Direct spectral Chebyshev method for solving the generalized wing equation
https://doi.org/10.29235/1561-2430-2026-62-1-15-26
Abstract
The generalized Prandtl integro-differential equation is considered, in which the singular integrals contain both the unknown solution and its derivative. Direct numerical methods for solving this problem are constructed, based on representing the solution and the variable coefficients of the equation in the form of Chebyshev interpolation polynomials. The developed approach, which utilizes known spectral relations, makes it possible to obtain analytical expressions for the singular component of the problem and reduces its solution to a system of linear algebraic equations for the coefficients of the interpolation polynomial. Two computational schemes for the solution are constructed using Chebyshev polynomials of the first and second kind, respectively. Using a model problem as an example, it is shown that both schemes ensure good conditioning of the algebraic problem and possess high accuracy, demonstrating an exponential convergence rate of the error.
Keywords
singular equation,
integro-differential equation,
orthogonal basis of Chebyshev polynomials,
spectral Chebyshev method,
generalized Prandtl equation
Supporting agencies: This work was supported by the State Program of Scientific Research “Convergence-2025”, subprogram “Mathematical models and methods”, task 1.4.01.2.
About the Authors
G. A. Rasolko
Belarusian State University
Belarus
Belarus
Galina A. Rasolka – Ph. D. (Physics and Mathematics), Associate Professor, Associate Professor of Web-Technologies and Computer Simulation Department
4, Nezavisimosti Ave., 220030, Minsk
V. M. Volkov
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Vasily M. Volkov – Dr. Sc. (Physics and Mathematics), Associate Professor, Chief Researcher
11, Surganov Str., 220072, Minsk
M. V. Ignatenko
Belarusian State University
Belarus
Belarus
Marina V. Ignatenko – Ph. D. (Physics and Mathematics), Associate Professor, Head of Web-Technologies and Computer Simulation Department
4, Nezavisimosti Ave., 220030, Minsk
References
1. Golubev V. V. Lectures on Wing Theory. Moscow, Nauka Publ., 1949. 480 p. (in Russian).
2. Gabdulkhaev B. G. Direct methods for solving the equation of wing theory. Izvestiya vysshikh uchebnykh zavedenii. Matematika = Russian Mathematics, 1974, no. 2, pp. 29–44 (in Russian).
3. Ivanov V. V. Theory of Approximate Methods and its Application to the Numerical Solution of Singular Integral Equations. Kyiv, Naukova dumka Publ., 1968. 287 p. (in Russian).
4. Elliott D. A comprehensive approach to the approximate solution of singular integral equations over the arc (−1,1). The Journal of Integral Equations and Applications, 1989, vol. 2, no. 1, pp. 59–94. https://doi.org/10.1216/jie-1989-2-1-59
5. Sahlan M. N., Feyzollahzadeh H. Operational matrices of Chebyshev polynomials for solving singular Volterra integral equations. Mathematical Sciences, 2017, vol. 11, no. 2, pp. 165–171. https://doi.org/10.1007/s40096-017-0222-4
6. Rasolko G. A. Numerical solution of singular integro-differential Prandtl equation by the method of orthogonal polynomials. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika = Journal of the Belarusian State University. Mathematics and Informatics, 2018, no. 3, pp. 68–74 (in Russian).
7. Rasolko G. A. To the numerical solution of singular integro-differential Prandtl equation by the method of orthogonal polynomials. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika = Journal of the Belarusian State University. Mathematics and Informatics, 2019, no. 1, pp. 58–68 (in Russian). https://doi.org/10.33581/2520-6508-2019-1-58-68
8. Rasolko G. A., Sheshko S. M., Sheshko M. A. Numerical Method for Some Singular Integro-Differential Equations. Differential Equations, 2019. vol. 55, no. 9, pp. 1242–1249. https://doi.org/10.1134/S0012266119090118
9. Bateman G., Erdelyi A. Higher Transcendental Functions. Moscow, Nauka Publ., 1973. 297 p. (in Russian).
10. Muskhelishvili N. I. Singular Integral Equations. Moscow, Nauka Publ., 1968. 513 p. (in Russian).
11. Rasolko G. A., Volkov V. M. Chebyshev spectral method for one class of singular integro-differential equations. Computational Mathematics and Mathematical Physics, 2025, vol. 65, no. 2, pp. 339–348. https://doi.org/10.1134/s0965542524701963
12. Pashkovsky S. Computational Applications of Chebyshev Polynomials and Series. Moscow, Nauka Publ., 1983. 384 p. (in Russian).
13. Kolmogorov A. N., Fomin S. V. Elements of the Theory of Functions and Functional Analysis. Moscow, Nauka Publ., 1989. 543 p. (in Russian).
14. Ozhegova A. V., Surai L. A. On the convergence in the integral metric of approximate solutions of the generalized equation of wing theory. Vestnik KGTU im. A. N. Tupoleva = Bulletin of KSTU named after A. N. Tupolev, 2014. no. 1, pp. 64–67.
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