Numerical simulation and analysis for variable-order fractal-fractional derivatives of bio-ethanol production models with recycling and death rate effects

Bio-ethanol is a good choice for alternative energy instead of the use of fossil fuels and is still a hot topic for researchers at present.In this paper, a nonlinear mathematical model for bio-ethanol production under the fractal-fractional operators in the sense of Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu with variable order is considered.We use the fixed point theorem of Banach's type to prove the existence and uniqueness of the solution.The stability analysis of the proposed bioethanol model is examined by using the various types of Ulam's stability.The Adams-Bashforth method is applied to construct the numerical schemes for each type of kernel, as well as the graphically numerical simulations for some sets of fractional orders and fractal dimensions, which are conducted to support our analytic results.