Page 12 - Mathematical Techniques of Fractional Order Systems

2 Mathematical Techniques of Fractional Order Systems

fractional calculus has seen a rapid growth in its applications. As it can be
said that real objects are generally fractional, even if that fractionality is
very low (Petra ´s, 2009), fractional calculus is particularly well suited to
describe the nonlinear relationship with time of anomalous diffusion
(Margin, 2006). Indeed, noninteger derivatives occur most frequently, and
naturally, in physical problems where the essential mechanisms, reactions, or
interactions are governed by diffusion processes. Of course, seeing that
many biological processes often present anomalous diffusion, fractional cal-
culus is an eligible and powerful tool to describe these phenomena (e.g., sub-
threshold nerve conduction, viscoelasticity, bioelectrodes) (Magin, 2006).
But diffusion, being one of the most relevant applications of fractional calcu-
lus, is far from being the only one. Fractional derivatives are used to formu-
late and solve different physical models allowing a continuous transition
from relaxation to oscillation phenomena; to predict the nonlinear survival
and growth curves of food-borne pathogens; to adapt the viscoelasticity
equations (Hooke’s Law and the Newtonian fluids Law) (Rahimy, 2010;
Petra ´s, 2009); fractional control plays an important role in general physics,
thermodynamics, electrical circuits theory and fractances, mechatronic sys-
tems, signal processing, chemical mixing, chaos theory, and many others
(Petra ´s, 2009). And there are many possible applications of fractional deriva-
tives in control (Vale ´rio and Sa ´ da Costa, 2006, 2011a, 2012; Azar et al.,
2017).
Many physical processes, however, also appear to exhibit a fractional
order behavior that varies with time or space (Lorenzo and Hartley,
2002b). In what concerns the field of viscoelasticity of certain materials,
the temperature effect in small amplitude strains is known to induce
changes from an elastic to viscoelastic/viscous behavior, where real appli-
cations may require a time varying temperature to be analyzed. The relax-
ation processes and reaction kinetics of proteins, which are described by
fractional differential equations, have been found to have an order with a
temperature dependence. The behavior of some diffusion processes in
response to temperature changes can be better described using variable
order elements rather than time-varying coefficients, among other cases
(Lorenzo and Hartley, 2002b). These are natural applications where vari-
able order operators as a function of time (t)orsomeother variable (x)
can be introduced with profit and in a very natural way (Lorenzo and
Hartley, 2002b; Vale ´rio and Sa ´ da Costa, 2013)
This chapter presents one such application of variable order derivatives:
the development of simpler mathematical models of bone remodeling, both
for healthy bone, and for bone tissue affected by cancer. There are several
models published in the literature, but the novelty which is the use of vari-
able order derivatives will result in simpler models that are easier to
understand.