More and more attention has been paid to numerical methods for problems on unbounded domains. In this paper, a spectral method using generalized Laguerre functions is proposed for high dimensional problems of high order defined on unbounded domains. The new basis is introduced, and some orthogonal and interpolation approximation results are established, which serve as useful tools in spectral methods for high order problems. As examples of applications, the spectral schemes for the fourth order problems are presented. Efficient algorithms are implemented. The numerical results are shown to demonstrate the high accuracy of suggested algorithms, which are coincided with the theoretical analysis.

无界域问题的数值方法越来越受到关注。在本文中，针对定义在无界域上的高阶高维问题，提出了一种使用广义拉盖尔函数的谱方法。引入了新的基础，并建立了一些正交和插值近似结果，这些结果可以作为高阶问题的谱方法中的有用工具。作为应用示例，介绍了四阶问题的频谱方案。实现了高效的算法。数值结果证明了所提出算法的高精度，这与理论分析相吻合。