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In this paper, a maximum principle for the one-dimensional sub-diffusionequation with Atangana-Baleanu fractional derivative is formulated and proved.The proof of the maximum principle is based on an extremum principle for theAtangana-Baleanu fractional derivative that is given in the paper, too. Themaximum principle is then applied to show that the initial-boundary-valueproblem for the linear and nonlinear time-fractional diffusion equationspossesses at most one classical solution and this solution continuously dependson the initial and boundary conditions.

Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions