Numerical methods of optimal accuracy for weakly singular Volterra integral equations

Objective: the main aim of this paper is the construction of the optimal with respect to accuracy order methods for weakly singular Volterra integral equations of different types. Methods: since the question of construction of the accuracy-optimal numerical methods is closely related with the optimal approximation problem, the authors apply the technique of the Babenko and Kolmogorov n-widths of compact sets from appropriate classes of functions. Results: the orders of the Babenko and Kolmogorov n-widths of compact sets from some classes of functions for one-dimensional and multidimensional cases are evaluated. The special local splines realizing the optimal estimates are also constructed. The optimal (with respect to accuracy order) spline-collocation methods are suggested. Conclusions: the obtained theoretical estimates are verified by the numerical examples for 2-D Volterra integral equations adduced in the paper.