A collocation method of lines for two‐sided space‐fractional advection‐diffusion equations with variable coefficients

Abstract

We present the method of lines (MOL), which is based on the spectral collocation method, to solve space‐fractional advection‐diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left‐ and right‐sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method.

A Finite Difference Method for Space Fractional Differential Equations with Variable Diffusivity Coefficient

“…In this section, we return to the two-sided fractional differential equation (1). To construct our finite difference approximation, we simply combine the finite difference schemes introduced in the two previous sections for the LS and RS fractional derivatives.…”

“…For the truncation error analysis, we assume that f ∈ C 1 (Ω) and ∈ C 2 (Ω) . In (1), x denotes the first-order derivative, and the two-sided fractional order differential operator ,…”

“…x and x I 1− b the LS and RS Riemann-Liouville fractional integrals, respectively, with the kernel 1− (x) ∶= x − Γ(1− ) . In the limiting case = 1 , the fractional derivative x reduces to x and problem (1) reduces to the classical two-point elliptic boundary value problem, where − x u is the ordinary diffusion flux from Fick's law, Fourier's law, or the Newtonian constitutive equation. An implied assumption is that the rate of diffusion at a certain location is independent of the global structure of the diffusing field.…”

“…Similarly, space fractional derivatives, which are suitable for the modeling of superdiffusion processes, account for anomalously large particle jumps at a rate inconsistent with the classical Brownian motion model. At the macroscopic level, these jumps give rise to a spatial fractional diffusion equation [2,4] where the numerical solution of this model was recently considered by several authors, see for example [1,15,24,39] and related references therein.…”