Abstract
This paper considers a method of stochastic solution to the anomalous diffusion equation with a fractional derivative with respect to both time and coordinates. To this end, the process of a random walk of a particle is considered, and a master equation describing the distribution of particles is obtained. It has been shown that in the asymptotics of large times, this process is described by the equation of anomalous diffusion, with a fractional derivative in both time and coordinates. The method has been proposed for local estimation of the solution to the anomalous diffusion equation based on the simulation of random walk trajectories of a particle. The advantage of the proposed method is the opportunity to estimate the solution directly at a given point. This excludes the systematic component of the error from the calculation results and allows constructing the solution as a smooth function of the coordinate.
Highlights
Anomalous diffusion processes are characterized by a power-law dependence of the width of the diffusion packet on time ∆(t) ∝ Dα tγ, where Dα is the diffusion coefficient [1,2,3,4].Depending on the value of the exponent γ, different diffusion regimes are distinguished:γ < 1/2, γ = 1/2, γ > 1/2
The Continuous Time Random Walk (CTRW) model underlies the model of anomalous diffusion [9]
The CTRW model describes the random walk of a particle using a hopping-trap mechanism
Summary
Anomalous diffusion processes are characterized by a power-law dependence of the width of the diffusion packet on time ∆(t) ∝ Dα tγ , where Dα is the diffusion coefficient [1,2,3,4]. ∂t is the partial time derivative, and ∂|∂x|α is the fractional-differential Riesz operator (A4) Using this fractional-differential model of combustion, the paper investigates the effect on the damping phenomenon of a quantity of the order of the fractional derivative α, the spatial size of the area under study, and the initial conditions. In the article [46], a fractional-differential generalization of the kinetic equation was obtained that describes the dependence of the radius of the ball on time in the model of combustion of a fireball, which was theoretically predicted by the Soviet physicist. We will propose a method for the numerical solution to the anomalous diffusion equation with a fractional derivative in both time and coordinate and with a source of a special type. 0 < α 6 2, D is the generalized diffusion coefficient
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