Numerical Methods of Optimal Accuracy for Weakly Singular Volterra Integral Equations

TL;DR

This paper develops and analyzes numerical methods that achieve optimal accuracy for solving weakly singular Volterra integral equations, including theoretical approximation bounds and numerical examples.

Contribution

It introduces accuracy-optimal numerical methods for weakly singular Volterra equations and evaluates related approximation limits using n-widths.

Findings

Optimal numerical methods constructed for 1D and 2D equations

Orders of n-widths for function classes evaluated

Numerical illustrations provided for 2-D equations

Abstract

Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely related with the optimal approximation problem, the orders of the Babenko and Kolmogorov \(n-\)widths of compact sets from some classes of functions have been evaluated. In conclusion we adduce some numerical illustrations for 2-D Volterra equations.