Discretization of Fractional Differential Equations by a Piecewise Constant Approximation

TL;DR

This paper corrects previous errors in discretizing fractional differential equations using a piecewise constant approximation, presenting a convergent numerical method that accurately captures the dynamics and can be implemented with non-uniform time steps.

Contribution

It provides a corrected formulation of the piecewise approximation for fractional differential equations, establishing convergence and a practical numerical method.

Findings

The method converges to the original fractional differential equation as the parameter tends to zero.

The derived difference equations can accurately capture the dynamics of the fractional equations.

The numerical method is equivalent to a quadrature-based approach and supports non-uniform time steps.

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. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. 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.