The Newton method for solving the Theodorsen integral equation

摘要

The Newton method for the solution of the Theodorsen integral equation in conformal mapping is studied. One step of this method consists of solving a linear integral equation, the solution of which is given explicitly as the result of a Riemann-Hilbert problem. Quadratic convergence of the Newton method is established under certain assumptions. Whereas in other methods a so-called ϵ-condition with ϵ < 1 is required to hold, our method converges also for ϵ ⩾ 1. We will also present a numerical implementation in which the result of one step of the Newton method is approximately by a vector in R2N which can be computed with 2N log N + O(N) multiplications. In comparison, one step of the Newton method for the discrete Theodorsen equation requires O(N3) multiplications.