Special versions of the subdomain method for integro-differential equations in the singular case (Q2388016)

The author investigates the approximate solution of the integro-differential equation \[ x(t) \prod_{j=1}^q(t-t_j)^{m_j}+ \sum_{j=0}^p \int_{-1}^1 K_j(t,s)x^{(j)}(s)\, ds= y(t)\quad (-1\leq t\leq 1) \] using new versions of the subdomain method based on special polynomial and spline methods, proves existence and uniqueness theorems for the corresponding approximate equation with the convergence of the sequence of approximate solutions, and estimates their error, where \(t_j\in(-1,1)\) and \(m_j\in{\mathbb N}\;(j=1,\ldots,q),\) \(K_j\;(j=0,\ldots,p)\) and \(y\) are given smooth functions, and \(x\) is the unknown function.