TY - JOUR
T1 - Highly Efficient and Accurate Spectral Approximation of the Angular Mathieu Equation for any Parameter Values $q$
AU - Alıcı , Haydar
AU - Shen , Jie
JO - Journal of Mathematical Study
VL - 2
SP - 131
EP - 149
PY - 2018
DA - 2018/06
SN - 51
DO - http://doi.org/10.4208/jms.v51n2.18.02
UR - https://global-sci.org/intro/article_detail/jms/12459.html
KW - Mathieu function, spectral methods, Jacobi polynomials, Laguerre polynomials.
AB - <p style="text-align: justify;">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 $q.$</p>