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Fractional Ince equation with a Riemann–Liouville fractional derivative

Applied Mathematics and Computation
Publication Date
DOI: 10.1016/j.amc.2013.04.044
  • Ince Equation
  • Fractional Calculus
  • Ince Polynomials
  • Eigenvalue Curves
  • Stability
  • Hill Equation


Abstract We extend the classical treatment of the Ince equation to include the effect of a fractional derivative term of order α>0 and amplitude c. A Fourier expansion is used to determine the eigenvalue curves a(∊) in function of the parameter ∊, the stability domains, and the periodic stable solutions of the fractional Ince equation. Two important observations are the detachment of the eigenvalue curves from the a-axis in the (∊,a)-plane, as well as the appearance of degenerate eigenvalues for suitable selections of the parameters. The fractional solutions, valid for the steady state of the system, are not orthogonal and have no defined parity. We also introduce a discrete numerical method to evaluate the Riemann–Liouville fractional derivative with lower terminal at -∞ for a class of functions. The case α=1 represents the Ince equation with an additional constant damping.

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