Positivity of discrete time-fractional operators with applications to phase-field equations

SAMIR KARAA*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We present general criteria ensuring the positivity of quadratic forms of convolution type generated by sequences of real numbers. A sharp result is obtained in the case of completely monotone sequences. Applications to widely used approximations of fractional integral and differential operators, including convolution quadrature and L1 formula on uniform temporal meshes, are presented and new inequalities are established. The results are found to be fundamental in the investigation of the numerical stability for time-fractional phase-field models. It is shown through a standard energy stability analysis and without the use of a fractional Grönwall inequality that several numerical schemes satisfy discrete energy dissipation laws.

Original languageEnglish
Pages (from-to)2040-2053
Number of pages14
JournalSIAM Journal on Numerical Analysis
Volume59
Issue number4
DOIs
Publication statusPublished - 2021
Externally publishedYes

Keywords

  • Completely monotone sequence
  • Convolution
  • Discrete fractional operators
  • Energy stable scheme
  • positive quadratic forms
  • Time-fractional phase-field equation

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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