In this paper, we investigate a spectral Petrov-Galerkin method for fractional initial value problems. Singularities of the solution at the origin inherited from the weakly singular kernel of the fractional derivative are considered, and the regularity is constructed for the solution in weighted Sobolev space. We present an optimal error estimate of the spectral Petrov-Galerkin method, and prove that the convergence order of the method in the 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 solve the produced linear system. Numerical experiments verify the theoretical findings and show the efficiency of the proposed algorithm, and exhibit that the presented numerical method works well for some time-fractional diffusion equations after suitable temporal semi-discrete.

翻译：在本文中,我们调查了用于分数初始值问题的光谱Petrov-Galerkin方法。 考虑了分数衍生物微弱单核内核所继承的源的解决方案的特性,并且为加权Sobolev空间的解决方案构建了常规性。 我们对Petrov-Galerkin光谱方法进行了最佳误差估计,并证明加权美元-诺尔姆方法的趋同顺序在光源术语中为3\alpha+1美元,平滑源术语中为3\alpha+1美元,其中1美元是分数衍生物的顺序。 一种具有准线性复杂性的迭代算法被视为解决产生的线性系统。 数字实验核实了理论结果并展示了拟议算法的效率,并展示了所提出的数字方法在适当的时间半分解后对一些时间折射方程十分有效。