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Generalized Hermite Spectral Method Matching Different Algebraic Decay at infinities

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Abstract

In this paper, we propose a new generalized Hermite spectral method. We introduce an orthogonal family of new generalized Hermite functions, with the weight function \((1+\frac{2}{\pi } \arctan x)^{\alpha }(1-\frac{2}{\pi }\arctan x)^{\gamma }\), \(\alpha \) and \(\gamma \) being arbitrary real numbers. The basic results on the corresponding orthogonal approximation and interpolation are established. As examples of applications, we provide the spectral schemes for a linear problem and the Fisher equation, which possess the spectral accuracy in space and match the different algebraic decay at infinities reasonably. Numerical results demonstrate their high efficiency and coincide well with the analysis.

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Authors and Affiliations

  1. Shanghai Normal University, Shanghai, 200234, China

    Ben-yu Guo

  2. Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science of E-institute of Shanghai Universities, Shanghai, China

    Ben-yu Guo

  3. Jiangsu Normal University, Xuzhou, 221116, China

    Chao Zhang

Authors
  1. Ben-yu Guo
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  2. Chao Zhang
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Corresponding author

Correspondence to Ben-yu Guo.

Additional information

The work of Chao Zhang is supported in part by NSF of China N.11171227 and N.11371123, Research Fund for Young Teachers of Jiangsu Normal University N.11XLR27, and Priority Academic Program Development of Jiangsu Higher Education Institutions.

The work of Ben-yu Guo is supported in part by NSF of China N.11171227, Fund for Doctoral Authority of China N.20123127110001, Fund for E-institute of Shanghai Universities N.E03004, and Leading Academic Discipline Project of Shanghai Municipal Education Commission N.J50101.

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Guo, By., Zhang, C. Generalized Hermite Spectral Method Matching Different Algebraic Decay at infinities. J Sci Comput 65, 648–671 (2015). https://doi.org/10.1007/s10915-014-9981-3

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  • DOI: https://doi.org/10.1007/s10915-014-9981-3

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