A discrete Grönwall inequality and energy estimates in the analysis of a discrete model for a nonlinear time-fractional heat equation

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.