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

  • Tetsuro Yamamoto
  • Shin’ichi Oishi
Article

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,axb, α0 u(a) − α1 u′(a) = α, β0 u(b) + β1 u′(b) = β, α0, α1, β0, β1 ≥ 0, 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 onp and ƒ, the solution for the discrete problem has the second order accuracyO(h 2), whereh denotes the maximum mesh size. Finally, observations are given.

Key words

two-point boundary value problems existence of solution error estimate fixed point theorems finite difference methods 

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Copyright information

© JJIAM Publishing Committee 2006

Authors and Affiliations

  1. 1.School of Science and EngineeringWaseda UniversityTokyoJapan

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