Particle tracking for fractional diffusion with two time scales

Abstract. Previous work [51] showed how to solve time-fractional diffusion equations by particle tracking. This paper extends the method to the case where the order of the fractional time derivative is greater than one. A subordination approach treats the fractional time derivative as a random time change of the corresponding Cauchy problem, with a first derivative in time. One novel feature of the time fractional case of order greater than one is the appearance of clustering in the operational time subordinator, which is non-Markovian. Solutions to the time-fractional equation are probability densities of the underlying stochastic process. The process models movement of individual particles. The evolution of an individual particle in both space and time are captured in a pair of stochastic differential equations, or Langevin equations. Monte Carlo simulation yields particle location, and the ensemble density approximates the solution to the variable coefficient time-fractional diffusi...

• Type: Journal
• Year: 2010
• Where: CMA
• Authors: Mark M. Meerschaert, Yong Zhang, Boris Baeumer
• Keywords: CMA 2010, Fractional Time, Time-fractional Diffusion Equation, Variable Coefficient