NUMERICAL METHODS OF OPTIMAL ACCURACY FOR WEAKLY SINGULAR
VOLTERRA INTEGRAL EQUATIONS 

Boykov Il'ya Vladimirovich, Doctor of physical and mathematical sciences, professor, head of sub-department of higher and applied mathematics, Penza State University (Penza, 40 Krasnaya str.), math@pnzgu.ru
Tynda Aleksandr Nikolaevich, Candidat of physical and mathematical sciences, associate professor, sub-department of higher and applied mathematics, Penza State University (Penza, 40 Krasnaya str.), tynda@pnzgu.ru
Krasnoshchekov Pavel Sergeevich, Doctor of physical and mathematical sciences, professor, full member of the Russian Academy of Sciences, principal scientist, Computing center named after A.A. Dorodnitsyn of the Russian Academy of Sciences (Moscow, 40 Vavilova str.), math@pnzgu.ru 

517.392 

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 accuracyoptimal 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. 

Volterra integral equations, optimal algorithms, Babenko and Kolmogorov n-widths, weakly singular kernels, collocation method. 

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