  Fractional calculus has been used to describe physical systems with
complexity. Here, we show that a fractional calculus approach can restore or
include complexity in any physical systems that can be described by partial
differential equations. We argue that the dispersion relation contains the
required information relating the energy and momentum space of the system and
thus fully describes their dynamics. The approach is demonstrated by two
examples: the Landau-Lifshitz equation in a 1D ferromagnetic chain, an example
of a periodic crystal system with a bounded dispersion relation; and a modified
KdV equation supporting surface gravity waves or Euler dispersion, an example
of an unbounded system in momentum space. The presented approach is applicable
to fluids, soft matter, and solid-state matter and can be readily generalized
to higher dimensions and more complex systems. While numerical calculations are
needed to determine the fractional operator, the approach is analytical and can
be utilized to determine analytical solutions and investigate nonlinear
problems.
