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Analytic Solutions of the Eigenvalues of Mathieu’s Equation

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  • Chein-Shan Liu

Abstract

Mathieu’s eigenvalue problem −y′′(x) + 2e_0 cos(2x)y(x) = λy(x), 0 < x < ℓ is symmetric if cos(2x) = cos(2ℓ − 2x) for ℓ = k0π, k0 ∈ N, and skew-symmetric if cos(2x) = − cos(2ℓ − 2x) for ℓ = π/2. Two typical boundary conditions are considered. When the eigenfunctions are expanded by the orthonormal bases of sine functions or cosine functions, we can derive an n-dimensional matrix eigenvalue problem, endowing with a special structure of the symmetric coefficient matrix A -= [a_ij], a_ij = 0 if i + j is an odd integer. Based on it, we can obtain the eigenvalues easily and analytically. When ℓ = k_0π, k_0 ∈ N, we have a_ij = 0 if |i − j| > 2k_0. Besides the diagonal band, A has two off-diagonal bands, and furthermore, a cross band appears when k_0 ≥ 2. The product formula, the recursion formulas of characteristic functions and a fictitious time integration method (FTIM) are developed to find the eigenvalues of Mathieu’s equation.

Suggested Citation

  • Chein-Shan Liu, 2020. "Analytic Solutions of the Eigenvalues of Mathieu’s Equation," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 12(1), pages 1-1, February.
  • Handle: RePEc:ibn:jmrjnl:v:12:y:2020:i:1:p:1
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    References listed on IDEAS

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    1. Gadella, M. & Giacomini, H. & Lara, L.P., 2015. "Periodic analytic approximate solutions for the Mathieu equation," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 436-445.
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    Cited by:

    1. Liu, Chein-Shan & Chen, Yung-Wei & Chang, Chih-Wen, 2024. "Precise eigenvalues in the solutions of generalized Sturm–Liouville problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 354-373.
    2. Liu, Chein-Shan & Li, Botong, 2023. "Solving Sturm–Liouville inverse problems by an orthogonalized enhanced boundary function method and a product formula for symmetric potential," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 640-660.

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    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
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