Collocation-Variational Approaches to the Numerical Solution of Volterra Integral Equations of the First Kind

Linear Volterra equations of the first kind are considered. A class of problems with a unique solution is identified, for which collocation-variational solution methods are proposed. According to the proposed algorithms, an approximate solution is found at nodes of a uniform grid (collocation condition), which yields an underdetermined system of linear algebraic equations. The system thus obtained is supplemented with the minimization condition for the objective function, which approximates the squared norm of the approximate solution. As a result, we obtain a quadratic programming problem with a quadratic objective function (squared norm of the approximate solution) and equality constraints (collocation conditions). This problem is solved by applying the Lagrange multiplier method. Fairly simple third-order methods are considered in detail. Numerical results for test problems are presented. Further development of this approach for the numerical solution of other classes of integral equations is discussed.