Chebyshev spectral method for solving complete generalised Prandtl equation - Rasolko, Galina A.; Volkov, Vasily M.; Ignatenko, Marina V. (Journal of the Belarusian State University. Mathematics and Informatics, 2026)
This article is devoted to the problem of constructing computational schemes for solving Prandtl integro-differential equations that arise in many problems in mechanics. An approximate numerical method for solving singular integro-differential equations of the generalised Prandtl equation type has been developed. The proposed approximate computational schemes are based on representing the solution of the equation as an expansion over an orthogonal basis of Chebyshev polynomials. The use of known spectral relations has made it possible to obtain an analytical expression for the singular component of the equation. As a consequence, the developed method demonstrates excellent accuracy and exponential rate of convergence of the approximate solution in relation to the degree of interpolation polynomials. The computational qualities of this method are demonstrated using a test example. In particular, it is shown that a discrete model based on the representation of the solution as a decomposition by Chebyshev polynomials leads to a well-conditioned system of linear algebraic equations for the decomposition coefficients, and the convergence rate of the approximate solution error can reach a linear speed in relation to the degree of the interpolation polynomial.
Keywords: approximate numerical algorithm; singular equation; integro-differential equation; orthogonal basis of Chebyshev polynomials; Chebyshev spectral method; generalised Prandtl equation