua Deng,et al.  CHAOS SYNCHRONIZATION OF FRACTIONAL-ORDER DIFFERENTIAL SYSTEMS , 2006 .

[104]  Yang Zhang,et al.  A finite difference method for fractional partial differential equation , 2009, Appl. Math. Comput..

[105]  Fawang Liu,et al.  Fractional high order methods for the nonlinear fractional ordinary differential equation , 2007 .

[106]  Hermann Brunner,et al.  Numerical simulations of 2D fractional subdiffusion problems , 2010, J. Comput. Phys..

[107]  Om P. Agrawal,et al.  Comparison of Five Numerical Schemes for Fractional Differential Equations , 2007 .

[108]  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 .

[109]  I. Turner,et al.  A fractional-order implicit difference approximation for the space-time fractional diffusion equation , 2006 .

[110]  Zaid M. Odibat,et al.  Approximations of fractional integrals and Caputo fractional derivatives , 2006, Appl. Math. Comput..

[111]  Roberto Garrappa,et al.  Explicit methods for fractional differential equations and their stability properties , 2009 .

[112]  Changpin Li,et al.  A note on the finite element method for the space-fractional advection diffusion equation , 2010, Comput. Math. Appl..

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

[114]  S. Momani,et al.  Numerical methods for nonlinear partial differential equations of fractional order , 2008 .

[115]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

[116]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

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

[118]  Lothar Gaul,et al.  On the numerical evaluation of fractional derivatives in multi-degree-of-freedom systems , 2006, Signal Process..

[119]  Fawang Liu,et al.  Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order , 2008 .

[120]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .