A comprehensive analysis for weakly singular nonlinear functional Volterra integral equations using discretization techniques

This study investigates weakly singular nonlinear functional Volterra integral equations (WSNFVIEs) of Urysohn type involving Riemann–Liouville operator. By imposing specific smoothness conditions on the involved functions, we establish both the existence and uniqueness of the solution using a fixed point approach. Subsequently, we employ discretization methods such as the trapezoidal and Euler methods to approximate the solution, resulting in a system of nonlinear algebraic equations. To ascertain the convergence order of the Euler method (first order) and the trapezoidal method (second order), we utilize the Grönwall inequality and its discrete counterpart. Additionally, we introduce a novel Grönwall inequality to establish the convergence of the trapezoidal method. Thoroughly we examine the Hyers–Ulam–Rassias and Hyers–Ulam stability of the integral equations within the specified domain. Finally, the efficacy of the proposed methods is validated through numerical examples accompanied by comparative analyses.