Page 116 - Mathematical Techniques of Fractional Order Systems

104 Mathematical Techniques of Fractional Order Systems

systems of differential equations are getting more attention in engineering as
well, due to the fact that their behavior is sometimes irreplaceable by com-
mon fractional or integer order dynamics. For instance, one could refer to
distributed order filters which are studied in Chapter 4 of Jiao et al. (2012).
Distributed order operators have also been successfully incorporated in PID
controllers (Zhou et al., 2013; Jakovljevic et al., 2014) leading to an
improvement of modeling of uncertainties tolerance and lead-lag compensa-
tors (Li et al., 2010). This, along with the fact that nowadays more physical
phenomena are being modeled by using distributed order operators (Lazovi´ c
et al., 2014; Saxena et al., 2014; Petrovic et al., 2015; Caputo and Carcione,
2013; Su, 2012), more attention is paid to differential equations of distrib-
uted order in the literature in the recent years. The major part of the litera-
ture however attends numerical methods for solving distributed order
differential equations (Hu et al., 2016; Ye et al., 2015; Gao et al., 2015; Li
and Wu, 2016; Gao and Sun, 2016), whereas there are less analytic
approaches towards the solution of these equations (Kochubei, 2009; Naber,
2004; Mainardi and Pagnini, 2007). Thus, the analytical solution of distrib-
uted order equations is the main concern of the present chapter, where we
focus on the case of exponential weight functions and present the exact solu-
tion of a system of linear time invariant (LTI) distributed order differential
equations associated with such weight functions. It shall be remarked that
the exact solution of a system of multiple distributed order differential
equations in the time domain has been rarely discussed before. Of course
the conventional Laplace transform approach used for the scalar case (with
a single differential equation and a single unknown variable) in the litera-
ture may be still extendable to the problem here which would eventually
lead to an integral representation of the solution through the Mellin’s
inverse formula. However in this chapter, we consider the problem as two
separate Volterra integral equations of the second type and use convolution
calculus to tackle the problem. This leads to aseriestyperepresentationof
the solution which is applicable to both scalar and matrix cases of the prob-
lem. Besides, some new results in calculation of convolution powers show
up by the unprecedented use of convolution calculus for this problem.
Moreover, as a special case, a relatively simple expression of the solution
is presented for the equations with a negative scalar as the dynamic matrix.
The expression presented here involves Gamma functions and is different
from the regular Laplace-type integral expression derived by using Laplace
transform in Naber (2004). Investigation of the role of fractional calculus
special functions in distributed order differential equations has previously
intrigued scientists seeking to construct a clearer relationship between frac-
tional calculus and its extension, i.e., distributed order calculus (Mainardi
and Pagnini, 2007). This new simple representation contributes to this
problem to some extent by providing the exact solution of a single-term
distributed order differential equation with a constant coefficient which