Abstract
Two methods based on quadrature formulas are proposed for the direct numerical integration of Prandtl’s singular integrodifferential equation. In the first method, Prandtl’s equation is solved directly by applying the method of mechanical quadrature and the circulation along an airfoil section is determined. In the second method, Prandtl’s equation is rewritten for the circulation derivative, which is determined by applying mechanical quadratures, and the circulation is then reconstructed using the same quadrature formulas. Both methods are analyzed numerically and are shown to converge. Their convergence rates are nearly identical, while the second method requires much more CPU time than the first one.
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References
N. I. Muskhelishvili, Singular Integral Equations (Nauka, Moscow, 1968; Wolters-Noordhoff, Groningen, 1972).
V. M. Aleksandrov and S. M. Mkhitaryan, Contact Problems for Bodies with Thin Coatings and Layers (Nauka, Moscow, 1985) [in Russian].
I. N. Vekua, “Prandtl’s integrodifferential equation,” Prikl. Mat. Mekh. 9(2), 143–156 (1945).
L. G. Magnaradze, “On a new integral equation in airfoil theory,” Soobshch. Akad. Nauk. Gruz. SSR 3, 503–508 (1942).
N. N. Shavlakadze, “Contact problem of bending for a plate with a thin reinforcement,” Mech. Solids 36(3), 122–127 (2001).
S. M. Belotserkovskii and I. K. Lifanov, Numerical Methods in Singular Integral Equations (Nauka, Moscow, 1985) [in Russian].
A. V. Sahakyan, “Method of discrete singularities as applied to solving singular integrodifferential equations,” Proceeding of the 6th International Conference on Problems in the Interaction Dynamics of Deformable Media (Goris, Armenia, 2008) pp. 383–387; http://www.mechins.sci.am/publ/avetik-sahakyan/Goris2008.doc.
A. V. Sahakyan, “Quadrature formulas for computing integrals with a variable upper limit,” Proceeding of the 2nd International Conference on Topical Problems in Continuum Mechanics (Dilijan, Armenia, 2010) pp. 107–111; http://www.mechins.sci.am/publ/avetik-sahakyan/Dilijan2010.doc.
A. A. Korneichuk, “Quadrature formulas for singular integrals,” in Numerical Methods for Solving Differential Equations and Quadrature Formulas (Nauka, Moscow, 1964), pp. 64–74 [in Russian].
A. V. Sahakyan, “Quadrature formulas of maximum algebraic accuracy for Cauchy type integrals with complex exponents of the Jacobi weight function,” Mech. Solids. 47, 695–699 (2012).
G. Szegö, Orthogonal Polynomials (Am. Math. Soc., New York, 1959; Fizmatgiz, Moscow, 1962).
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Original Russian Text © A.V. Sahakyan, N.N. Shavlakadze, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 8, pp. 1281–1288.
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Sahakyan, A.V., Shavlakadze, N.N. Two methods for direct numerical integration of the Prandtl equation and comparative analysis between them. Comput. Math. and Math. Phys. 54, 1244–1250 (2014). https://doi.org/10.1134/S0965542514080119
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DOI: https://doi.org/10.1134/S0965542514080119