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

Fingerprint

Semilinear Heat Equation
Computing
Numerical Verification
Analytic Semigroup
Evolution Operator
Heat Equation
Hot Temperature
Error Estimates
Approximate Solution
Uniqueness
Exact Solution
Fixed point
Nonlinearity
Propagation
Numerical Examples
Interval
Formulation

Keywords

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

ASJC Scopus subject areas

  • Software
  • Computational Mathematics
  • Applied Mathematics

Cite this

Accurate method of verified computing for solutions of semilinear heat equations. / Takayasu, Akitoshi; Mizuguchi, Makoto; Kubo, Takayuki; Oishi, Shinichi.

In: Reliable Computing, Vol. 25, 2017, p. 74-99.

Research output: Contribution to journalArticle

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