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

Fingerprint

Semigroup Theory
Semilinear Parabolic Equation
Initial-boundary-value Problem
Boundary value problems
Banach Fixed Point Theorem
Analytic Semigroup
Sufficient Conditions
Unique Solution
Numerical methods
Uniqueness
Fixed point
Numerical Methods
Numerical Solution
Interval
Formulation
Theorem
Form

Keywords

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

ASJC Scopus subject areas

  • Numerical Analysis

Cite this

A method of verified computations for solutions to semilinear parabolic equations using semigroup theory. / Mizuguchi, Makoto; Takayasu, Akitoshi; Kubo, Takayuki; Oishi, Shinichi.

In: SIAM Journal on Numerical Analysis, Vol. 55, No. 2, 2017, p. 980-1001.

Research output: Contribution to journalArticle

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