Exploring Fractional-Order Nonlinear Dynamics in Biodiesel Production with Optimal Control

Biodiesel, a sustainable and renewable energy source, is a promising alternative to fossil fuels. The transesterification process of biodiesel production effectively captures memory effects in reaction kinetics. In this study, we developed two fractional order models of the chemical catalytic transesterification reaction to explore the memory effects of the reaction kinetics utilizing two different non-singular kernel methods: Caputo-Fabrizio and Atangana-Baleanu in the Caputo sense. We compared the results with experimental data of biodiesel production and demonstrated the existence and uniqueness of the solution for the fractional system. A sensitivity analysis is performed using the Latin hypercube sampling method to evaluate the impact of various parameters on biodiesel production, followed by the computation of partial rank correlation coefficients based on Pearson’s correlation coefficient. We exhibit the dynamic behavior of all reactants corresponding to these fractional models with the variation of fractional order and the memory rate parameter. Additionally, we display the memory effect through the surface plots for biodiesel production by varying fractional order, molar ratio, and ultrasound frequency. Our numerical comparison with experimental data identifies the fractional-order value for the best fit of biodiesel production and can be increased by applying optimal control on ultrasound frequency.