The usual classical polynomials-based spectral Galerkin and Petrov–Galerkin methods enjoy high-order accuracy for problems with smooth solutions. However, their accuracy and fidelity can be deteriorated when the solutions exhibit weakly singular behaviors and this issue becomes much more severe for polynomial-based spectral methods. The eigenfunctions of the Sturm–Liouville problems of fractional order serve as basis functions for constructing efficient spectral approximations for fractional differential models with nonsmooth solutions. In this paper, the Petrov–Galerkin spectral method is adopted to deal with the initial singularity in the temporal direction in which the first kind Jacobi poly-fractonomials are utilized as temporal trial functions and the second kind Jacobi poly-fractonomials as temporal test functions. Along the spatial direction, the Galerkin spectral method is adopted for the first time to deal with the boundary singularity in the spatial direction in which weighted Jacobi functions are utilized as bases in multi-dimensions. Various numerical experiments are provided to demonstrate the performance of the proposed schemes.

通常基于经典多项式的谱 Galerkin 和 Petrov-Galerkin 方法对于具有平滑解的问题具有高阶精度。然而，当解决方案表现出弱奇异行为时，它们的准确性和保真度可能会下降，并且这个问题对于基于多项式的谱方法变得更加严重。分数阶 Sturm-Liouville 问题的本征函数用作构造具有非光滑解的分数阶微分模型的有效谱近似的基函数。本文采用Petrov-Galerkin谱方法处理时间方向的初始奇异性，其中第一类Jacobi多项式作为时间试函数，第二类Jacobi多项式作为时间测试函数。在空间方向上，首次采用伽辽金谱法处理空间方向的边界奇异性，在多维上以加权雅可比函数为基。提供了各种数值实验来证明所提出方案的性能。