On some explicit Adams multistep methods for fractional differential equations

In this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order. The proposed methods are obtained by modifying, in a suitable way, Fractional-Adams-Moulton methods and they represent a way for extending classical Adams-Bashforth multistep methods to the fractional case. The attention is hence focused on the investigation of stability properties. Intervals of stability for k-step methods, k=1,...,5, are computed and plots of stability regions in the complex plane are presented.

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

[2]  C. Lubich,et al.  Runge-Kutta theory for Volterra and Abel integral equations of the second kind , 1983 .

[3]  I. Podlubny.  Fractional differential equations , 1998 .

[4]  Neville J. Ford,et al.  Comparison of numerical methods for fractional differential equations , 2006 .

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

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

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

[8]  C. Lubich,et al.  On the Stability of Linear Multistep Methods for Volterra Convolution Equations , 1983 .

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

[10]  C. Lubich,et al.  Fractional linear multistep methods for Abel-Volterra integral equations of the second kind , 1985 .

[11]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[12]  W. Ames.  Mathematics in Science and Engineering , 1999 .

[13]  Peter Henrici,et al.  Automatic Computations with Power Series , 1956, JACM.

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

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

[16]  Roberto Garrappa,et al.  Fractional Adams-Moulton methods , 2008, Math. Comput. Simul..

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

[18]  P. Wolkenfelt,et al.  The Construction of Reducible Quadrature Rules for Volterra Integral and Integro-differential Equations , 1982 .

[19]  C. Lubich,et al.  A Stability Analysis of Convolution Quadraturea for Abel-Volterra Integral Equations , 1986 .

[20]  Christian Lubich,et al.  Fast and Oblivious Convolution Quadrature , 2006, SIAM J. Sci. Comput..

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