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A nodal spline collocation method for the solution of Cauchy singular integral equations. (English) Zbl 1176.65153
The authors present a collocation method, based on optimal nodal spline approximation, for solving the following singular integral equation with constant coefficients
$aw_{\alpha ,\beta } (x)f(x) + \frac{b}{\pi }\int_{ - 1}^1 {\frac{w_{\alpha ,\beta } (t)f(t)}{t - x}dt} + \int_{ - 1}^1 {w_{\alpha ,\beta } (t)k(x,t)f(t)dt = g(x),\quad - 1 < x < 1},$
where the first integral is defined in the sense of Cauchy principal value, $$k$$ is a Fredholm kernel and $$w_{\alpha ,\beta }$$ is the Jacobi weight function
$w_{\alpha ,\beta } (t) = (1 - t)^\alpha (1 + t)^\beta ,\,\,\alpha ,\beta > - 1.$

##### MSC:
 65R20 Numerical methods for integral equations 45E05 Integral equations with kernels of Cauchy type