Periodic Solutions, Eigenvalue Curves, and Degeneracy of the Fractional Mathieu Equation

Quantum Fest 2015

Parra-Hinojosa, A.^1 ; Gutiérrez-Vega, J.C.^2

Department of Informatics, Technische Universität München, Boltsmannstr 3, Garching

85748, Germany^1

Photonics and Mathematical Optics Group, Tecnológico de Monterrey, Monterrey

64849, Mexico^2

关键词: Choice of parameters;    Differential operators;    Eigenvalue curves;    Fractional derivatives;    Matrix representation;    Method of harmonic balances;    Orthogonality properties;    Periodic solution;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/698/1/012005/pdf
DOI  :  10.1088/1742-6596/698/1/012005

来源: IOP

【 摘 要 】

We investigate the eigenvalue curves, the behavior of the periodic solutions, and the orthogonality properties of the Mathieu equation with an additional fractional derivative term using the method of harmonic balance. The addition of the fractional derivative term breaks the hermiticity of the equation in such a way that its eigenvalues need not be real nor its eigenfunctions orthogonal. We show that for a certain choice of parameters the eigenvalue curves reveal the appearance of degenerate eigenvalues. We offer a detailed discussion of the matrix representation of the differential operator corresponding to the fractional Mathieu equation, as well as some numerical examples of its periodic solutions.