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Discretization of Fractional Differential Equations by a Piecewise Constant Approximation
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
Consistency Conditions for Fundamentally Discrete Theories
The dynamics of physical theories is usually described by differential
equations. Difference equations then appear mainly as an approximation which
can be used for a numerical analysis. As such, they have to fulfill certain
conditions to ensure that the numerical solutions can reliably be used as
approximations to solutions of the differential equation. There are, however,
also systems where a difference equation is deemed to be fundamental, mainly in
the context of quantum gravity. Since difference equations in general are
harder to solve analytically than differential equations, it can be helpful to
introduce an approximating differential equation as a continuum approximation.
In this paper implications of this change in view point are analyzed to derive
the conditions that the difference equation should satisfy. The difference
equation in such a situation cannot be chosen freely but must be derived from a
fundamental theory. Thus, the conditions for a discrete formulation can be
translated into conditions for acceptable quantizations. In the main example,
loop quantum cosmology, we show that the conditions are restrictive and serve
as a selection criterion among possible quantization choices.Comment: 33 page
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