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February 2006 A Mathematical Theory for Numerical Treatment of Nonlinear Two-Point Boundary Value Problems
Tetsuro Yamamoto, Shin'ichi Oishi
Japan J. Indust. Appl. Math. 23(1): 31-62 (February 2006).

Abstract

This paper gives a unified mathematical theory for numerical treatment of two--point boundary value problems of the form $-(p(x)u')'+f(x,u,u')=0,\ a\le x \le b,\ \alpha_0 u(a)-\alpha_1 u'(a)=\alpha,\ \beta_0 u(b)+\beta_1 u'(b)=\beta,\ \alpha_0,\alpha_1,\beta_0,\beta_1 \ge 0,\ \alpha_0+\alpha_1>0,\ \beta_0+\beta_1>0,\ \alpha_0 +\beta_0 >0$. Firstly, a unique existence of solution is shown with the use of the Schauder fixed point theorem, which improves Keller's result \cite{Keller}. 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 \cite{Keller} and Ortega--Rheinboldt \cite{Ortega}. Furthermore, it is shown that, under some assumptions on $p$ and $f$, the solution for the discrete problem has the second order accuracy $O(h^2)$, where $h$ denotes the maximum mesh size. Finally, observations are given.

Citation

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Tetsuro Yamamoto. Shin'ichi Oishi. "A Mathematical Theory for Numerical Treatment of Nonlinear Two-Point Boundary Value Problems." Japan J. Indust. Appl. Math. 23 (1) 31 - 62, February 2006.

Information

Published: February 2006
First available in Project Euclid: 19 June 2006

zbMATH: 1105.34009
MathSciNet: MR2210295

Keywords: error estimate , Existence of solution , finite difference methods , ‎fixed point theorems , two-point boundary value problems

Rights: Copyright © 2006 The Japan Society for Industrial and Applied Mathematics

Vol.23 • No. 1 • February 2006
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