Highly Efficient and Accurate Spectral Approximation of the Angular Mathieu Equation for any Parameter Values $q$ (Journal of Mathematical Study, 2018)
The eigenpairs of the angular Mathieu equation under the periodicity conditionare accurately approximated by the Jacobi polynomials in a spectral-Galerkinscheme for small and moderate values of the parameter $q.$ On the other hand, the periodicMathieu functions are related with the spheroidal functions of order $±1/2.$ It iswell-known that for very large values of the bandwidth parameter, spheroidal functionscan be accurately approximated by the Hermite or Laguerre functions scaled bythe square root of the bandwidth parameter. This led us to employ the Laguerre polynomialsin a pseudospectral manner to approximate the periodic Mathieu functionsand the corresponding characteristic values for very large values of $q.$