Abstract
In this paper, we consider a spectral method based on generalized Hermite functions in multiple dimensions. We first introduce three normed spaces and prove their equivalence, which enables us to develop and to analyze generalized Hermite approximations efficiently. We then establish some basic results on generalized Hermite orthogonal approximations in multiple dimensions, which play important roles in the relevant spectral methods. As examples, we consider an elliptic equation with a harmonic potential and a class of nonlinear wave equations. The spectral schemes are proposed, and the convergence is proved. Numerical results demonstrate the spectral accuracy of this approach.
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Acknowledgments
This work is supported in part by NSF of China, N.11171225, Innovation Program of Shanghai Municipal Education Commission, N.12ZZ131, Shuguang Project of Shanghai Municipal Education Commission, N.08SG45, and The Fund for E-institute of Shanghai Universities, N.E03004.
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Xiang, Xm., Wang, Zq. Generalized Hermite Approximations and Spectral Method for Partial Differential Equations in Multiple Dimensions. J Sci Comput 57, 229–253 (2013). https://doi.org/10.1007/s10915-013-9703-2
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DOI: https://doi.org/10.1007/s10915-013-9703-2