Inverse problem for determining the fractional order in oscillatory systems by moving mass coordinates

This paper addresses an inverse problem for determining the fractional order of an oscillatory system operating in a Newtonian fluid. The governing fractional differential equation is first transformed into a Volterra integral equation of the second kind with respect to the displacement of the mass coordinate. Subsequently, the solution of the resulting integral equation is represented in the form of a Neumann series using successive approximations. Based on available statistical data, a quadratic functional describing the deviation between theoretical and observed displacements is constructed. The unknown fractional order is then identified through the minimization of this functional. An efficient computational algorithm for estimating the fractional order is proposed, and its performance is illustrated by a numerical example. The obtained results demonstrate that the identified fractional order satisfies the first-order optimality condition, with the first variation of the quadratic functional approaching zero with an accuracy of 10 - 8 {10⁻⁸} , thereby confirming the effectiveness and high numerical accuracy of the proposed approach.