The spectral method for high order problems with proper simulations of asymptotic behaviors at infinity

In this paper, we investigate new spectral and multidomain spectral methods for high order problems. We introduce a family of new generalized Laguerre functions, which are mutually orthogonal with the weight function xα(δ+x)−γxα(δ+x)−γ, δ>0δ>0,αα and γγ being arbitrary real numbers. The corresponding quasi-orthogonal approximation and Laguerre–Gauss–Radau type interpolation are proposed. The spectral and multidomain spectral schemes are provided for several model problems, which not only fit the mixed inhomogeneous boundary conditions on the fixed boundary exactly, but also match the asymptotic behaviors at infinity reasonably. Numerical results demonstrate the efficiency of suggested algorithms, and confirm the analysis well.