On an explicit finite difference method for fractional diffusion equations

A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Fick's law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov definition of the fractional derivative operator to obtain an explicit fractional FTCS scheme for solving the fractional diffusion equation. The resulting method is amenable to a stability analysis a la von Neumann. We show that the analytical stability bounds are in excellent agreement with numerical tests. Comparison between exact analytical solutions and numerical predictions are made.Comment: 22 pages, 6 figure

Stabilized leachates: sequential coagulation–flocculation + chemical oxidation process

The combined sedimentation-chemical oxidation treatment of medium-stabilized landfill leachates has been investigated. The sequence of stages implemented was: (a) coagulation–flocculation by pH decrease (pH 2) to acidic conditions (COD removal ≈ 25% related to COD0 ≈ 7500 ppm); (b) coagulation–flocculation by Fe(III) addition (0.01 M) at pH 3.5 (COD removal ≈ 40% related to COD of supernatant after step (a); (c) Fenton (Fe(III) = 0.01 M; H2O2 = 1.0 M) oxidation (COD removal ≈ 80% related to COD of supernatant after step (a); and (d) coagulation–flocculation of Fenton’s effluent at pH 3.5 (COD removal ≈ 90% related to COD of supernatant after step (a). The use of Kynch theory allows for the design of clarifiers based on the amount of solids fed. For a general example of 1000 m3 day−1 of a feeding stream, clarifier area values of 286, 111 and 231 m2 were calculated for compacting indices of 3.7, 2.67 and 2.83 corresponding to the first, second and third consecutive sedimentation processes, respectively, (steps (a), (b) and (d))

Stabilized leachates: Ozone-activated carbon treatment and kinetics

Ozone has been used as a pre-oxidation step for the treatment of stabilized leachates. Given the refractory nature of this type of effluents, the conversion of some wastewater quality parameters has been moderate after 1 h of ozonation (i.e. 30% chemical oxygen demand (COD) depletion). Ozone uptake was calculated in the interval 1.3–1:5 g of ozone per gram of COD degraded. An optimum dose of ozone has been experienced in terms of biodegradability of the processed effluent (60 min of treatment, 1 103 mol L1 ozone inlet feeding concentration and 50 L h1 gas flowrate). pH and other typical hydroxyl radical generator systems exerted no influence on the efficiency of the process, suggesting the negligible role played by the indirect route of oxidation (generation of hydroxyl radicals). The ozonated effluent was thereafter treated in a second adsorption stage by using a commercial activated carbon. Removal levels up to 90% of COD in approximately 120 h were experienced for adsorbent dosages of 30 g L1 : Both steps, the single ozonation and the adsorption stage have been modelled by using different pseudoempirical models. r 2003 Elsevier Ltd. All rights reserved

Survival probability and order statistics of diffusion on disordered media

We investigate the first passage time t_{j,N} to a given chemical or Euclidean distance of the first j of a set of N>>1 independent random walkers all initially placed on a site of a disordered medium. To solve this order-statistics problem we assume that, for short times, the survival probability (the probability that a single random walker is not absorbed by a hyperspherical surface during some time interval) decays for disordered media in the same way as for Euclidean and some class of deterministic fractal lattices. This conjecture is checked by simulation on the incipient percolation aggregate embedded in two dimensions. Arbitrary moments of t_{j,N} are expressed in terms of an asymptotic series in powers of 1/ln N which is formally identical to those found for Euclidean and (some class of) deterministic fractal lattices. The agreement of the asymptotic expressions with simulation results for the two-dimensional percolation aggregate is good when the boundary is defined in terms of the chemical distance. The agreement worsens slightly when the Euclidean distance is used.Comment: 8 pages including 9 figure

The flyby anomaly: a multivariate analysis approach

[EN] The flyby anomaly is the unexpected variation of the asymptotic post-encounter velocity of a spacecraft with respect to the pre-encounter velocity as it performs a slingshot manoeuvre. This effect has been detected in, at least, six flybys of the Earth but it has not appeared in other recent flybys. In order to find a pattern in these, apparently contradictory, data several phenomenological formulas have been proposed but all have failed to predict a new result in agreement with the observations. In this paper we use a multivariate dimensional analysis approach to propose a fitting of the data in terms of the local parameters at perigee, as it would occur if this anomaly comes from an unknown fifth force with latitude dependence. Under this assumption, we estimate the range of this force around 300 km .Acedo Rodríguez, L. (2017). The flyby anomaly: a multivariate analysis approach. Astrophysics and Space Science. 362(2):1-7. doi:10.1007/s10509-017-3025-zS173622Acedo, L.: Adv. 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Modal Series Expansions for Plane Gravitational Waves

