Comparison of the methods for the calculation of fractional-order differential equations

Real objects in general are fractional-order systems, although in some types of systems the order is very close to an integer order. Since major advances have been made in this area in the last decades, it is possible to consider also the real order of the dynamical systems by using fractional order of the differential equations. Such models are more adequate for the description of dynamical systems than integer-order models. Appropriate methods for the numerical calculations of fractional-order differential equations are needed. In this contribution we will compare some previous methods used for simulation purposes with the methods based on approximate formulas for numerical inversion of Laplace transforms. The verification and comparison will be based mainly on the accuracy and computing time which is very important e.g. in the tasks of simulation or identification using optimization methods where too many calculations are needed and faster methods can save time very significantly.