Page 131 - Mathematical Techniques of Fractional Order Systems
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Exact Solution of Linear Fractional Distributed Order Systems Chapter | 4 119
exponential weight function. However, it is possible to obtain a simpler
expression for the solution of (4.82) which is done in the next Theorem.
This system describes anomalous nonexponential distributed order relaxa-
tion processes (Kochubei, 2009). Thetypicalmethodusedto treat (4.82) in
the literature is through the Laplace transform. In this method, the solution
is first derived in the Laplace domain. Then the Mellin’s inverse formula
is utilized to return the obtained solution back to time domain, which
involves solving a complex integral (Naber, 2004). This approach leads to
a solution expressed by a Laplace-type integral eventually (Naber, 2004;
Mainardi and Pagnini, 2007). For instance, in case of a unitary weight func-
tion, substituting w αðÞ 5 1; αA 0; 1 in the solution obtained by (Naber,
½
2004), gives
e λ r 1 1Þ
ð 1N 2rt
ð
xtðÞ 5 x 0ðÞ dr ð4:83Þ
2 2 2
0 rλ π 1 rr112λlnrð Þ
In the following Theorem we present an alternative representation of this
solution expressed in terms of Gamma functions, which may be more coher-
ent with the concept of distributed order differential equations as the general-
ization of the fractional order counterparts.
Exact solution of (4.82) under the assumption
Theorem 2:
w αðÞ 5 1; αA 0; 1 is given by
½
t
ð λτ21 λτ
ð t2τÞ ð t2τÞ τ
xtðÞ 5 x 0ðÞ 2 e dτ; t . 0 ð4:84Þ
ð
ð
0 ΓλτÞ Γλτ 1 1Þ
Proof: Let us define the auxiliary function
ð t λτ
τ
vtðÞ 5 ð t2τÞ e dτ ð4:85Þ
ð
0 Γλτ 1 1Þ
In terms of (4.85) we will derive the actual solution of (4.82) as follows.
At first, note that it is possible to write (4.85) in the form
ð 1N λτ
τ
vtðÞ 5 ð t2τÞ e Ht 2 τð Þdτ ð4:86Þ
ð
0 Γλτ 1 1Þ
in which H :ðÞ is the Heaviside unit step function. Using the variable change
τ 52 lnα in the integral (4.86) gives
ð 1 2λlnα
ð t1lnαÞ 22lnα
vtðÞ 5 e Ht 1 lnαÞdα ð4:87Þ
ð
ð
0 Γ 2λlnα 1 1Þ
