The convergence of a quadrature-difference method for a class of linear singular integro-differential equations on a segment (Q1390424)

The author analyzes the discretization of boundary-value problems for singular integro-differential equations \[ \sum^n_{\nu= 0}[a_\nu(t) x^{(\nu)}(t)+ b_\nu(t)(Sx^{(\nu)})(t)+ (Th_\nu x^{(\nu)})(t)]= f(t),\quad t\in I:= (-1,1), \] \(m\geq 1\), where the given functions \(a_\nu\), \(b_\nu\), \(f\), and \(h_\nu\) are continuous on \(\overline I\) and \(\overline I\times\overline I\), respectively, by quadrature-difference methods. Here, the integral operators are given by \[ (Sx^{(\nu)})(t):= {1\over\pi} \int^1_{-1} (\tau- t)^{-1} x^{(\nu)}(\tau)d\tau\quad(\text{Cauchy integral}), \] and \[ (Th_\nu x^{(\nu)})(t):= {1\over \pi} \int^1_{-1} h_\nu(t,\tau) x^{(\nu)}(\tau)d\tau \quad(\text{regular integral}). \] These methods are based on nodal points given by the zeros of appropriate Jacobi polynomials, and on suitable difference approximations to the derivatives. The analysis of the resulting convergence rates for equations of index \(0\) or \(\pm 1\) assumes Hölder conditions for the coefficients. There are no numerical examples or applications.