Geodesic
Numerical method for systems of nonlinear Volterra integral equations of the first kind with discontinuous kernels
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 1, pp. 55-63

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In 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 Fréchet 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.
Keywords: systems of nonlinear Volterra integral equations, discontinuous kernels, Newton-Kantorovich method, adaptive meshes, approximation of the integrals. 
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     title = {Numerical method for systems of nonlinear {Volterra} integral equations of the first kind with discontinuous kernels},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
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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. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 1, pp. 55-63. http://geodesic.mathdoc.fr/item/SVMO_2018_20_1_a5/