Accuracy Estimation of the Discrete, Approximated Atangana-Baleanu Operator

Abstract

In the paper the accuracy analysis of the discrete approximations of the Atangana-Baleanu (AB) operator is addressed. The AB operator is the nonsingular kernel operator proposed by Atangana and Baleanu in the papers [1, 2]. It is obtained by replacing the exponential function in the Caputo-Fabrizio operator by the Mittag-Leffler function. The use of the the AB operator at a digital platform (PLC, microcontroller) reuqires to apply the discrete approximation of the factor $s^\alpha$. This can be done using discrete ORA or CFE approximations. The step responses of the both approximations are compared to the analytical response of operator. As the cost function the FIT function available in MATLAB is employed. Results of simulations show that the discrete ORA approximation gives better results than the CFE approximation calculated at the same time grid.