[EN] Propagation of gravitational disturbances at the speed of light is one of the key predictions of the General Theory of Relativity. This result is now backed indirectly by the observations of the behavior of the ephemeris of binary pulsar systems. These new results have increased the interest in the mathematical theory of gravitational waves in the last decades, and severalmathematical approaches have been developed for a better understanding of the solutions. In this paper we develop a modal series expansion technique in which solutions can be built for plane waves from a seed integrable function. The convergence of these series is proven by the Raabe-Duhamel criteria, and we show that these solutions are characterized by a well-defined and finite curvature tensor and also a finite energy content.Acedo Rodríguez, L. (2016). Modal Series Expansions for Plane Gravitational Waves. Gravitation and Cosmology. 22(3):251-257. doi:10.1134/S0202289316030026S251257223A. Einstein and N. Rosen, Journal of the Franklin Institute 223, 43–54 (1937).N. Rosen, Gen. Rel. Grav. 10, 351–364 (1979).C. Sivaram, Bull. Astr. Soc. India 23, 77–83 (1995).J. M. Weisberg, D. J. Nice, and J. H. Taylor, Astroph. J. 722, 1030–1034(2010); arXiv: 1011.0718.B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).J. B. Griffiths, Colliding waves in general relativity (Clarendon, Oxford, 1991).S. Chandrasekhar, The mathematical theory of black holes (Clarendon, Oxford, 1983).D. Bini, V. Ferrari and J. Ibañez, Nuovo Cim. B 103, 29–44 (1989).L. Acedo, G. González-Parra, and A. J. Arenas, Nonlinear Analysis: Real World Applications 11, 1819–1825 (2010).L. Acedo, G. González-Parra, and A. J. Arenas, Physica A 389, 1151–1157 (2010).G. González-Parra, L. Acedo, and A. J. Arenas, Numerical Algorithms, published online 2013. doi 10.1007/s11075-013-9776-xW. Rindler, Relativity: Special, General and Cosmological, 2nd ed. (Oxford Univ., New York, 2006).G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, Orlando, Florida, 1985).L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd ed. (Pergamon, New York, 1971).O. Costin, “Topological construction of transseries and introduction to generalized Borel summability,” in Analyzable Functions and Applications, Ed. by O. Costin, M. D. Kruskal, and A. Macintyre, Contemp. Math. 373 (Providence, RI, USA: Am. Math. Soc., 2005); arXiv: math/0608309.S. R. Coleman, Phys. Lett. B 70, 59–60 (1977).W. B. Campbell and T. A. Morgan, Phys. Lett. B 84, 87–88 (1979).A. S. Rabinowitch, Int. J. Adv. Math. Sciences 1 (3), 109–121 (2013).A. Feinstein and J. Ibañez, Phys. Rev. D 39 (2), 470–473 (1989)

Learning about knowledge: A complex network approach

This article describes an approach to modeling knowledge acquisition in terms of walks along complex networks. Each subset of knowledge is represented as a node, and relations between such knowledge are expressed as edges. Two types of edges are considered, corresponding to free and conditional transitions. The latter case implies that a node can only be reached after visiting previously a set of nodes (the required conditions). The process of knowledge acquisition can then be simulated by considering the number of nodes visited as a single agent moves along the network, starting from its lowest layer. It is shown that hierarchical networks, i.e. networks composed of successive interconnected layers, arise naturally as a consequence of compositions of the prerequisite relationships between the nodes. In order to avoid deadlocks, i.e. unreachable nodes, the subnetwork in each layer is assumed to be a connected component. Several configurations of such hierarchical knowledge networks are simulated and the performance of the moving agent quantified in terms of the percentage of visited nodes after each movement. The Barab\'asi-Albert and random models are considered for the layer and interconnecting subnetworks. Although all subnetworks in each realization have the same number of nodes, several interconnectivities, defined by the average node degree of the interconnection networks, have been considered. Two visiting strategies are investigated: random choice among the existing edges and preferential choice to so far untracked edges. A series of interesting results are obtained, including the identification of a series of plateaux of knowledge stagnation in the case of the preferential movements strategy in presence of conditional edges.Comment: 18 pages, 19 figure