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Quadrature methods for solving Theodorsen’s singular integral equation. (English. Russian original) Zbl 1074.65152
Russ. Math. 47, No. 8, 79-82 (2003); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2003, No. 8, 82-85 (2003).
Introduction: Summary: We investigate the quadrature methods for solving Theodorsen’s singular integral equation \[ \varphi(s)+ {\lambda\over 2\pi} \int^{2\pi}_0 \ln\rho(\varphi(\sigma))\text{ctg} {\sigma- s\over 2} d\sigma= y(s),\tag{1} \] where \(\lambda\) is the numerical parameter, \(\rho(s)\) and \(y(s)\) are the known \(2\pi\)-periodic functions, and \(\varphi(s)\) is the desired function. The singular integral is understood here as the Cauchy-Lebesgue principal value. This equation is used in theory of conformal mappings and in a number of applications. Numerous results exist concerning the approximate approaches for solving equation (1).
65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
30C20 Conformal mappings of special domains