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Title: A discrete Grönwall inequality and energy estimates in the analysis of a discrete model for a nonlinear time-fractional heat equation
Authors: Hendy, A. S.
Macías-Díaz, J. E.
Issue Date: 2020
Publisher: MDPI AG
Citation: Hendy A. S. A discrete Grönwall inequality and energy estimates in the analysis of a discrete model for a nonlinear time-fractional heat equation / A. S. Hendy, J. E. Macías-Díaz. — DOI 10.3390/math8091539 // Mathematics. — 2020. — Vol. 8. — Iss. 9. — 1539.
Abstract: In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. 16: 892-910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order O(τ2-α + h4), in the case that 0 < α < 1 satisfies 3α ≥ 3/2, which means that 0.369 α ≤ 1. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order 0 < α < 1 used for that scheme at tk+1/2. © 2020 by the authors.
Access: info:eu-repo/semantics/openAccess
SCOPUS ID: 85091373602
PURE ID: 13913283
ISSN: 22277390
DOI: 10.3390/math8091539
metadata.dc.description.sponsorship: The first author wishes to acknowledge the support of RFBR Grant 19-01-00019. Meanwhile, the second author would like to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT). The second author acknowledges financial support from CONACYT through grant A1-S-45928. Acknowledgments: The authors wish to thank the guest editors for their kind invitation to submit a paper to the special issue of Mathematics MDPI on "Computational Mathematics and Neural Systems". They also wish to thank the anonymous reviewers for their comments and criticisms. All of their comments were taken into account in the revised version of the paper, resulting in a substantial improvement with respect to the original submission.
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