TY - JOUR
T1 - Positivity of discrete time-fractional operators with applications to phase-field equations
AU - KARAA, SAMIR
N1 - Funding Information:
\ast Received by the editors September 22, 2020; accepted for publication (in revised form) April 7, 2021; published electronically July 20, 2021. https://doi.org/10.1137/20M1368641 Funding: This work was supported by Sultan Qaboos University under grant IG/SCI/ MATH/20/04. \dagger FracDiff Research Group, Department of Mathematics, Sultan Qaboos University, Al-Khod 123, Muscat, Oman (skaraa@squ.edu.om).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
KW - Completely monotone sequence
KW - Convolution
KW - Discrete fractional operators
KW - Energy stable scheme
KW - Time-fractional phase-field equation
KW - positive quadratic forms
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U2 - 10.1137/20M1368641
DO - 10.1137/20M1368641
M3 - Article
AN - SCOPUS:85112672057
SN - 0036-1429
VL - 59
SP - 2040
EP - 2053
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
IS - 4
ER -