Modeling and Computing of Fractional Convection Equation

Abstract

In this paper, we derive the fractional convection (or advection) equations (FCEs) (or FAEs) to model anomalous convection processes. Through using a continuous time random walk (CTRW) with power-law jump length distributions, we formulate the FCEs depicted by Riesz derivatives with order in (0, 1). The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in (0, 1) are constructed too. Then the numerical approximations to FCEs are studied in detail. By adopting the implicit Crank-Nicolson method and the explicit Lax-Wendroff method in time, and the secondorder numerical method to the Riesz derivative in space, we, respectively, obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space. The accuracy and efficiency of the derived methods are verified by numerical tests. The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.

Numerical Approaches to Fractional Integrals and Derivatives: A Review

“…4,l+1 for l ≥ 7, and Recently, the above conjecture for (α) 3,l with 0 < α < 1 has been proved in Ref. [107].…”

Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

“…Anomalous diffusion is the theory of diffusing particles in environments that are not locally homogeneous. [3][4][5][6] A physical-mathematical model to anomalous diffusion may be based on FPDEs containing derivatives of fractional order in both space and time, where the subdiffusion appears in time and the superdiffusion occurs in space simultaneously. 7,8 On the other hand, although most of time-space fractional diffusion models are initially defined with the spatially integral fractional Laplacian (IFL), [9][10][11][12] many previous studies (cf., e.g., previous works 4,5,[13][14][15] ) always substitute the space Riesz fractional derivative 1 for the IFL.…”

“…In physics, fractional derivatives are used to model anomalous diffusion. Anomalous diffusion is the theory of diffusing particles in environments that are not locally homogeneous 3‐6 . A physical–mathematical model to anomalous diffusion may be based on FPDEs containing derivatives of fractional order in both space and time, where the subdiffusion appears in time and the superdiffusion occurs in space simultaneously 7,8 .…”

Multi-term time fractional diffusion equations and novel parameter estimation techniques for chloride ions sub-diffusion in reinforced concrete

“…In recent decades, the theory and application of fractional calculus have drawn increasing attentions from researchers in different disciplines [22,26,27]. Fractional differential equations have been successfully applied to diverse anomalous phenomena occurring in various scientific and engineering fields [28][29][30]. The time-fractional diffusion model (TFD model) was proposed to describe chloride ions transport in concrete.…”