Mixed Legendre-Hermite Spectral Methods And Their Applications

Many problems in science and engineering are set in unbounded domains. The simplest method for solving them numerically is to set certain artificial boundaries, impose some artificial boundary conditions and then resolve the corresponding approximate problems. Whereas these treatments may cause additional errors. Thus it seems better to solve such problems directly.The purpose of this work is to develop the spectral method associated with some orthogonal systems of polynomials for non-isotropic heat transfer in an infinite plate. Firstly, a weak formulation of this problem is derived. The existence, uniqueness and regularity of its solution are discussed. Next, the mixed Legendre-Hermite polynomial approximation in non-isotropic Sobolev space is proposed. A mixed Legendre-Hermite spectral scheme is constructed. Its convergence is proved. Numerical results show the efficiency of this new approach and coincide well with theoretical analysis. Furthermore, we establish some results on the mixed Legendre-Hermite interpolation in certain non-isotropic Sobolev space. They are the theoretical foundation of the mixed Legendre-Hermite pseu-dospectral method. The corresponding mixed Legendre-Hermite pseudospectral scheme is constructed. Its convergence is proved. We also present some numerical results which demonstrate the spectral accuracy in space of proposed mixed scheme.Some results and techniques used in this work are also applicable to other nonlinear partial differential equations in unbounded domains.