Improving solution convergence of boundary problems for ordinary differential equations by splitting basic operator into components

A method to solve self-conjugated boundary problems for ordinary differential equations being polynomials of differential operators of second order, is proposed. The method is based on representing the solution in terms of generalised Fourier’s series by eigenfunctions of differential operators of second order. A technique to improve the Fourier’s series depending on approach to split the basic operator into components, is described. This approach is generalised for the case when eigenfunctions of the operator of second order do not meet or meet only partially boundary conditions of the initial problem.