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so that evaluating the integral on the rhs yields
∞
∞
k
k
α
(
−
1
)
λ
(
1
+
k
α)
e
−
ut
E
)
α
)
α
(
−
(λ
t
dt
=
u
k
α
+
1
(
1
+
k
α)
0
k
=
0
k
u
k
α
∞
u
α
−
1
λ
α
+
1
u
=
0
(
−
1
)
=
u
α
.
k
=
Note that when
α
=
1 this expression is just the Laplace transform for the exponential
and, indeed,
)
α
)
α
=
1
=
e
−
λ
t
E
α
(
−
(λ
t
(5.47)
and consequently the MLF is considered a generalization of the exponential function.
This limit of the MLF is consistent with the fractional relaxation equation reducing to
the ordinary relaxation equation in this limit since the inhomogeneous term in (
5.43
)
vanishes at
.
The MLF has been used to model the stress relaxation of materials for over a century
and has a number of interesting properties in both the short-time and the long-time
limits. In the short-time limit it yields the Kohlrausch-Williams-Watts law for stress
relaxation in rheology given by
α
=
1
e
−
(λ
t
)
α
)
α
)
=
lim
t
→
E
α
(
−
(λ
t
,
(5.48)
0
which is also known as the stretched exponential. This name is a consequence of the
fact that the fractional index is less than unity, so the exponential relaxation is stretched
out in time, with the relaxation being slower than ordinary exponential relaxation for
α<
1. In a probability context the MLF is known as the Weibull distribution and is
used to model crack growth within materials, as mentioned previously.
In the long-time limit the MLF yields an inverse power law, known as the Nutting law,
1
)
α
)
=
lim
E
α
(
−
(λ
t
)
α
.
(5.49)
(λ
t
t
→∞
Consequently, the initial relaxation for the MLF is slower than the ordinary exponential,
but asymptotically in time the relaxation slows down considerably, becoming an inverse
power law. Here again, the long-time limiting form of the MLF is the well-known proba-
bility distribution observed for multiple phenomena in the first few chapters. We discuss
the probability implications of this solution later.
Figure
5.1
displays the general MLF as well as its two asymptotes; the dashed curve is
the stretched exponential and the dotted curve depicts the inverse power law. It is appar-
ent from the discussion that there is a long-time memory associated with the fractional
relaxation process characterized by the inverse power law. This asymptotic behavior
replaces the short-time memory of the exponential relaxation in ordinary relaxation.
Moreover, it is apparent that the MLF smoothly joins the two empirically determined
asymptotic distributions of Kohlrausch-Williams-Watts and Nutting.
Magin [
14
] gives an excellent discussion of the properties of the MLF in both the
time and the Laplace-transform domains. Moreover, he discusses the solutions to the
