Page 128 - Mathematical Techniques of Fractional Order Systems

116 Mathematical Techniques of Fractional Order Systems

Proof: First of all, it is possible to write (4.66) in the form

k ð 1N
X b k;i
τ k21
i
ðiÞ
p k tðÞ 5 k 2 1Þ! e τ ln t 2 τÞ 2 Γ ðÞ Ht 2 τÞdτ ð4:67Þ
1
ð
ð
i51 0 ð
in which H :ðÞ denotes the Heaviside unit step function. Taking the Laplace
transform of (4.67) and using the time shift property of the Laplace trans-
form yields
k ð 1N
τ k21 2sτ
X b k;i
i
P k sðÞ 5 k 2 1Þ! e τ e L t-s ln tðÞ 2 Γ ðÞ HtðÞ dτ
ðiÞ
1
i51 0 ð
ð4:68Þ
k
1N 1 τ k21 2sτ X i
5 e τ e b k;i L t-s ln tðÞ 2 Γ ðÞ=s dτ
Ð
1
ðiÞ
0 k 2 1Þ!
ð
i51
Using Lemma 6 gives
τ k21
ð 1N 1 k k ð 1N e τ
τ k21 2sτ
P k sðÞ 5 e τ e ð lnsÞ dτ 5 ð lnsÞ e 2sτ dτ
0 ð k 2 1Þ! s s 0 ð k 2 1Þ!
ð4:69Þ
Note that the integral in the right side of (4.69) is actually the Laplace
n o
e τ
τ k21
transform L τ-s k 2 1Þ! . Therefore, using Lemma 2 gives
ð
k
ð lnsÞ
P k sðÞ 5 ð4:70Þ
ss21Þ k
ð
On the other hand, note that taking the Laplace transform of p k tðÞ by
directly using its definition (i.e., (4.52)) would result
Ð t τ k τ
L t-s e E ðÞdτ 5
1
0
k
1 k 1 k ln sðÞ ð4:71Þ
t
L t-s E ðÞ 5 L t-s E 1 tðÞ 5
1
s js-s21 s js-s21 ss21Þ k
ð
By comparing (4.70) and (4.71), the proof of this Lemma is
completed.
Prior to proceeding, it is worth mentioning that calculation of convolu-
tions involving exponential integral functions are of interest in some papers
(Fisher and Al-Sirehy, 2015; Geller and Ng, 1969). This problem is treated
in the following Remark as a side result of Lemma 7, which concerns the
k-fold convolution of exponential integral function with itself. This gener-
alizes the work done in Geller and Ng (1969).