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.
Highlights
Mathieu’s equation is a well known second-order ordinary differential equation (ODE), endowing with periodic coefficient (McLachlan 1947; Bateman & Erdelyi, 1955; Meixner, Schafke & Wolf, 1980)
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 := [ai j], ai j = 0 if i + j is an odd integer
Problems 1 and 2 are solved in Section 4, where we develop a fictitious time integration method (FTIM) to obtain eigenvalues, based on the recursion formula to compute the characteristic function
Summary
Mathieu’s equation is a well known second-order ordinary differential equation (ODE), endowing with periodic coefficient (McLachlan 1947; Bateman & Erdelyi, 1955; Meixner, Schafke & Wolf, 1980). A number of physical phenomena and engineering problems can be described by Mathieu’s equation, which appears in the solution of the Helmholtz equation of an elliptic membrane by using the method of separation of variables (Plestenjak, Gheorghiu & Hochstenbach, 2015). In the paper we consider the following Mathieu’s eigenvalue problem (Bujurke, Salimath & Shiralashetti, 2008; Gheorghiu, Hochstenbach, Plestenjak & Rommes, 2012):. Problems 1 and 2 are solved, where we develop a fictitious time integration method (FTIM) to obtain eigenvalues, based on the recursion formula to compute the characteristic function.
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