On the convergence of collocation methods for Volterra–Riccati integral equations with weakly singular kernel

This paper studies the Volterra–Riccati integral equation with a weakly singular kernel, which is a special case of Volterra–Hammerstein equations. The solution of this equation may exhibit blow-up phenomena. We first establish two sufficient conditions for the global existence and uniqueness of the exact solution based on the Schauder–Tychonoff fixed-point theorem, then we analyze the regularity of the exact solution. Subsequently, the piecewise polynomial collocation scheme on graded meshes is applied to discretize the Volterra–Riccati integral equation, and the optimal convergence result is obtained. Several numerical experiments are carried out to illustrate the obtained theoretical result.