Discretization of fractional differential equations by a piecewise constant approximation

C. N. Angstmann, B. I. Henry, B. A. Jacobs, A. V. McGann

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


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 languageEnglish
Pages (from-to)23-36
Number of pages14
JournalMathematical Modelling of Natural Phenomena
Issue number6
Publication statusPublished - 2017
Externally publishedYes


  • Caputo derivatives
  • Discretization
  • Fractional differential equations
  • Integrablization

ASJC Scopus subject areas

  • Modeling and Simulation


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