Information Technology Reference
In-Depth Information
properties of migraine pathology through the regulation of cerebral blood flow, as well
as the weak multifractality of walking.
The fractional calculus has also been shown to describe the heterogeneity of ran-
dom fields and the propagation of probability densities in inhomogeneous anisotropic
physical webs. The CTRW of Montroll and Weiss has experienced a renaissance in
the modern explanation of the behavior of complex networks. In particular, the scal-
ing of phenomena in space and time has been shown to be a consequence of fractional
diffusion equations, one of the more popular solutions of which is the Lévy distribution.
5.6
Problems
5.1 Generalized Weierstrass Function
The fractional derivatives of the GWF can yield deep insight into the utility of the
fractional calculus for understanding complex phenomena and consequently it is worth-
while going through the algebra of the analysis at least once. Carry out the integration
indicated by (
5.35
) explicitly to derive (
5.36
), as well as the integration indicated by
(
5.37
) to derive (
5.38
). The details of this analysis as well as additional discussion are
given in [
22
].
5.2 Scaling of the second moment
Complete the discussion of the second moment of the solution to the fractional Langevin
equation (
5.72
) by evaluating the integral for the variance (
5.76
) using the delta-
correlated property of the Gaussian random force. If this proves to be too difficult, carry
out the calculation for the
0 case, but do it for the autocorrelation function rather
than the second moment. If you really want to impress your professor, do both.
λ
=
5.3 The fractal autocorrelation function
Of course, the solution (
5.111
) when measured is often too erratic to be useful with-
out processing. The quantity typically constructed in a geophysical context is the
autocorrelation function or covariance function
x
)
=
Y
x
)
ξ
.
Construct an analytic expression for this function on the basis of the above problem
assuming the statistical properties (
5.104
). Note that there is a simple way to do this and
one that is not so simple.
C
(
x
−
(
x
)
Y
(
References
[1] E. Bacry, J. Delour and J. F. Muzy, “Multifractal random walk,”
Phys.Rev.E
64
, 026103-1
(2001).
[2] J. B. Bassingthwaighte, L. S. Liebovitch and B. J. West,
Fractal Physiology
, Oxford: Oxford
University Press (1994).
[3] K. V. Bury,
Statistical Models in Applied Science
, New York: John Wiley (1975).
