Steady-state solutions of the Whittaker–Hill equation of fractional order

Abstract

We investigate the steady-state solutions of the Whittaker–Hill equation, including a fractional derivative term. Using appropriate Fourier series, we characterize the behavior of the eigenvalue surfaces as a function of the differential equation parameters and describe the corresponding eigensolutions. We also examine the situation where the fractional derivative is zero to serve as a comparison for the fractional solutions. For some special combinations of the parameters, the fractional derivative term leads to the appearance of degenerate complex eigenvalues.