### 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, α_{0}u(a) - α_{1}u′(a) = α, β_{0}u(b) + β_{1}u′(b) = β, α_{0}, α_{1}, β_{0}, β_{1} ≥ _{0}, α_{0} + α_{1} > 0, β_{0} + β_{1} > 0, α_{0} + β_{0} > 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(h^{2}), where h denotes the maximum mesh size. Finally, observations are given.

Original language | English |
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Pages (from-to) | 31-62 |

Number of pages | 32 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 23 |

Issue number | 1 |

Publication status | Published - 2006 Feb |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**A mathematical theory for numerical treatment of nonlinear two-point boundary value problems.** / Yamamoto, Tetsuro; Oishi, Shinichi.

Research output: Contribution to journal › Article

*Japan Journal of Industrial and Applied Mathematics*, vol. 23, no. 1, pp. 31-62.

}

TY - JOUR

T1 - A mathematical theory for numerical treatment of nonlinear two-point boundary value problems

AU - Yamamoto, Tetsuro

AU - Oishi, Shinichi

PY - 2006/2

Y1 - 2006/2

N2 - 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, 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 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.

AB - 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, 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 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.

KW - Error estimate

KW - Existence of solution

KW - Finite difference methods

KW - Fixed point theorems

KW - Two-point boundary value problems

UR - http://www.scopus.com/inward/record.url?scp=33645527992&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645527992&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33645527992

VL - 23

SP - 31

EP - 62

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

IS - 1

ER -