Parametric Identification of the Fractional-Derivative Order in the Bagley–Torvik Model

We consider a second-order differential equation that contains a fractional derivative (the Bagley–Torvik equation); here, the order of the derivative is in the range from 1 to 2 and is not known in advance. This model is used for describing oscillation processes in a viscoelastic medium. In order to study the equation, we use the Laplace transform; this makes it possible to obtain (in the explicit form) the image of the solution of the corresponding Cauchy problem. Numerical solutions for the various values of the parameter are constructed. Based on this solution, we propose a numerical technique for the parametric identification of an unknown order of the fractional derivative from the available experimental data. In the range of possible values of the parameter, the deviation function is determined by the least-squares method. The minimum of this function determines the search value of the parameter. The developed technique is tested on the experimental data for samples of polymer concrete, the fractional-derivative parameter in the model is determined, the theoretical and experimental curves are compared, and the accuracy of the parametric identification and the adequacy of the technique are established.