Geodesic
The rate of convergence of hypersingular equations numerical computation
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 2, pp. 139-146 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{VYURU_2018_11_2_a10,
     author = {S. I. Eminov and S. Yu. Petrova},
     title = {The rate of convergence of hypersingular equations numerical computation},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {139--146},
     year = {2018},
     volume = {11},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2018_11_2_a10/}
}
TY  - JOUR
AU  - S. I. Eminov
AU  - S. Yu. Petrova
TI  - The rate of convergence of hypersingular equations numerical computation
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2018
SP  - 139
EP  - 146
VL  - 11
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VYURU_2018_11_2_a10/
LA  - en
ID  - VYURU_2018_11_2_a10
ER  - 
%0 Journal Article
%A S. I. Eminov
%A S. Yu. Petrova
%T The rate of convergence of hypersingular equations numerical computation
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2018
%P 139-146
%V 11
%N 2
%U http://geodesic.mathdoc.fr/item/VYURU_2018_11_2_a10/
%G en
%F VYURU_2018_11_2_a10
S. I. Eminov; S. Yu. Petrova. The rate of convergence of hypersingular equations numerical computation. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 2, pp. 139-146. http://geodesic.mathdoc.fr/item/VYURU_2018_11_2_a10/

[1] Eminova V. S., Eminov S. I., “Justification of the Galerkin Method for Hypersingular Equations”, Journal of Computational Mathematics and Mathematical Physics, 56:3 (2016), 417–425 | DOI | DOI | MR | Zbl

[2] W. Rudin, Functional Analysis, McGRAW-HILL, N.Y., 1973 | MR | Zbl

[3] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and Series. Special Functions, Nauka, M., 1983 (in Russain)

[4] Computer Techniques in Electromagnetics, Pergamon, 1973

[5] Sukacheva T. G., Matveeva O. P., “Taylor Problem for the Zero-Order Model of an Incompressible Viscoelastic Fluid”, Differential Equations, 51:6 (2015), 783–791 | DOI | DOI | MR | Zbl