A two-stage approach for nonlinear weakly singular Volterra integral equations: Local psi-series expansion and piecewise spectral collocation

This paper develops a two-stage approach for a general class of nonlinear weakly singular Volterra integral equations of the second kind. The core of the approach is a domain decomposition strategy designed to overcome the challenges posed by the solution’s lack of smoothness. Stage I employs a psi-series expansion to precisely resolve the singular behavior of the solution within a small subinterval near the origin. Stage II then applies a piecewise Legendre collocation method to efficiently compute the smooth component of the solution on the remaining regular subinterval. We provide a rigorous error analysis for the two-stage method, and prove its optimal convergence rate in L2-norm. The analysis illustrates that this framework can simulate the solution’s long-time behavior with high accuracy. Numerical experiments validate our theoretical results, demonstrating the method’s robust performance for solving weakly singular Volterra integral equations with low-regularity solutions. Furthermore, the experiments highlight a key practical insight: optimal accuracy and fast computation can be achieved by carefully balancing the singular subinterval size (Stage I) against the step size and number of collocation points (Stage II).