Zhao, LijingDeng, WeihuaHesthaven, Jan S.2016-04-032016-04-032016-04-032016https://infoscience.epfl.ch/handle/20.500.14299/125518In this paper, we first introduce fractional integral spaces, which possess some features: (i) when 0 < α < 1, functions in these spaces are not required to be zero on the boundary; (ii)the tempered fractional operators are equivalent to the Riemann-Liouville operator in the sense of the norm. Spec- tral Galerkin and Petrov-Galerkin methods for tempered fractional advection problems and tempered fractional diffusion problems can be developed as the classical spectral Galerkin and Petrov-Galerkin methods. Error analysis is provided and numerically confirmed for the tempered fractional advection and diffusion problems.Tempered fractional operatorsfractional integral spacesspectral methoderror estimate.Spectral methods for tempered fractional differential equationstext::preprint