A general collocation analysis for weakly singular Volterra integral equations with variable exponent.

The article focuses on the analysis and numerical solution of variable-exponent weakly singular Volterra integral equations (VIEs) of the second kind, where the kernel singularity exponent \(\alpha(t)\) varies with time. It develops a comprehensive theoretical framework establishing existence, uniqueness, and sharp regularity results for solutions, extending classical constant-exponent theory to the variable-exponent setting. The paper introduces a novel error analysis for piecewise polynomial collocation methods of arbitrary degree, applicable on graded or uniform meshes, and demonstrates convergence rates including superconvergence at collocation points without relying on classical resolvent techniques. Extensions to certain nonlinear VIEs and systems of VIEs are also presented, supported by numerical experiments that confirm theoretical error bounds and reveal enhanced performance for Radau IIA collocation points, suggesting directions for future research.