Details of Research Outputs

We present three kinds of time-dependent Hermite spectral approximations for linear diffusion equations and the generalized viscous Burgers' equation in unbounded domains. The methods use the Hermite function approximation, Hermite polynomial approximation, and symmetrical Hermite approximation, respectively. A time-dependent scaling factor is employed in the Hermite expansions, which yields a positive bilinear form avoiding instability of tlie classical Hermite spectral approximation. The time-dependent approach plays a similar stability role to the similarity transformation technique proposed by Funaro and Kavian [Math. Comput., 57 (1991), pp. 597-619]. However, since co-ordinate transformations are not required, the present method is more efficient and is easier to implement. In fact, with the time-dependent scaling the resulting discretization system is of the same form as that associated with the classical Hermite spectral method. Numerical experiments are carried out to support the theoretical stability and convergence results for the proposed approach.