A new generalized Laguerre function-generalized Hermite function mixed approximation and its application to nonlinear differential equation

The generalized Laguerre-generalized Hermite mixed spectral method is developed. Various orthogonal projections are investigated. Some mixed approximation results are established. As an example of their important applications, a mixed spectral scheme is proposed for a two-dimensional nonlinear problem. Its convergence is proved. Numerical results demonstrate the high accuracy of this new spectral method.