Algorithmic analysis demonstrates rapid, accurate solving of fractional Volterra equations, indicating substantial speedups over standard direct and fast Fourier transform methods.
Key Points
Develop an accurate and computationally efficient numerical framework combining product integration and sum-of-exponentials compression to solve second-kind fractional Volterra equations with weakly singular kernels.
Combined analytical product integration for local algebraic kernel singularities with sum-of-exponentials (SOE) history compression to transform Abel-type convolution memory into causal recurrences.
Applied diagonal splitting and separated rank-1 approximations to manage smooth non-convolution modulations, achieving O(N_t log N_t) work complexity and O(log N_t) auxiliary storage.
Established an error bound of order O(Δt^(m+1)) for degree-m methods under smooth data, with piecewise linear schemes demonstrating near second-order convergence.
Achieved runtimes of 0.052 s on an Abel benchmark with N_t = 2^17, compared to 18.2 s for direct product integration (~350-fold speedup) and 1.15 s for an FFT-accelerated implementation (~22-fold speedup).
Observed zero growth in maximum nodal error during fixed-step long-time stability benchmark evaluations.