Abstract
Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.
Highlights
A confusion or rather an implicit link exists between fractional behaviors and fractional models [1]
This infinite dimension or these infinitely large time constants are responsible for the infinite memory of fractional models
Fractional operators and the resulting fractional models are known for their memory generates power-law behaviors
Summary
A confusion or rather an implicit link exists between fractional behaviors (of physical, biological, thermal, etc., origin) and fractional models (a tool to model fractional behaviors) [1]. Fractional calculus has become a common modeling tool in many fields [4,5,6,7] This link between power-law behaviors and fractional models often found in the literature does not result from physical justification, as noted in the present paper. The infinite spatial dimension is the conclusion reached in this work after some mathematical transformations that lead to a representation of a fractional model by a diffusion equation. The diffusive representation obtained exhibits infinitely large time constants that are a consequence of the definition of fractional models in an infinite spatial domain. This infinite dimension or these infinitely large time constants are responsible for the infinite memory of fractional models. A list of alternative solutions to fractional models to produce power-law behaviors is given
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