Equivalence of the Fundamental Spline and Green’s Function in Constructing an Exact Finite-Dimensional Analog of the Boundary-Value Problem for an Ordinary Differential Equation of the Fourth Order

The author considers a problem with the main and natural boundary conditions on an interval. A new method for constructing an exact discrete analog of the problem is proposed. The method deals with the projection of the differential equation on local splines formed by the fundamental system of solutions to the Cauchy problems for the homogeneous equation of the original problem. A system of linear algebraic equations with a 5-diagonal matrix is obtained for the values of the exact solutions of the original problem at the points of a uniform grid. To implement an exact analog, we recommend using high-order accuracy schemes formed by partial sums of series in even powers of the grid step for the solutions to Cauchy problems.