High‐Order Singularity‐Corrected Sum‐of‐Exponentials Product Integration for Second‐Kind Fractional Volterra Equations With Modulated Weakly Singular Kernels

ABSTRACT Weakly singular fractional Volterra equations require both accurate treatment of the algebraic kernel singularity and efficient evaluation of the growing history term. We develop a singularity‐corrected sum‐of‐exponentials product‐integration framework for second‐kind equations with kernels , . Product integration treats the local singularity analytically, while sum‐of‐exponentials (SOE) compression converts the resolved Abel‐type history into causal recurrences. For smooth data, the degree‐ method satisfies an error bound of order ; a reduced‐regularity estimate is also established for algebraically singular solutions. Smooth non‐convolution modulations are handled by diagonal splitting, and a separated rank‐ approximation yields a fully accelerated variant with work and auxiliary memory. Numerical tests show nearly second‐order convergence for the piecewise linear scheme, the predicted rate reduction for weakly regular solutions, and no observed growth of the maximum nodal error in a fixed‐step long‐time benchmark. In the Abel benchmark with , SOE‐PI requires s, compared with s for direct product integration and s for an FFT‐accelerated implementation of the same discretization, corresponding to speedups of about and , respectively.