An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

•An ADI spectral method for the 2D multi-term time fractional anomalous equation is proposed.•The stability and convergence theorems of the ADI spectral method are proved.•The non-smooth solution case for this equation is also considered.•We add some correction terms to the ADI spectral method to achieve high accuracy.