Numerical analysis of multi-term time-fractional nonlinear subdiffusion equations with time delay: What could possibly go wrong?

Due to the lack of a discrete fractional Grönwall-type inequality, the techniques of analyzing the L2−1_σ difference schemes would not be correct to apply directly to the nonlinear multi-term fractional subdiffusion equations with time delay, especially when the maximum order of the fractional derivatives is not an integer. The purpose of this paper is twofold. First, we introduce a discrete form of fractional Grönwall-type inequality, which in turn fills a gap in the proofs of convergence and stability analyses of such difference schemes. Second, some examples of improper apply of classical convergence and stability techniques are introduced. Moreover, detailed proofs for the convergence and stability theorems are provided departing from the proposed discrete fractional Grönwall-type inequalities.