Page 129 - Mathematical Techniques of Fractional Order Systems
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Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 117
Remark 1: Let E be defined as in (4.55). From (4.52), it is revealed that
k 2t d
E ðÞ 5 e p k tðÞ. Thus, according to Lemma 7, it is deduced that
t
1 dt
k ð t
i
k
τ
E ðÞ 5 X b k;i e τ2t k22 ð τ 1 k 2 1Þln t 2 τÞdτ 2 t k21 Γ ðÞ
1
t
ðiÞ
ð
1 k 2 1Þ!
i51 ð 0
ð4:72Þ
where k $ 2 and b k;i are the elements of matrix E 21 5 b k;i .
The results of the previous Lemmas are now gathered to obtain the exact
solution of (4.17) in the time domain as stated in the following theorem.
2π21
Theorem 1: Assume :A: .2 1=lW 21 2e holds where lW 21 :ðÞ
denotes the lower branch of the Lambert W function. Then the exact solution
of (4.17) with a unitary weight function w αðÞ 5 1;αA 0; 1 is given by
½
1N k ð t
XX
i
k τ k21
xtðÞ 5 c k;i A e τ ln t 2 τÞ 2 Γ ðÞ dτx 0ðÞ
ðiÞ
1
ð
k50 i51 0
! ð4:73Þ
1N k ð t
XX
e τ
i
1 c k;i A k21 τ k21 ln t 2 τÞ 2 Γ ðÞ dτ TBu tðÞ
0
ðiÞ
1
ð
k51 i51 0
in which
b k;i
c k;i 5 ð4:74Þ
ð k 2 1Þ!
Coefficients b k;i are the elements of the matrix E 21 5 b k;i in which E is
given by (4.55) and u tðÞ is the derivative of utðÞ defined by (4.44).
0
Proof: Substituting p k tðÞ obtained from Lemma 7 in (4.53) and (4.54)
gives φ tðÞ and φ tðÞ, respectively. Solution (4.74) is deduced by replacing
2
1
these values in (4.41).
Theorem 1 can be immediately extended to the case in which the weight
function w αðÞ is of exponential type. This is stated in the following
corollary.
2π21
Corollary 1: Assume :A: .2 1=lW 21 2e holds where lW 21 :ðÞ
denotes the lower branch of Lambert W function. Then the exact solution of
α
system (4.17) with exponential weight function w αðÞ 5 ca ; αA 0; 1 in
½
which aAR . 0 and cAR 2 0fg is given by
1N k ð t=a τ
k
XX e i
xtðÞ 5 c k;i Aτ=c τ ln t=a 2 τ 2 Γ ðÞ dτx 0ðÞ
ðiÞ
1
k50 i51 0
ð4:75Þ
0 1
1N k ð t=a
XX c k;i k21 τ i
1 @ Aτ=c e ln t=a 2 τ 2 Γ ðÞ dτ TBu tðÞ
1
0
ðiÞ
A
k51 i51 0 c
0
in which c k;i is defined as (4.74) and u tðÞ is defined by (4.44).
