Highly Efficient and Accurate Spectral Approximation of the Angular Mathieu Equation for any Parameter Values $q$

Abstract

The eigenpairs of the angular Mathieu equation under the periodicity condition are accurately approximated by the Jacobi polynomials in a spectral-Galerkin scheme for small and moderate values of the parameter $q.$ On the other hand, the periodic Mathieu functions are related with the spheroidal functions of order $±1/2.$ It is well-known that for very large values of the bandwidth parameter, spheroidal functions can be accurately approximated by the Hermite or Laguerre functions scaled by the square root of the bandwidth parameter. This led us to employ the Laguerre polynomials in a pseudospectral manner to approximate the periodic Mathieu functions and the corresponding characteristic values for very large values of $q.$