Parameter identification of fractional derivative order in Bagley–Torvik model

We consider a second-order differential equation containing a derivative of a fractional order (the Bagley-Torvik equation) in which the order of the derivative is in the range from 1 to 2 and is not known in advance. This model is used to describe oscillation processes in a viscoelastic medium. To study the equation, we use the Laplace transform, which allows us to obtain in an explicit form the image of the solution of the corresponding Cauchy problem. Numerical solutions are constructed for different values of the parameter. On the basis of the solution obtained, a numerical technique is proposed for parametric identification of an unknown order of a fractional derivative from the available experimental data. On the range of possible values of the parameter, the least-squares deviation function is determined. The minimum of this function determines the desired value of the parameter. The approbation of the developed technique on experimental data for polymer concrete samples was carried out, the fractional derivative parameter in the model was determined, the theoretical and experimental curves were compared, the accuracy of the parametric identification and the adequacy of the technique were established.