Estudo e Aplicações do Cálculo Fracionário em Engenharia Química

Abstract

In recent decades, interest in the study of fractional differential equations has been growing. This is due to the large number of applications that can be developed from this concept. Although the number of contributions has increased significantly, several aspects related to the influence and physical meaning of a fractional order still require further investigation. Moreover, it is important to highlight the numerical difficulties arising from the substitution of an integer-order derivative by a fractional one. In this work, the Fractional Adams-Moulton Methods, Fractional Backward Euler Method, and Fractional Forward Euler Method are investigated in chemical engineering applications. For this purpose, simulation and inverse problems and phase plane analysis are evaluated. To ensure dimensional consistency in the fractional model, a correction factor is considered. In general, it was concluded that all numerical approaches were efficient for integrating fractional differential models with different levels of complexity. However, the best strategy in terms of the computational cost versus accuracy trade-off was the Fractional Adams-Moulton Method. As expected, the obtained results demonstrated that the fractional order significantly influences the simulated profiles in each application. In this case, it is important to note that depending on the considered fractional order, physically infeasible profiles may be obtained. When evaluating two inverse problems in the fractional context, it was concluded that the objective function value can be improved, as the inclusion of a new parameter (fractional order) increases the degrees of freedom of the optimization problem. Finally, regarding phase plane analysis, the results show that the variation of the fractional order modifies the time required for the process to reach a steady state.