Legendre–Galerkin Methods for Third Kind VIEs and CVIEs

The main purpose of this paper is to present a spectral Legendre–Galerkin method for solving Volterra integral equations of the third kind. When the operator associated with the equivalent Volterra integral equations of second kind is compact, the resulting system produced by this spectral method is uniquely solvable and the approximate solution attains the optimal convergence order. While the related operator is noncompact, that brings a serious challenge in numerical analysis. In order to overcome this difficulty, we first decompose the original operator into three operators, one is the identity operator, the other is the contraction operator and the third one is compact. Under this decomposition, we show that the proposed method guarantees the unique solvability of the approximate equation. Moreover, we establish that the approximate solution arrives at the quasi-optimal order of global convergence. In addition, we extend this spectral method to solve the associated cordial Volterra integral equations. Finally, to confirm the theoretical results, two numerical examples are presented.