The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Guido Germano - One of the best experts on this subject based on the ideXlab platform.
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monte carlo simulation of uncoupled continuous time random walks yielding a Stochastic Solution of the space time fractional diffusion equation
Physical Review E, 2008Co-Authors: Daniel Fulger, Enrico Scalas, Guido GermanoAbstract:We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy α-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the Stochastic Solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Levy α-stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.
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Monte Carlo simulation of uncoupled continuous-time random walks yielding a Stochastic Solution of the space-time fractional diffusion equation.
Phys Rev E Stat Nonlin Soft Matter Phys, 2008Co-Authors: Guido GermanoAbstract:We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Lévy alpha -stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the Stochastic Solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Lévy alpha -stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.
Paolo Paradisi - One of the best experts on this subject based on the ideXlab platform.
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A Stochastic Solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
arXiv: Statistical Mechanics, 2016Co-Authors: Gianni Pagnini, Paolo ParadisiAbstract:The Stochastic Solution with Gaussian stationary increments is establihsed for the symmetric space-time fractional diffusion equation when $0 < \beta < \alpha \le 2$, where $0 < \beta \le 1$ and $0 < \alpha \le 2$ are the fractional derivation orders in time and space, respectively. This Solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental Solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuos time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental Solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a Stochastic Solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal L\'evy stable densities. This Stochastic Solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure. Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental Solution of the symmetric space-time fractional diffusion equation.
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A Stochastic Solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
Fractional Calculus and Applied Analysis, 2016Co-Authors: Gianni Pagnini, Paolo ParadisiAbstract:AbstractThe Stochastic Solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 <Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental Solution of the symmetric space-time fractional diffusion equation.
Mark M Meerschaert - One of the best experts on this subject based on the ideXlab platform.
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Correlation structure of fractional Pearson diffusions
Computers & mathematics with applications (Oxford England : 1987), 2013Co-Authors: Nikolai Leonenko, Mark M Meerschaert, Alla SikorskiiAbstract:The Stochastic Solution to a diffusion equations with polynomial coefficients is called a Pearson diffusion. If the first time derivative is replaced by a Caputo fractional derivative of order less than one, the Stochastic Solution is called a fractional Pearson diffusion. This paper develops an explicit formula for the covariance function of a fractional Pearson diffusion in steady state, in terms of Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long-range dependent, with a correlation that falls off like a power law, whose exponent equals the order of the fractional derivative.
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Stochastic Solution to a time-fractional attenuated wave equation.
Nonlinear dynamics, 2012Co-Authors: Mark M Meerschaert, Peter Straka, Yuzhen Zhou, Robert J. McgoughAbstract:The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical Solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal Solution to this fractional wave equation is developed.
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Stochastic Solution of space time fractional diffusion equations
Physical Review E, 2002Co-Authors: Mark M Meerschaert, David A Benson, Hanspeter Scheffler, Boris BaeumerAbstract:Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original Stochastic Solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit Solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.
Gianni Pagnini - One of the best experts on this subject based on the ideXlab platform.
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A Stochastic Solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
arXiv: Statistical Mechanics, 2016Co-Authors: Gianni Pagnini, Paolo ParadisiAbstract:The Stochastic Solution with Gaussian stationary increments is establihsed for the symmetric space-time fractional diffusion equation when $0 < \beta < \alpha \le 2$, where $0 < \beta \le 1$ and $0 < \alpha \le 2$ are the fractional derivation orders in time and space, respectively. This Solution is provided by imposing the identity between two probability density functions resulting (i) from a new integral representation formula of the fundamental Solution of the symmetric space-time fractional diffusion equation and (ii) from the product of two independent random variables. This is an alternative method with respect to previous approaches such as the scaling limit of the continuos time random walk, the parametric subordination and the subordinated Langevin equation. A new integral representation formula for the fundamental Solution of the space-time fractional diffusion equation is firstly derived. It is then shown that, in the symmetric case, a Stochastic Solution can be obtained by a Gaussian process with stationary increments and with a random wideness scale variable distributed according to an arrangement of two extremal L\'evy stable densities. This Stochastic Solution is self-similar with stationary increments and uniquely defined in a statistical sense by the mean and the covariance structure. Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental Solution of the symmetric space-time fractional diffusion equation.
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A Stochastic Solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
Fractional Calculus and Applied Analysis, 2016Co-Authors: Gianni Pagnini, Paolo ParadisiAbstract:AbstractThe Stochastic Solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 <Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental Solution of the symmetric space-time fractional diffusion equation.
Boris Baeumer - One of the best experts on this subject based on the ideXlab platform.
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Stochastic Solution of space time fractional diffusion equations
Physical Review E, 2002Co-Authors: Mark M Meerschaert, David A Benson, Hanspeter Scheffler, Boris BaeumerAbstract:Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original Stochastic Solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit Solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.
