A method of verified computations for solutions to semilinear parabolic equations using semigroup theory

Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo, Shinichi Oishi

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem is transformed into a fixed-point form using an analytic semigroup. The sufficient condition is derived from Banach's fixed-point theorem. This paper also introduces a recursive scheme to extend a time interval in which the validity of the solution can be verified. As an application of this method, the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation.

Original languageEnglish
Pages (from-to)980-1001
Number of pages22
JournalSIAM Journal on Numerical Analysis
Volume55
Issue number2
DOIs
Publication statusPublished - 2017 Jan 1

Keywords

  • Existence and local uniqueness
  • Semilinear parabolic initial-boundary value problems
  • Verified numerical computations

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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