Optimal by the order methods of solving integral equations in a special case (Q1695287)

The following linear third-kind integral equation with fixed singularities of the kernel is considered \[ Ax\equiv x(t)\prod_{j=1}^{l}(t-t_j)^{m_j} + \int\limits_{-1}^{1}K(t,s)\left[(s+1)^{p_1}(1-s)^{p_2} \right]^{-1} x(s)ds=y(t), \tag{1} \] where \(t\in [-1,1]\), \(t_j\in(-1,1)\), \(m_j\in \mathbb{N} \; (j=\overline{1,l})\), \(p_1,p_2\in \mathbb{R}^{+}\), \(K\) and \(y\) are given continuous functions satisfying some smoothness properties. The integral in (1) is understood in the sense of Hadamard finite part. The authors suggest and substantiate special generalized versions of spline methods for (1) in the space of distributions. It is shown that these methods are optimal by the order.