Discontinuous piecewise polynomial collocation methods for integral-algebraic equations of Hessenberg type

This paper mainly focuses on the integral-algebraic equations (IAEs) of Hessenberg type. The tractability index is investigated. The existence, uniqueness, and regularity are analyzed, and the resolvent representation is given. First, the convergence theory of perturbed collocation methods in discontinuous piecewise polynomial space is established for first-kind Volterra integral equations, then it is used to derive the optimal convergence properties of discontinuous piecewise polynomial collocation methods for Hessenberg-type IAEs. Numerical examples illustrate the theoretical results.

Automated summary

This paper focuses on the integral-algebraic equations (IAEs) of Hessenberg type. It analyzes the tractability index, as well as the existence, uniqueness, and regularity of the equations, providing a resolvent representation. The paper establishes the convergence theory of perturbed collocation methods in discontinuous piecewise polynomial space for first-kind Volterra integral equations and applies it to derive the optimal convergence properties of discontinuous piecewise polynomial collocation methods for Hessenberg-type IAEs. Numerical examples are presented to demonstrate the theoretical results.