Abstract
We investigate for the diffusion equation the differences manifested by the solutions when three different types of spatial differential operators of noninteger (or fractional) order are considered for a limited and unlimited region. In all cases, we verify an anomalous spreading of the system, which can be connected to a rich class of anomalous diffusion processes.
Highlights
The fractional calculus represents an important tool which has been successfully applied to several contexts (DAS; MAHARATNA, 2013; DEBNATH, 2003; MACHADO et al, 2014; GLÖCKLE; NONNENMACHER, 1995; HILFER, 2000; SHLESINGER et al, 1994)
There is more than one definition of the fractional differential operators which have been used to investigate these situations
Let us start our analysis about the differences of these operators by investigating the behavior of the solutions when these fractional differential operators are incorporated in the diffusive term
Summary
The fractional calculus represents an important tool which has been successfully applied to several contexts (DAS; MAHARATNA, 2013; DEBNATH, 2003; MACHADO et al, 2014; GLÖCKLE; NONNENMACHER, 1995; HILFER, 2000; SHLESINGER et al, 1994). The last point has received much attention since that the usual approach (RISKEN, 1989; GARDINER, 2009) does not provide a suitable description of the experimental results and, requires extensions. In this sense, by using fractional calculus, the diffusion equation (or Fokker - Planck equation) and the Langevin equation have been extended and, used to investigate several situations such as the ones present in Refs. There is more than one definition of the fractional (or noninteger) differential operators which have been used to investigate these situations.
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