The perturbed Galerkin method for Cauchy singular integral equation with constant coefficients (Q1115167)

The equation which is considered in this paper has the form: \[ (*)\quad av(x)+b/\pi \int^{1}_{-1}v(t)/(t-x)dt+\int^{1}_{- 1}k(x,t)v(t)dt=f(x),\quad -1<x<1 \] where a and b are constants. A Galerkin method based on Jacobi polynomials, is proposed for an approximate solution of the equation and from this is developed a discrete Galerkin method. \(L_ 2\) and uniform convergence results are proved under Hölder continuity conditions on the kernel k, and right hand side f which take into account the quadrature errors when inner products are replaced by numerical quadratures. These results lead to proofs of classical collocation methods for the solution of (*). The approach is based on that due to \textit{G. Miel} [Integral Equations Oper. Theory 8, 268-275 (1985; Zbl 0557.47008) and SIAM J. Numer. Anal. 23, 135-143 (1986; Zbl 0593.65093)].