2016/05/06 by Christopher N. Angstmann, Angstmann, Christopher N, B. I. Henry +3
Engineering · Mathematics · #Advanced Control Systems Design #Computational Physics (physics.comp-ph) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical and Theoretical Analysis #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1605.01815
openalex publication_date 2016/05/06 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
There has recently been considerable interest in using a nonstandard\npiecewise approximation to formulate fractional order differential equations as\ndifference equations that describe the same dynamical behaviour and are more\namenable to a dynamical systems analysis. Unfortunately, due to mistakes in the\nfundamental papers, the difference equations formulated through this process do\nnot capture the dynamics of the fractional order equations. We show that the\ncorrect application of this nonstandard piecewise approximation leads to a one\nparameter family of fractional order differential equations that converges to\nthe original equation as the parameter tends to zero. A closed formed solution\nexists for each member of this family and leads to the formulation of a\ndifference equation that is of increasing order as time steps are taken. Whilst\nthis does not lead to a simplified dynamical analysis it does lead to a\nnumerical method for solving the fractional order differential equation. The\nmethod is shown to be equivalent to a quadrature based method, despite the fact\nthat it has not been derived from a quadrature. The method can be implemented\nwith non-uniform time steps. An example is provided showing that the difference\nequation can correctly capture the dynamics of the underlying fractional\ndifferential equation.\n