  In this work, we study a new spectral Petrov-Galerkin approximation of
space-time fractional reaction-diffusion equations with viscosity terms built
by Riemann-Liouville fractional-order derivatives. The proposed method is
reliant on generalized Jacobi functions (GJFs) for our problems. The
contributions are threefold: First, thanks to the theoretical framework of
variational problems, the well-posedness of the problem is proved. Second, new
GJF-basis functions are established to fit weak solutions, which take full
advantages of the global properties of fractional derivatives. Moreover, the
basis functions conclude singular terms, in order to solve our problems with
given smooth source term. Finally, we get a numerical analysis of error
estimates to depend on GJF-basis functions. Numerical experiments confirm the
expected convergence. In addition, they are given to show the effect of the
viscosity terms in anomalous diffusion.
