Numerical solution of a singular integral equation related with a dynamic contact interaction problem

Authors

  • V. G. Popov National University «Odesa maritime academy», 65029, Odessa, Didrikhson str., 8
  • O. I. Kyrylova National University «Odesa maritime academy», 65029, Odessa, Didrikhson str., 8

DOI:

https://doi.org/10.17721/1812-5409.2021/3.17

Keywords:

singular integral equation, fixed singularity, contact stresses

Abstract

A singular integral equation with a fixed singularity to which the problem of contact interaction of two quarters of spaces in the conditions of harmonic oscillations of longitudinal shear is reduced is considered. A quarters of the space is situated so that the half-space composed of them has a stepped boundary. In the contact area, the conditions for ideal coupled are satisfied. The unknown function in this equation is the contact stresses. For the numerical solution of this equation, a method that takes into account the asymptotic behavior of contact stresses at the edge point is proposed. The basis of this method is the use of special quadrature formulas for singular integrals obtained in the article. When obtaining these formulas, the unknown function was approximated by an interpolation polynomial, in which the roots of the Laguerre polynomials are the points of interpolation. The values of the unknown function at the interpolation points are found by the collocation method, herewith the collocation points of collocationare the roots of the special function. An approximate formula for calculating contact stresses can have practical application. The effectiveness of the proposed method is demonstrated by the numerical example.

Pages of the article in the issue: 93 - 96

Language of the article: Ukrainian

References

POPOV G. (1969) O metode ortogonal’nih mnogotshlenov v contactnih zadachah teorii uprugosty. PMM. Т.33. № 3. P. 518-533.

BELOTSERKOVSKIJ S., LIFANOV I. (1985) Chislennije metodi v singularnih integralnih uravnenijah i ih primenenije v aerodinamike, teoriji uprugosti, elektrodinamike. Moskva: Nauka.

POPOV V. (2020) Dvovimirni zadachi teoriji prugnosti, hsto zvodjatsja do singularnih integral’nih rivn’an’ z neruhomimi osoblivoctjami Mat.metodi ta fiz.-meh. polja. № 63 (1). P. 94-105.

DUDUCHAVA R. (1979) Integralnie uravnenija svertki s razrivnimi predsimvolami, singularnie integral’nie uravnenija s nepodvignimi osobennostami i ih prilogenija k zadacham mechaniki. Trudi Tbilisskogo matem. instituta AN SSSR. T. 60. P.313.

SEGE G. (1962) Ortogonal’nie mnogotshleni. Moskva: Fizmatgiz.

KRILOV V. (1967) Pribligennoje vitshislenije integralov. Moskva: Nauka.

No 3 (2021)

Downloads

Published

2021-12-07

How to Cite

Popov, V. G., & Kyrylova, O. I. (2021). Numerical solution of a singular integral equation related with a dynamic contact interaction problem. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (3), 93–96. https://doi.org/10.17721/1812-5409.2021/3.17

Issue

No. 3 (2021)

Section

Differential equations, mathematical physics and mechanics

License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).