On numerical solution of one class of integro-differential equations in a special case

A complete theory of solvability of a linear integro-differential equation with a coefficient having power-law zeros is developed. For its approximate solution in the space of generalized functions, special generalized versions of the collocation method based on the use of standard polynomials and cubic splines of minimal defect are proposed and justified. Optimality in the order of accuracy of the method is established.