A mathematical theory for numerical treatment of nonlinear two-point boundary value problems

Tetsuro Yamamoto, Shin'ichi Oishi

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

This paper gives a unified mathematical theory for numerical treatment of two-point boundary value problems of the form -(p(x)u′)′ + f(x,u,u′)=0, a ≤ x ≤ b, α0u(a) - α1u′(a) = α, β0u(b) + β1u′(b) = β, α0, α1, β0, β10, α0 + α1 > 0, β0 + β1 > 0, α0 + β0 > 0. Firstly, a unique existence of solution is shown with the use of the Schauder fixed point theorem, which improves Keller's result [6]. Next, a new discrete boundary value problem with arbitrary nodes is proposed. The unique existence of solution for the problem is also proved by using the Brouwer theorem, which extends some results in Keller [6] and Ortega-Rheinboldt [10]. Furthermore, it is shown that, under some assumptions on p and f, the solution for the discrete problem has the second order accuracy O(h2), where h denotes the maximum mesh size. Finally, observations are given.

Original languageEnglish
Pages (from-to)31-62
Number of pages32
JournalJapan Journal of Industrial and Applied Mathematics
Volume23
Issue number1
DOIs
Publication statusPublished - 2006 Feb

Keywords

  • Error estimate
  • Existence of solution
  • Finite difference methods
  • Fixed point theorems
  • Two-point boundary value problems

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

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