一类分数阶对流扩散方程差分格式的理论分析

A Difference Scheme and a Theorem Analysis for a Kind of Fractional Convection-Dispersion Equation

考虑一般的对流扩散方程,将一阶的时间导数用Caputo分数阶导数替换,二阶的空间导数用Riemann-Liouville分数阶导数替换,得到了一个Riemann-Liouville-Caputo分数阶对流扩散方程.给出了这个方程的一种计算有效的隐式差分格式,并证明了该差分格式是无条件稳定、无条件收敛的,其收敛阶为O(l+h).最后给出了数值例子.

A classical convection-dispersion equation in which the first-order time derivative was replaced by a Caputo derivative and the second-order space derivative was replaced by a Riemann-Liouville derivative was considered, and a Riemann-Liouville-Caputo fractional convection-dispersion equation was obtained. A computationally effective implicit difference approximation was presented. It was shown that the scheme was unconditionally stable and convergent respectively. The convergence order of the scheme was （ ）Oτ＋h . Finally, some numerical examples were given.