Asymptotics on the Fredholm integral equation with a highly oscillatory and weakly singular kernel

This paper focuses on the asymptotic behaviors of the solutions to the second-kind Fredholm integral equations with a highly oscillatory and weakly singular kernel. We employ van der Corput lemmas, resolvents of the Volterra integral equations, and the contraction mapping principle to analyse these behaviors. Numerical results are provided to show how the singularity and oscillatory parameters affect the solution.