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In this paper, we investigate a spectral Petrov-Galerkin method forfractional initial value problems. Singularities of the solution at the origininherited from the weakly singular kernel of the fractional derivative areconsidered, and the regularity is constructed for the solution in weightedSobolev space. We present an optimal error estimate of the spectralPetrov-Galerkin method, and prove that the convergence order of the method inthe weighted $L^2$-norm is $3\alpha+1$ for smooth source term, where $\alpha$is the order of the fractional derivative.  An iteration algorithm with a quasi-linear complexity is considered to solvethe produced linear system. Numerical experiments verify the theoreticalfindings and show the efficiency of the proposed algorithm, and exhibit thatthe presented numerical method works well for some time-fractional diffusionequations after suitable temporal semi-discrete.

On spectral Petrov-Galerkin method for solving fractional initial value problems in weighted Sobolev space