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96
Finite Difference Approximations For TwoSided SpaceFractional Partial Differential Equations
 APPLIED NUMERICAL MATHEMATICS, APPL. NUMER. MATH
, 2006
"... Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical methods to solve a class of initialboundary v ..."
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Cited by 53 (6 self)
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Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical methods to solve a class of initialboundary value fractional partial differential equations with variable coefficients on a finite domain. We examine the case when a lefthanded or a righthanded fractional spatial derivative may be present in the partial differential equation. Stability, consistency, and (therefore) convergence of the methods are discussed. The stability (and convergence) results in the fractional PDE unify the corresponding results for the classical parabolic and hyperbolic cases into a single condition. A numerical example using a finite difference method for a twosided fractional PDE is also presented and compared with the exact analytical solution.
Fractional diffusion processes: Probability distribution and continuous time random walk
 LECTURE NOTES IN PHYSICS 621
, 2007
"... A physicalmathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the spacetime fractional diffusion equation we mean an evolution equation obtained from the sta ..."
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Cited by 37 (8 self)
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A physicalmathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the spacetime fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the secondorder space derivative with a RieszFeller derivative of order α ∈ (0,2] and skewness θ (θ  ≤ min {α,2 − α}), and the firstorder time derivative with a Caputo derivative of order β ∈ (0,1]. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar selfsimilar stochastic process. We view it as a generalized diffusion process that we call fractional diffusion process, and present an integral representation of the fundamental solution. A more general approach to anomalous diffusion is however known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.
A fractional generalization of the Poisson processes
, 2007
"... It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like wa ..."
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Cited by 33 (10 self)
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It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a nonMarkovian renewal process with a waiting time distribution described by the MittagLeffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a powerasymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.
Five years of ContinuousTime Random Walks in Econophysics
, 2005
"... This paper is a short review on the application of continuoustime random walks to Econophysics in the last five years. ..."
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Cited by 29 (2 self)
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This paper is a short review on the application of continuoustime random walks to Econophysics in the last five years.
HOMOGENEOUS TEMPORAL ACTIVITY PATTERNS IN A LARGE ONLINE COMMUNICATION SPACE
"... The manytomany social communication activity on the popular technologynews website Slashdot has been studied. We have concentrated on the dynamics of message production without considering semantic relations and have found regular temporal patterns in the reaction time of the community to a news ..."
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Cited by 14 (7 self)
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The manytomany social communication activity on the popular technologynews website Slashdot has been studied. We have concentrated on the dynamics of message production without considering semantic relations and have found regular temporal patterns in the reaction time of the community to a newspost as well as in single user behavior. The statistics of these activities follow lognormal distributions. Daily and weekly oscillatory cycles, which cause slight variations of this simple behavior, are identified. The findings are remarkable since the distribution of the number of comments per users, which is also analyzed, indicates a great amount of heterogeneity in the community. The reader may find surprising that only two parameters, those of the lognormal law, allow a detailed description, or even prediction, of social manytomany information exchange in this kind of popular public spaces.
Towards a statistical theory of texture evolution in polycrystals
 SIAM Journal Sci. Comp
"... Abstract. Most technologically useful materials possess polycrystalline microstructures composed of a large number of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network play a crucial role in determining the properties of a mater ..."
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Cited by 11 (7 self)
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Abstract. Most technologically useful materials possess polycrystalline microstructures composed of a large number of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network play a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains—a texture—that provides for a desired set of material properties. One of the most challenging aspects of this problem is to understand the role of topological reconfigurations during coarsening. Here we propose an upscaling procedure suitable for large complex systems. The procedure is based on numerical experimentation combined with stochastic tools and consists of largescale numerical simulations of a system at a microscopic level, statistical analysis of the microscopic data, and formulation of the model based on stochastic characteristics predicted by the statistical analysis. This approach promises to be valuable in establishing the effective model of microstructural evolution in realistic two and threedimensional systems. To test ideas we use our upscaling procedure to study the mesoscopic behavior of a reduced onedimensional network of grain boundaries. Despite the simplicity of its formulation, this model exhibits highly nontrivial behavior characterized by growth and disappearance of grain boundaries and develops probability distributions similar to those observed in higherdimensional simulations. Here we focus on the grain deletion events which are common to all coarsening systems.
From power laws to fractional diffusion: the direct way
 Vietnam Journal of Mathematics 32 SI
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Correlated continuous time random walks
"... Abstract. Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Spacefractional derivatives describe heavy tailed jumps, and the timefractional version codes h ..."
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Cited by 10 (4 self)
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Abstract. Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Spacefractional derivatives describe heavy tailed jumps, and the timefractional version codes heavy tailed waiting times. This paper develops scaling limits and governing equations in the case of correlated jumps. For longrange dependent jumps, this leads to fractional Brownian motion or linear fractional stable motion, with the time parameter replaced by an inverse stable subordinator in the case of heavy tailed waiting times. These scaling limits provide an interesting class of nonMarkovian, nonGaussian selfsimilar processes. 1.