a b s t r a c tThis paper deals with a concept and description of a RC network as an electro-analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions. It is based on well-known analogy between heat and electrical conduction. In the paper are compared analytical solution together with numerical solution and experimentally measured data. For the first time a fractional order model of diffusion process and its modeling via lumped RC network has been used. Simple examples of simulations, measurements and their comparison are shown.

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The paper is devoted to variable order estimation process when measurements are obtained in two different ways: directly and by lossy network. Since the problem of fractional order estimation is highly nonlinear, dual estimation algorithm based on Unscented Fractional Order Kalman filter has been used. In dual estimation process, state variable and order estimation have been divided into two sub-processes. For estimation state variables and variable fractional order, the Fractional Kalman filter and the Unscented Fractional Kalman filter have been used, respectively. The order estimation algorithms were applied to numerical examples and to real fractional variable order inertial system realized as an analog circuit.

The paper presents derivation and interpretation of one type of variable order derivative definitions. For mathematical modelling of considering definition the switching and numerical scheme is given. The paper also introduces a numerical scheme for a variable order derivatives based on matrix approach. Using this approach, the identity of the switching scheme and considered definition is derived. The switching scheme can be used as an interpretation of this type of definition. Paper presents also numerical examples for introduced methods. Finally, the idea and results of analog (electrical) realization of the switching fractional order integrator (of orders 0.5 and 1) are presented and compared with numerical approach. *

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