Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation

The paper investigates an inverse problem for finding the order of the fractional derivative in the sense of Gerasimov–Caputo in the wave equation with an arbitrary positive self-adjoint operator $A$ having a discrete spectrum. By means of the classical Fourier method, it is proved that the value of the projection of the solution onto some eigenfunction at a fixed time uniquely restores the order of the derivative. Several examples of the operator $A$ are discussed, including a linear system of fractional differential equations, fractional Sturm–Liouville operators, and many others.