Solvability and Convergence Analysis of The Multi-Step Numerical Method for Third-Kind Volterra Integral equations involving a vanishing delay

Abstract

We investigate the application of collocation schemes to a class of Volterra delay integral equations that feature compact and noncompact operators. This class of integral equations with vanishing delays is solved numerically using multi-step collocation techniques implemented on various mesh types. The study includes an analysis of the associated operators and establishes the existence, uniqueness, and regularity of the exact solution. We prove the existence and uniqueness of the collocation solutions on different meshes. Additionally, the convergence criteria and the corresponding order of convergence are presented for each mesh type. The validity of the convergence analysis is verified through numerical examples.