Approximation properties of solutions to multipoint boundary-value problems

2020/12/31 by A. A. Murach, Aleksandr Murach, O. B. Pelekhata +5 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #math.CA #msc:34B08 #msc:34B10

paper · pdf · doi:10.48550/arxiv.2012.15604

13 pages, revised version, Russian

arxiv created 2021/01/26 · arxiv updated 2021/01/27

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]→ ℂ^m belong to the Sobolev space (W₁^r)^m, and the boundary conditions are given in the form By=q where B:(C^(r-1))^m→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 (W₁^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 (W₁^r)^m and (C^(r-1))^m.

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• Parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces

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