Abstract
The prolate spheroidal wave equation (PSWE) is transformed, using suitable mappings, into three different canonical forms which resemble the Jacobi, Laguerre and the Hermite differential equations. The eigenpairs of the PSWE are approximated with the corresponding classical orthogonal polynomial as a basis set. It is observed that for any zonal wavenumber m the Jacobi type pseudospectral methods are well suited for small bandwidth parameters c whereas the Hermite and Laguerre pseudospectral methods are appropriate for very large c values. Moreover, Jacobi pseudospectral methods work well for any parameter values such that \(m\ge c\). Our numerical results confirm that for any values of m, the Jacobi \(\left[ (\alpha ,\beta )=(\pm 1/2,m)\right] \) and the Laguerre \(({\upgamma }=\pm 1/2)\) pseudospectral methods formulated in this article for the numerical solution of the PSWE with small and very large bandwidth parameters, respectively, are highly efficient both from the accuracy and fastness point of view.
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Acknowledgements
The first author would like to dedicate this research to the memory of his father who suddenly passed away just two days later than the author’s arrival to USA. He feels a hearthfelt sadness for being thousands of miles away at the moment he passed away.The first author’s research was supported by a grant from TUBITAK, the Scientific and Technological Research Council of Turkey. The second author’s research was partially supported by NSF DMS-1419053 and AFOSR FA9550-16-1-0102.
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Alıcı, H., Shen, J. Highly Accurate Pseudospectral Approximations of the Prolate Spheroidal Wave Equation for Any Bandwidth Parameter and Zonal Wavenumber. J Sci Comput 71, 804–821 (2017). https://doi.org/10.1007/s10915-016-0321-7
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DOI: https://doi.org/10.1007/s10915-016-0321-7