In the paper a new, fractional order, discrete transfer function model of an elementary inertial plant is proposed. The model uses Atangana-Baleanu and discrete Fractional Order Backward Difference operators to describe of the fractional derivative. Such a transfer models have not be presented yet. The analytical formula of the step response for time-continuous transfer function is given. The similarity of the proposed model to “classic” one using Caputo operator is also considered. The stability and the convergence of the discrete transfer function are analyzed. Theoretical results are expanded by simulations. The proposed discrete, approximated model is accurate and its numerical complexity is low. It can be useful in modeling of different physical phenomena, for example thermal processes.

In the paper a new, fractional order, discrete model of a two-dimensional temperature field is addressed. The proposed model uses Grünwald-Letnikov definition of the fractional operator. Such a model has not been proposed yet. Elementary properties of the model: practical stability, accuracy and convergence are analysed. Analytical conditions of stability and convergence are proposed and they allow to estimate the orders of the model. Theoretical considerations are validated using exprimental data obtained with the use of a thermal imaging camera. Results of analysis supported by experiments point that the proposed model assures good accuracy and convergence for low order and relatively short memory length.