Linear Volterra equations of the first kind are considered. A class of problems that have a single solution is identified, and collocation-variational methods are proposed to solve them numerically. The essence of these algorithms is that the approximate solution is found at the nodes of a uniform grid (the collocation condition) that yield an underdetermined system of linear algebraic equations. The system thus obtained is supplemented by the condition of minimum of the objective function, which approximates the squared norm of the approximate solution. As a result, a quadratic programming problem is obtained, viz. the objective function (the squared norm of the approximate solution) is quadratic, and the constraints (the collocation conditions) are equalities. This problem is solved by the method of Lagrange multipliers. Sufficiently simple third-order methods are considered in detail. The calculation results for test problems are given. Further development of this approach to solve other classes of integral equations numerically is discussed.
M. V. Bulatov (Wed,) studied this question.