Approximation properties of solutions to multipoint boundary-value problems

TL;DR

This paper demonstrates that solutions to general linear boundary-value problems for systems of differential equations can be approximated arbitrarily well by solutions to explicitly constructed multipoint boundary-value problems, with error estimates provided.

Contribution

It introduces a method to approximate solutions of general boundary-value problems using explicit multipoint problems, with error bounds in Sobolev and continuous function spaces.

Findings

Solutions can be approximated arbitrarily closely in Sobolev space.

Explicit multipoint boundary-value problems are constructed for approximation.

Error estimates for solutions are established in Sobolev and continuous spaces.

Abstract

We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$, known as general boundary-value problems. Their solutions $y:[a,b]\to \mathbb{C}^{m}$ belong to the Sobolev space $(W_1^{r})^m$, and the boundary conditions are given in the form $By=q$ where $B:(C^{(r-1)})^{m}\to\mathbb{C}^{rm}$ is an arbitrary continuous linear operator. We prove that a solution to such a problem can be approximated with an arbitrary precision in $(W_1^{r})^m$ by solutions to multipoint boundary-value problems with the same right-hand sides. These multipoint problems are built explicitly and do not depend on the right-hand sides of the general boundary-value problem. For these problems, we obtain estimates of errors of solutions in the normed spaces $(W_1^{r})^m$ and $(C^{(r-1)})^{m}$.