Fractional Adams-Moulton methods

In the simulation of dynamical systems exhibiting an ultraslow decay, differential equations of fractional order have been successfully proposed. In this paper we consider the problem of numerically solving fractional differential equations by means of a generalization of k-step Adams-Moulton multistep methods. Our investigation is focused on stability properties and we determine intervals for the fractional order for which methods are at least A(@p/2)-stable. Moreover we prove the A-stable character of k-step methods for k=0 and k=1.

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

[2]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

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

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

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

[6]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

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

[8]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

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

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

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

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

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

[14]  Syvert P. Norsett.  A criterion forA(α)-stability of linear multistep methods , 1969 .

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