The Grünwald-Letnikov method for fractional differential equations

This paper is devoted to the numerical treatment of fractional differential equations. Based on the Grunwald-Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given.

[1]  N. Ford,et al.  Numerical Solution of the Bagley-Torvik Equation , 2002, BIT Numerical Mathematics.

[2]  Shyam L. Kalla,et al.  Some boundary value problems of temperature fields in oil strata , 2003, Appl. Math. Comput..

[3]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

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

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

[6]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[7]  O. Agrawal,et al.  Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than 1 , 2005 .

[8]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[9]  Pankaj Kumar,et al.  An approximate method for numerical solution of fractional differential equations , 2006, Signal Process..

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

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

[12]  L. Boyadjiev.  Some Fractional Extensions of the Temperature Field Problem in Oil Strata , 2007 .

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

[14]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

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

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

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

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

[19]  Ram K. Saxena,et al.  On the solutions of certain fractional kinetic equations , 2008, Appl. Math. Comput..

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

[21]  Diego A. Murio,et al.  Stable numerical evaluation of Grünwald–Letnikov fractional derivatives applied to a fractional IHCP , 2009 .

[22]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[23]  Shyam L. Kalla,et al.  On the Lauwerier formulation of the temperature field problem in oil strata , 2005, Int. J. Math. Math. Sci..

[24]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .