Abstract
In this paper, a time-space fractional advection diffusion equation is considered for the natural extension of the convection diffusion equation. An explicit difference scheme and an implicit difference scheme are presented. The stability and convergence of the two difference schemes are discussed. It is shown that the explicit difference scheme is conditionally stable and convergent, and the implicit difference scheme is unconditionally stable and convergent. The convergence order of the two methods is O (tau+h).
Highlights
The research and application of fractional differential equations have attracted the attention of many scholars in the last few decades
The fractional order convection diffusion equations are the generalization of the integral order convection diffusion equations
5 Conclusions A time-space fractional advection diffusion equation is studied in this paper
Summary
The research and application of fractional differential equations have attracted the attention of many scholars in the last few decades. Yu studied the implicit difference approximation of the time fractional reaction diffusion equation in [11]. Zhuang [12] and Tan [13] separatively studied the explicit and implicit difference schemes for time-space fractional reactiondiffusion equations, and the stability and convergence were discussed. Two approximate results in two different generalizations of the space-time-fractional advection diffusion equation were discussed in [18]. A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection diffusion equation was obtained in [19]. In this paper, based on the finite difference method, we discuss a time-space fractional advection diffusion equation, where the time term, the advection term, and the diffusion term are all fractional order derivatives. Let us consider the numerical solution of the equations based on the difference method
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