Accurate method of verified computing for solutions of semilinear heat equations

Akitoshi Takayasu, Makoto Mizuguchi, Takayuki Kubo, Shinichi Oishi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact solution in a neighborhood of a numerically computed approximate solution. Our method is based on a fixed-point formulation using the evolution operator, an iterative numerical verification scheme to extend a time interval in which the validity of the solution can be verified, and rearranged error estimates for avoiding the propagation of an overestimate. As a result, compared with the previous verification method using the analytic semigroup, our method can enclose the solution for a longer time. Some numerical examples are presented to illustrate the efficiency of our verification method.

Original languageEnglish
Pages (from-to)74-99
Number of pages26
JournalReliable Computing
Volume25
Publication statusPublished - 2017

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Keywords

  • Evolution operator
  • Interval analysis
  • Parabolic partial differential equation
  • Verified numerical computation

ASJC Scopus subject areas

  • Software
  • Computational Mathematics
  • Applied Mathematics

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