Generalized Hermite spectral method matching asymptotic behaviors

In this paper, we propose the generalized Hermite spectral method by using a family of new generalized Hermite functions, which are mutually orthogonal with the weight function (1+x2)−γ, γ being an arbitrary real number. We establish the basic results on the corresponding orthogonal approximation and interpolation, which simulate the asymptotic behaviors of approximated functions at infinity reasonably. As examples of applications, the spectral schemes are provided for two model problems. Numerical results demonstrate their spectral accuracy in space.