Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

We present optimal error estimates for spectral Petrov--Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov--Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates.