Analysis of collocation methods for a class of third-kind auto-convolution Volterra integral equations

Abstract

In this paper, we deal with piecewise polynomial collocation methods for a class of third-kind auto-convolution Volterra integral equations (AVIEs). A weighted exponential norm is introduced to prove the existence and uniqueness of the analytic solution, and the regularity is discussed based on the theory for linear cordial Volterra integral equations (CVIEs). The solvability of collocation methods with uniform meshes is discussed following the step-by-step approach, and the uniform boundedness of collocation solutions is proved by introducing an analogous discrete weighted exponential norm. The optimal convergence property is obtained by some new techniques. Some numerical experiments are given to illustrate the theoretical results.