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MSC2010 65R20

Numerical method for systems of nonlinear Volterra integral equations of the first kind with discontinuous kernels

A. N. Tynda1, D. N. Sidorov2, I. R. Muftahov3

AnnotationIn this paper we investigate the systems of nonlinear Volterra integral equations of the first kind with kernels having jump discontinuities along the set of smooth curves. The necessary theory concerning the existence and uniqueness of solutions of such systems is described. An iterative numerical method is proposed, based on the linearization of integral operators using the modified Newton-Kantorovich scheme. For this purpose, we calculate the Fr\'{e}chet derivatives of the components of integral vector-operator at the initial approximation point. The kernels of the integral equations in the linear systems remain constant for each iteration. This allows to reduce the computational expenses in numerical realization of the method. For linear systems of integral equations arising at each step of the iterative process, we use a piecewise-constant approximation of the exact solution and special adaptive grids that take into account kernels discontinuities. The error of the method is estimated. Suggested numerical approach also allows the application of some more accurate approximations of the exact solution in aggregate with the corresponding quadrature formulas. The accuracy order increases by unity when the piecewise-linear approximation is used.
Keywordssystems of nonlinear Volterra integral equations, discontinuous kernels, Newton-Kantorovich method, adaptive meshes, approximation of the integrals.

1Aleksandr N. Tynda, Associate Professor, Department of Higher and Applied Mathematics, Penza State University (40 Krasnaya St., Penza 440026, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-6023-9847, tyndaan@mail.ru

2Denis N. Sidorov, Leading Research Fellow Energy Systems Institute Russian Academy of Sciences (130 Lermontov St. Irkutsk 664033, Russia), D.Sc. (Physics and Mathematics), ORCID: https://orcid.org/0000-0002-3131-1325, contact.dns@gmail.com

3I. R. Muftahov, Postgraduate student, Main Computing Center of Joint Stock Company <<Russian Railways>> (25 Mayakovaskogo St., Irkutsk 664005, Russia), ORCID: https://orcid.org/0000-0003-2516-459X, ildar_sm@mail.ru

Citation: A. N. Tynda, D. N. Sidorov, I. R. Muftahov, "[Numerical method for systems of nonlinear Volterra integral equations of the first kind with discontinuous kernels]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:1 (2018) 55–63 (In Russian)

DOI 10.15507/2079-6900.20.201801.55-63