Skip to main content
Log in

Method of Orthogonal Polynomials for an Approximate Solution of Singular Integro-Differential Equations as Applied to Two-Dimensional Diffraction Problems

  • NUMERICAL METHODS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We consider a mathematical model of scattering of \(H \)-polarized electromagnetic waves by a screen with a curvilinear boundary based on a singular integro-differential equation with a Cauchy kernel and a logarithmic singularity. The integrands contain both the unknown function and its first derivative. For the numerical analysis of this model, two computational schemes are constructed based on the representation of the unknown function in the form of a linear combination of orthogonal Chebyshev polynomials and spectral relations, which permit one to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the basis of Chebyshev polynomials are calculated as a solution of the corresponding system of linear algebraic equations. The results of numerical experiments show that the error in the approximate solution on a grid of 20–30 nodes does not exceed the roundoff error.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Rasol’ko, G.A., Numerical solution of a singular integro-differential Prandtl equation by the method of orthogonal polynomials, Zh. Belorus. Gos. Univ. Mat. Inf., 2018, no. 3, pp. 68–74.

  2. Rasol’ko, G.A., To the numerical solution of a singular integro-differential Prandtl equation by the method of orthogonal polynomials, Zh. Belorus. Gos. Univ. Mat. Inf., 2019, no. 1, pp. 58–68.

  3. Rasol’ko, G.A., Sheshko, S.M., and Sheshko, M.A., Numerical method for some singular integro-differential equations, Differ. Equations, 2019, vol. 55, no. 9, pp. 1242–1249.

    Article  MathSciNet  Google Scholar 

  4. Rasol’ko, G.A. and Sheshko, S.M., Approximate solution of one singular integro-differential equation by the method of orthogonal polynomials, Zh. Belorus. Gos. Univ. Mat. Inf., 2020, no. 2, pp. 10–20.

  5. Panasyuk, V.V., Savruk, M.P., and Nazarchuk, Z.T., Metod singulyarnykh integral’nykh uravnenii v dvumernykh zadachakh difraktsii (Method of Singular Integral Equations in Two-dimensional Diffraction Problems), Kiev: Naukova Dumka, 1984.

    Google Scholar 

  6. Bateman, H. and Erdélyi, A., Higher Transcendental Functions, New York: McGraw-Hill, 1953. Translated under the title: Vysshie transtsendentnye funktsii. T. 2 , Moscow: Nauka, 1966.

    MATH  Google Scholar 

  7. Pashkovskii, S., Vychislitel’nye primeneniya mnogochlenov i ryadov Chebysheva (Computational Applications of Chebyshev Polynomials and Series), Moscow: Nauka, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to G. A. Rasol’ko or V. M. Volkov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rasol’ko, G.A., Volkov, V.M. Method of Orthogonal Polynomials for an Approximate Solution of Singular Integro-Differential Equations as Applied to Two-Dimensional Diffraction Problems. Diff Equat 57, 814–823 (2021). https://doi.org/10.1134/S0012266121060094

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266121060094

Navigation