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Fractional dynamics of tracer transport in fractured media from local to regional scales

100%
Zhang Y., Reeves D., Pohlmann K., Chapman J., Russell C.
Open Physics
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2013
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vol. 11
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issue 6
634-645
EN
Tracer transport through fractured media exhibits concurrent direction-dependent super-diffusive spreading along high-permeability fractures and sub-diffusion caused by mass transfer between fractures and the rock matrix. The resultant complex dynamics challenge the applicability of conventional physical models based on Fick’s law. This study proposes a multi-scaling tempered fractional-derivative (TFD) model to explore fractional dynamics for tracer transport in fractured media. Applications show that the TFD model can capture anomalous transport observed in small-scale single fractures, intermediate-scale fractured aquifers, and two-dimensional large-scale discrete fracture networks. Tracer transport in fractured media from local (0.255-meter long) to regional (400-meter long) scales therefore can be quantified by a general fractional-derivative model. Fractional dynamics in fractured media can be scale dependent, owning to 1) the finite length of fractures that constrains the large displacement of tracers, and 2) the increasing mass exchange capacity along the travel path that enhances sub-diffusion.
2
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Fractional order modelling of zero length column desorption response for adsorbents with variable particle sizes

100%
Machado J., Zaman S., Baleanu D.
Open Physics
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2013
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vol. 11
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issue 6
881-885
EN
This manuscript analyses the data generated by a Zero Length Column (ZLC) diffusion experimental set-up, for 1,3 Di-isopropyl benzene in a 100% alumina matrix with variable particle size. The time evolution of the phenomena resembles those of fractional order systems, namely those with a fast initial transient followed by long and slow tails. The experimental measurements are best fitted with the Harris model revealing a power law behavior.
3
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Analysis of football player’s motion in view of fractional calculus

100%
Couceiro M., Clemente F., Martins F.
Open Physics
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2013
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vol. 11
|
issue 6
714-723
EN
Accurately retrieving the position of football players over time may lay the foundations for a whole series of possible new performance metrics for coaches and assistants. Despite the recent developments of automatic tracking systems, the misclassification problem (i.e., misleading a given player by another) still exists and requires human operators as final evaluators. This paper proposes an adaptive fractional calculus (FC) approach to improve the accuracy of tracking methods by estimating the position of players based on their trajectory so far. One half-time of an official football match was used to evaluate the accuracy of the proposed approach under different sampling periods of 250, 500 and 1000 ms. Moreover, the performance of the FC approach was compared with position-based and velocity-based methods. The experimental evaluation shows that the FC method presents a high classification accuracy for small sampling periods. Such results suggest that fractional dynamics may fit the trajectory of football players, thus being useful to increase the autonomy of tracking systems.
4
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Time-fractional extensions of the Liouville and Zwanzig equations

88%
Lukashchuk S.
Open Physics
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2013
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vol. 11
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issue 6
740-749
EN
This paper presents extensions to the classical stochastic Liouville equation of motion that contain the Riemann-Liouville and Caputo time-fractional derivatives. At first, the dynamic equations with the time-fractional derivatives are formally obtained from the classical Liouville equation. A feature of these new equations is that they have the same common formal solution as the classical Liouville equation and therefore may be used for study of the Hamiltonian system dynamics. Two cases of the time-dependent and time-independent Hamiltonian are considered separately. Then, the time-fractional Liouville equations are deduced from the short- and long-time asymptotic expansions of the obtained dynamic equations. The physical meaning of the resulting equations is discussed. The statements of the Cauchy-type problems for the derived time-fractional Liouville equations are given, and the formal solutions of these problems are presented. At last, the projection operator formalism is employed to derive the time-fractional extensions of the Zwanzig kinetic equations and the corresponding formal statistical operators from the time-fractional Liouville equations.
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