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

Tetsuro Yamamoto, Shinichi 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 . 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  and Ortega-Rheinboldt . 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 language English 31-62 32 Japan Journal of Industrial and Applied Mathematics 23 1 Published - 2006 Feb

### Fingerprint

Nonlinear Boundary Value Problems
Two-point Boundary Value Problem
Boundary value problems
Existence of Solutions
Brouwer's theorem
Discrete Boundary Value Problem
Second-order Accuracy
Schauder Fixed Point Theorem
Mesh
Denote
Arbitrary
Vertex of a graph
Observation
Form

### 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

In: Japan Journal of Industrial and Applied Mathematics, Vol. 23, No. 1, 02.2006, p. 31-62.

Research output: Contribution to journalArticle

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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 . 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  and Ortega-Rheinboldt . 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.

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