  Tempered fractional derivatives originated from the tempered fractional
diffusion equations (TFDEs) modeled on the whole space R (see [23]). For
numerically solving TFDEs, two kinds of generalized Laguerre functions were
defined and some important properties were proposed to establish the
approximate theory. The related prototype tempered fractional differ- ential
problems was proposed and solved as the guidance. TFDEs are numerically solved
by two domains Laguerre spectral method and the numerical experiments show some
properties of the TFDEs and verify the efficiency of the spectral scheme.
