Page 119 - Mathematical Techniques of Fractional Order Systems
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Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 107
Let ftðÞ:R . 0 -R. We denote iterated self-convolution of function ftðÞ by
ð t
k
f ðÞ 5 f τðÞf ð k21Þ ð t 2 τÞdτ; kAN ð4:6Þ
t
0
0
t
and for the initial function we define f ðÞ 5 δ tðÞ. Definition (4.6) is some-
times referred to as the convolution power of function ftðÞ. Fractional inte-
gral operator is defined by the convolution integral (Podlubny, 1998, p. 65)
t α21
ð ð t2τÞ
α
I ftðÞ 5 fðτÞdτ
0 t ð4:7Þ
0 ΓðαÞ
The fractional derivatives considered in the present chapter is in the
Caputo sense, which is defined as (Podlubny, 1998, p. 78)
ftðÞ; α 5 0
8
<
c α 1 2 α _
I
t
0 D fðtÞ 5 0 t ftðÞ; 0 , α , 1 ð4:8Þ
: _
ftðÞ; α 5 1
Caputo also introduced the distributed order differential operator, which
is in fact defined by a weighted integration of fractional order differential
operator over the order of differentiation. This definition generalizes com-
mon fractional order derivatives and is represented by (Jiao et al., 2012,p.6)
ð 1
α
c wðαÞ xtðÞ 5 w αðÞ D xtðÞdα
c
0 D t 0 t ð4:9Þ
0
In the above definition, w α ðÞ (αA½0; 1) denotes the weight function. In
this chapter, an exponential weight function is considered, though the meth-
odology and some of the results and procedures could be applied to more
generally defined weight functions equally well. As a matter of fact, a gener-
alized calculus has emerged by introduction of definition (4.9), which is
called distributed order calculus. Prior to any attempt at investigation of the
problems that this calculus potentially has to offer, it is needed to review dis-
tributed order integrals and derivatives properties more closely. In order to
start, let us write (4.9) in the Laplace domain as
ð 1N ð 1N
c wðαÞ α α21 x 0ðÞdα
L t-s 0 D t xtðÞ 5 ^ w αðÞs XsðÞdα 2 ^ w αðÞs ð4:10Þ
0 0
where L t-s :fg is the Laplace transform operator, XsðÞ is the Laplace trans-
form of xtðÞ, and
^ w αðÞ 5 w αðÞ αA½0; 1 ð4:11Þ
0; α=2½0; 1
Rewriting (4.10) in the form
1N 1N
ð ð
c wðαÞ αlnðsÞ αlnðsÞ
xð0Þ
L t-s 0 D xðtÞ 5 ^ wðαÞe XðsÞdα 2 ^ wðαÞe dα ð4:12Þ
t
0 s 0
