This paper presents a novel Petrov-Galerkin method for a class of linear systems of fractional differential equations. New fractional-order generalized Jacobi functions are introduced, and their approximation properties are investigated. We show that these functions satisfy the given supplementary conditions and have the same asymptotic behavior as the exact solution, which are essential properties to design a high-order spectral Petrov-Galerkin method. For implementing our scheme, we represent the approximate solution by a linear combination of fractional-order generalized Jacobi functions and minimize the residual using shifted fractional Jacobi functions. The numerical solvability of the resultant algebraic system are justified as well as convergence and stability properties of the proposed scheme are explored. Finally, the reliability of the method is evaluated using various analytical and realistic problems.

本文针对一类分数阶微分方程的线性系统提出了一种新的 Petrov-Galerkin 方法。引入了新的分数阶广义雅可比函数，并研究了它们的近似性质。我们表明这些函数满足给定的补充条件，并具有与精确解相同的渐近行为，这是设计高阶谱 Petrov-Galerkin 方法的基本性质。为了实现我们的方案，我们通过分数阶广义雅可比函数的线性组合来表示近似解，并使用移位分数雅可比函数来最小化残差。对合成代数系统的数值可解性进行了论证，并探讨了所提出方案的收敛性和稳定性特性。最后，