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 Vestnik YuUrGU. Ser. Mat. Model. Progr., 2018, Volume 11, Issue 2, Pages 139–146 (Mi vyuru437)

Short Notes

The rate of convergence of hypersingular equations numerical computation

S. I. Eminov, S. Yu. Petrova

Novgorod State University Yaroslav the Wise, Veliky Novgorod, Russian Federation

Abstract: Numerical methods for solving hypersingular equations based on Chebyshev polynomials of the second kind with a weight taking into account the Meixner physical conditions on the edge are developed. We obtained estimates of the rate of convergence using the analytical form of the matrix of an integral operator with a logarithmic singularity. Authors considered a delta function model, and its inapplicability in diffraction problems and vibrator antennas are shown. Previously, a numerical-analytical method for solving the excitation problems of vibrator antennas was proposed, but in the present work, the rationale for the numerical-analytical method is given for the first time. Unlike the reduction method, the numerical-analytical method demonstrates reliable convergence, not only in diffraction problems but also in antenna excitation problems. The specific feature of the excitation problems is that the right-hand side of the hypersingular equation is localized in a small region, in comparison with the characteristic dimensions of the antenna. Mathematically, this means that the right-hand side of the hypersingular equation decomposes into a slowly-convergent series. A similar property is also possessed by the solution of the equation. That is why the method of reduction is not effective enough. An example of a numerical solution is considered. Estimates of the rate of convergence are obtained. The applicability of developed methods for investigating a wide range of diffraction problems is shown.

Keywords: hypersingular integral; Chebyshev polynomial; rate of convergence; operator matrix; reduction method; Fredholm system of the second kind.

DOI: https://doi.org/10.14529/mmp180211

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Document Type: Article
UDC: 519.837
MSC: 41A50
Language: English

Citation: S. I. Eminov, S. Yu. Petrova, “The rate of convergence of hypersingular equations numerical computation”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 11:2 (2018), 139–146

Citation in format AMSBIB
\Bibitem{EmiPet18} \by S.~I.~Eminov, S.~Yu.~Petrova \paper The rate of convergence of hypersingular equations numerical computation \jour Vestnik YuUrGU. Ser. Mat. Model. Progr. \yr 2018 \vol 11 \issue 2 \pages 139--146 \mathnet{http://mi.mathnet.ru/vyuru437} \crossref{https://doi.org/10.14529/mmp180211}