On computational proofs of the existence of solutions to nonlinear parabolic problems

Mitsuhiro T. Nakao, Yoshitaka Watanabe

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

This paper is an extension of the preceding study (Nakao, this journal, 1991) in which we described a numerical verification method of the solution for one-space dimensional parabolic problems, to the several-space dimensional case. Here, numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer. We reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed-point problem of a compact operator on certain function spaces. As in the preceding study based upon a simple C0 finite-element approximation and its constructive a priori error estimates, a numerical verification procedure is presented with some numerical examples.

Original languageEnglish
Pages (from-to)401-410
Number of pages10
JournalJournal of Computational and Applied Mathematics
Volume50
Issue number1-3
DOIs
Publication statusPublished - 1994 May 20
Externally publishedYes

Fingerprint

Nonlinear Parabolic Problems
Numerical Verification
Existence of Solutions
Parabolic Problems
A Priori Error Estimates
Fixed Point Problem
Compact Operator
Finite Element Approximation
Numerical Techniques
Function Space
Initial-boundary-value Problem
Boundary value problems
Mathematical operators
Numerical Examples

Keywords

  • Error estimates
  • Finite-element method
  • Fixed-point theorem
  • Parabolic problem

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

On computational proofs of the existence of solutions to nonlinear parabolic problems. / Nakao, Mitsuhiro T.; Watanabe, Yoshitaka.

In: Journal of Computational and Applied Mathematics, Vol. 50, No. 1-3, 20.05.1994, p. 401-410.

Research output: Contribution to journalArticle

Nakao, Mitsuhiro T. ; Watanabe, Yoshitaka. / On computational proofs of the existence of solutions to nonlinear parabolic problems. In: Journal of Computational and Applied Mathematics. 1994 ; Vol. 50, No. 1-3. pp. 401-410.
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