Capturing Multiple Singularities with Spectral Accuracy for Multi-Term Fractional Differential Equations

This paper develops a robust numerical scheme based on a frame collocation method for solving multi-term fractional ordinary differential equations (FODEs) whose solutions exhibit multiple singularities at the origin. To adaptively capture the singular behavior, we construct a hybrid basis-function frame by combining shifted fractional Legendre polynomials. An efficient computational formula for the Caputo fractional derivative is derived, which transforms the original problem into a nonlinear algebraic system at the collocation points. Due to the over-completeness of the fractional polynomial frame, the resulting linear system becomes rank-deficient, with only a small subset of singular components carrying meaningful solution information. To eliminate the adverse effects of numerical null-space components, we employ truncated singular value decomposition (TSVD) regularization, thereby enabling stable and high-precision solutions. Extensive numerical experiments on several benchmark problems, including the fractional Bagley–Torvik equation, linear multi-term FODEs, and nonlinear cases, demonstrate that the proposed method achieves exponential convergence rates. Notably, when the singular exponent of the solution matches a tunable parameter (δ) in the basis functions, superconvergence is observed, significantly outperforming standard spectral methods. Compared with traditional spectral approaches, the proposed frame collocation framework retains spectral accuracy while exhibiting superior capability in handling complex singular structures, providing a powerful and reliable tool for high-precision simulations of multi-term fractional differential equations.