Collocation-variational approaches to the solution to Volterra integral equations of the first kind

Volterra integral equations of the first kind on a bounded interval are considered. It is assumed that the kernel and the right-hand side of an equation are sufficiently smooth functions, the kernel does not vanish on the diagonal, while the right-hand side vanishes at the initial integration moment. For the numerical solution to such equations, single-step methods based on two-step quadrature rules are proposed. Discretization of this type yields an underdetermined system of linear algebraic equations, which has infinitely many solutions. The system is supplemented with the condition that the norm of the approximate solution is minimal in some analogues of a Sobolev space to uniquely determine the approximate solution at discretization nodes. Such methods are always stable in the case of a second-order approximation and converge to the exact solution with the second order. Numerical results produced by the proposed methods as applied to well-known test examples are presented.