An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials
Abstract
Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed. This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.
References
- Panasyuk VV, Savruk MP, Nazarchuk ZT. Metod singulyarnykh integral’nykh uravnenii v dvumernykh zadachakh difraktsii [The method of singular integral equations in two-dimensional diffraction problems]. Kyiv: Naukova dumka; 1984. 344 p. Russian.
- Bakhvalov NS, Zhidkov NP, Kobel’kov GM. Chislennye metody [Numerical methods]. Moscow: Nauka; 1987. 598 p. Russian.
- Bateman Н. Higher transcendental functions. Volume 2. Erdélyi A, editor. New York: McGraw-Hill Book Company; 1953. XIV , 396 p. Russian edition: Bateman Н, Erdélyi A. Vysshie transtsendentnye funktsii. Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonal’nye mnogochleny. Vilenkin NYa, translator. Moscow: Nauka; 1966. 295 p. (Spravochnaya matematicheskaya biblioteka).
- Pashkovskii S. Vychislitel’nye primeneniya mnogochlenov i ryadov Chebysheva [Computational applications of polynomials and Chebyshev series]. Lebedev VI, editor; Kiro SN, translator. Moscow: Nauka; 1983. 384 p. Russian.
- Rasolko GA. Numerical solution of singular integro-differential Prandtl equation by the method of orthogonal polynomials. Journal of the Belarusian State University. Mathematics and Informatics. 2018;3:68–74. Russian.
Copyright (c) 2020 Journal of the Belarusian State University. Mathematics and Informatics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
The authors who are published in this journal agree to the following:
- The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
- The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
- The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)