Abstract
In this paper, we mainly study the solution and properties of the multiterm time-fractional diffusion equation. First, we obtained the stochastic representation for this equation, which turns to be a subordinated process. Based on the stochastic representation, we calculated the mean square displacement (MSD) and time average mean square displacement, then proved some properties of this model, including subdiffusion, generalized Einstein relationship, and nonergodicity. Finally, a stochastic simulation algorithm was developed for the visualization of sample path of the abnormal diffusion process. The Monte Carlo method was also employed to show the behavior of the solution of this fractional equation.
Highlights
The diffusion equations that generalize the usual one have received considerable attention due to the broadness of their physical applications, in particular, to the anomalous diffusion
Fractional diffusion equations and the nonlinear fractional diffusion equations have been successfully applied to several physical situations such as percolation of gases through porous media [1], thin saturated regions in porous media [2], standard solid-on-solid model for surface growth [3], thin liquid films spreading under gravity [4], in the transport of fluid in porous media and in viscous fingering [5], modeling of non-Markovian dynamical processes in protein folding [6], relaxation to equilibrium in a system with long temporal memory [7], and anomalous transport in disordered systems [8], diffusion on fractals [9], and the multiphysical transport in porous media, such as electroosmosis [10, 11]
The subordinated process YðtÞ = XðEtÞ is the stochastic representation of the multiterm time-fractional advection-diffusion equation (1), where the parent process and subordinator of Yt is defined as pffiffiffiffiffiffi dXðτÞ = vdτ + 2DdBðτÞ, ð3Þ
Summary
The diffusion equations that generalize the usual one have received considerable attention due to the broadness of their physical applications, in particular, to the anomalous diffusion. Jiang et al studied the multiterm time-space fractional advection-diffusion equation based on the spectral representation of the fractional Laplacian operator [22]. We introduce the stochastic representation method to solve this multiterm time-fractional diffusion equation.
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