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A discrete method for the logarithmic-kernel integral equation on an open arc. (English) Zbl 0782.65159

The third author and B. J. Burn [J. Integral Equations Appl. 4, No. 1, 117-151 (1992; Zbl 0760.65131)] introduced a family of quadrature methods for solving the singular integral equation \[ (-1/\pi)\int_ \Gamma v(y)\log| x-y| ds_ y = g(x),\quad x\in\Gamma, \] on a closed curve \(\Gamma\). The present paper analyses the adaption of this approach to the case where \(\Gamma\) is an open (smooth) arc: here, the quadrature methods are used, after employing the well-known cosine substitution, to discretize a certain Petrov-Galerkin approximation. It is shown that this family of quadrature methods includes schemes of arbitrary order \(p\). As an illustration, the authors describe a stable \({\mathcal O}(h^ 3)\) method on a uniform mesh.

MSC:

65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

Citations:

Zbl 0760.65131
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