Discretization of fractional differential equations by a piecewise constant approximation

¹ School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia
² School of Computer Science and Applied Mathematics, University of the Witswatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa
³ DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

b.henry@unsw.edu.au

Received: 22 October 2017
Accepted: 23 October 2017

Abstract

There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation.