A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation

Ahmed S. Hendy and Jorge E. Macías-Díaz
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Ahmed S. Hendy: Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., 620002 Yekaterinburg, Russia
Jorge E. Macías-Díaz: Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia

Mathematics, 2020, vol. 8, issue 9, 1-15

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 ( τ 2 − α + h 4 ) , in the case that 0 < α < 1 satisfies 3 α ≥ 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 t k + 1 / 2 .

Keywords: nonlinear fractional heat equation; discrete energy estimates; discrete fractional Grönwall inequality; convergence and stability analyses (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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