A numerical verification method for the existence of weak solutions for nonlinear boundary value problems

Mitsuhiro T. Nakao

Research output: Contribution to journalArticle

48 Citations (Scopus)

Abstract

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems (BVP). We formulate the problem as a fixed point of a Newton-like operator and present a verification algorithm by computer based on Sadovskii's fixed point theorem. In order to compute the Newton type iterative sequences in infinite dimensional space, two concepts, the rounding and the rounding error, are introduced. Also some numerical examples are illustrated. We emphasize that, since the present method uses the weak formulation of BVP and the finite element approximations instead of the existing validation techniques, it is aslo applicable, in principle, to the BVP for partial differential equations as it stands.

Original languageEnglish
Pages (from-to)489-507
Number of pages19
JournalJournal of Mathematical Analysis and Applications
Volume164
Issue number2
DOIs
Publication statusPublished - 1992 Mar 1
Externally publishedYes

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Numerical Verification
Existence of Weak Solutions
Nonlinear Boundary Value Problems
Boundary value problems
Boundary Value Problem
Weak Formulation
Infinite-dimensional Spaces
Rounding error
Rounding
Two-point Boundary Value Problem
Finite Element Approximation
Numerical Techniques
Fixed point theorem
Existence of Solutions
Partial differential equation
Fixed point
Verify
Numerical Examples
Partial differential equations
Mathematical operators

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

A numerical verification method for the existence of weak solutions for nonlinear boundary value problems. / Nakao, Mitsuhiro T.

In: Journal of Mathematical Analysis and Applications, Vol. 164, No. 2, 01.03.1992, p. 489-507.

Research output: Contribution to journalArticle

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