## 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.

Original language | English |
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Pages (from-to) | 23-36 |

Number of pages | 14 |

Journal | Mathematical Modelling of Natural Phenomena |

Volume | 12 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |

## Keywords

- Caputo derivatives
- Discretization
- Fractional differential equations
- Integrablization

## ASJC Scopus subject areas

- Modeling and Simulation