On the method of integrating matrices for solving boundary value problems for ordinary equations of the fourth order (Q1382674)

Further studies on the method of integrating matrices (MIM) for solving the boundary value problem (BVP) for ordinary differential equations of fourth order are presented. It is known that MIM consists of two phases: (i) reducing BVP to a second kind Volterra integral equation and (ii) approximating the Volterra operator by means of a collocation method. In the second phase the corresponding integrating matrices have to be determined. Thus the BVP reduces to solving a finite-dimensional algebraic system of equations. The solution of BVP is then restored using numerical integration and the computed integrating matrices. The most important advantage of MIM is that the algebraic system is well conditioned and this property does not depend on the choice of the collocation nodes. Till now MIM was combined with sliding cubic and cubic spline interpolations to obtain the integrating matrices. These techniques lead to dense matrices in the linear systems. Here the authors propose another approach, they take the zeros of the Legendre polynomials and use them as collocation nodes in the Lagrange interpolation. This produces a sparse matrix in the discrete problem, which is also symmetric if the BVP is self adjoint. The authors demonstrate their approach for a one-dimensional BVP of the form \[ \begin{aligned} &(mu'')'' - (pu')' +gu = f, \quad x \in (0,l);\\ &u=u_0,\quad u' =u_{0}' \text{ for } x=0; \\ &mu'' = -q,\quad (mu'')' - pu' = q' \text{ for } x=l.\end{aligned} \] Stability of the solution of the discrete problem is proved and estimates for the exactness of the approximate solution are given.