Page 127 - Mathematical Techniques of Fractional Order Systems

Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 115

k k k 1 1
It can be shown that the relation 1 5 holds
i 2 1 i i
true. By using this identity we obtain
s a11 Ð 1N a ð k11 2st dt 5
e
t lntÞ
0
k
Γ ðk11Þ ð a 1 1Þ 1 X k 1 1 Γ ð a 1 1Þ 2lnsÞ k2i11 1
ðiÞ
ð
i
i51
k11 ð4:61Þ
ð
Γ a 1 1Þ 2lnsÞ 5
ð
k11
X k 1 1 k2i11
ðiÞ
ð
Γ ð a 1 1Þ 2lnsÞ
i
i50
This indicates that (4.59) holds true for k 1 1 too, which in turn proves
the induction. Setting a 5 0 in expression (4.59) and a little calculation
results in the following formula
k
X
i
k 21Þ Γ ðk2iÞ i
k
ð
L t-s lntð Þ 5 ð 1 ðÞ lnsÞ =s ð4:62Þ
i
i50
or equivalently
k
X
k i i
k
1
ðkÞ
L t-s lntð Þ 2 Γ ðÞ=s 5 ð 21Þ Γ ðk2iÞ 1 ðÞ lnsÞ =s ð4:63Þ
ð
i
i51
for calculation of the Laplace transform of the logarithmic functions with
k k
integer powers. By defining y k 5 ð lnsÞ and c k 5 L t-s lntÞ 2 Γ ðÞ=s for
ðkÞ
1
ð
s
k 5 1; 2;...; N it is possible to write (4.63) in the form of a system of linear
algebraic equations as
c 5 Ey ð4:64Þ
T T

in which y 5 y 1 y 2 ? y N , c 5 c 1 c 2 ? c N and E is
k
defined by (4.55). Note that since e kk 521ð Þ , matrix E is always nonsingu-
lar which makes it possible to uniquely solve (4.64) for y to give
21
y 5 E c ð4:65Þ
From which the relation (4.56) is concluded.
Now Lemma 6 will be used in the next Lemma to obtain p k tðÞ in explicit
form. On the basis of this result, we would be prepared to rewrite the solu-
tion of (4.17) accordingly.
Lemma 7: Let p k tðÞ and E be defined as in (4.52) and (4.55), respectively.
We have
k ð t
X b k;i τ k21 i
p k tðÞ 5 e τ ln t 2 τÞ 2 Γ ðÞ dτ ð4:66Þ
ðiÞ
1
ð
i51 0 ð k 2 1Þ!

where b k;i denote the elements of matrix E 21 5 b k;i .