Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 293, Issue -, Pages 14-28Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2014.04.024
Keywords
Fractional calculus; Anomalous diffusion; Random walk
Funding
- NSF [DMS-1025486]
- Fujian Governmental Scholarship
- National Natural Science Foundation of China [11101344]
- Directorate For Geosciences
- Division Of Earth Sciences [1344280] Funding Source: National Science Foundation
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Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series. (C) 2014 Elsevier Inc. All rights reserved.
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