中文摘要：

设计了一个谱Galerkin方法来求解时间空间分数阶对流扩散方程初边值问题.证明了变分问题解的存在唯一性以及谱Galerkin方法的收敛性.并通过数值算例检验了理论分析结果.

英文摘要：

A spectral Galerkin method is designed for time-space fractional advection-diffusion equation with initial boundary conditions. The existence and uniqueness of the variational form are proved, and it is also proved that the proposed method has spectral accuracy both in temporal and spatial direction. Numerical experiments are provided to check the theoretical results.