Abstract
A numerical scheme has been constructed for solving a linear hypersingular integral equation on a segment with the integral treated in the sense of the Hadamard principle value by the method of piecewise linear approximations on an arbitrary nonuniform grid, with the hypersingular integral being regularized by approximating the unknown function with a constant in a small neighborhood of the singular point. The radius of the neighborhood can be chosen independently of the grid pitch, the latter understood as the maximum distance between the nodes. The uniform convergence of the obtained numerical solutions to the exact solution is proved as the grid pitch and the radius of the neighborhood in which the regularization is performed simultaneously tend to zero.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lifanov, I.K., Singular Integral Equations and Discrete Vortices, VSP, The Nethederlands, 1996.
Belotserkovskii, S.M. and Lifanov, I.K., Chislennye metody v singulyarnykh integral’nykh uravneniyakh (Numerical Methods in Singular Integral Equations), Moscow: Nauka, 1985.
Lifanov, I.K., Poltavskii, L.N., and Vainikko, G.M., Hypersingular Integral Equations and Their Applications, Chapman & Hall CRC, 2004.
Ryzhakov, G.V. and Setukha, A.V., On the convergence of a numerical method for a hypersingular integral equation on a closed surface, Differ. Equations, 2010, vol. 46, no. 9, pp. 1353–1363.
Ryzhakov, G.V. and Setukha, A.V., On the convergence of the vortex loop method with regularization for the Neumann boundary value problem on a plane screen, Differ. Equations, 2011, vol. 47, no. 9, pp. 1365–1371.
Ryzhakov, G.V., Setukha, A.V., and Ssatuf, I., On the numerical solution of the Neumann boundary value problem on a screen by the method of vortex loop with the use of a nonuniform grid, Uchen. Zap. Orlov. Gos. Univ. Estestv., Tekhnich. Meditsin. Nauki, 2010, no. 4, pp. 12–19.
Hadamard, J., Zadacha Koshi dlya lineinykh uravnenii s chastnymi proizvodnymi giperbolicheskogo tipa (Cauchy Problem for Partial Differential Equation of the Hyperbolic Type), Moscow: Nauka, 1978.
Muskhelishvilli, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Nauka, 1966.
Lebedeva, S.G. and Setukha, A.V., On the numerical solution of a complete two-dimensional hypersingular integral equation by the method of discrete singularities, Differ. Equations, 2013, vol. 49, no. 2, pp. 224–234.
Parton, V.Z. and Perlin, P.I., Integral’nye uravneniya teorii uprugosti (Integral Equations of Elasticity Theory), Moscow: Nauka, 1977.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Setukha, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 2, pp. 237–249.
Rights and permissions
About this article
Cite this article
Setukha, A.V. Convergence of a numerical method for solving a hypersingular integral equation on a segment with the use of piecewise linear approximations on a nonuniform grid. Diff Equat 53, 234–247 (2017). https://doi.org/10.1134/S0012266117020094
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266117020094