New Spectral Methods Based On Non-standard Basis On Unbounded Domains

The numerical solution of differential equation plays an extremely wide range of roles in engineering,science counting and other fields,finite difference method,finite element method and spectral method are three important methods of it.It is noteworthy that spectral methods are based on the combination of the orthogonal polynomials or functions which enjoy high accuracy,that is,smoother the exact solution is,better the approximation is.In this paper,we construct new sets of Fourier-type basis functions for the secondorder problems with constant coefficients on unbounded domains,and based on the new constructed basis functions,new space-time spectral methods are proposed.We also propose new collocation schemes based on the newly constructed non-standard basis functions for problems with variable coefficients or non-linear problems.We first introduce a family of new basis functions by matrix decomposition technique for secondorder boundary problems of constant coefficients(Dirichlet homogeneous boundary conditions)defined on the half unbounded domains.Based on the new basis functions,we propose new space-time spectral methods,and the designed algorithms not only ensure the high precision but also save the calculation time.We prove its spectral accuracy both in space direction and time direction.The numerical results also show the correctness of theoretical analysis.Next,we construct a set of non-standard basis functions based on Laguerre polynomials or functions from generalised Birkhoff interpolation problems by the technique of inverting the operators,which can be explicitly obtained by a fast and stable manner,and we provide some important properties of the new basis.Based on the new non-standard basis functions,we propose new collocation schemes whose finial coefficient matrices corresponding to the problems being well-conditioned.The numerical results also show the advantages of the new basis.Then,we also construct a new set of Fourier-type basis functions for the second-order problems with constant coefficients on the whole unbounded domains.And for the corresponding space-time spectral methods,the proposed algorithms not only guarantee high precision but also save the time.The numerical results also prove its high efficiency.Finally,for the whole unbounded domains of the variable coefficient or nonlinear problems,we also construct a new set of non-standard basis functions based on Hermite polynomials or functions.Also,we propose a new spectral collocation schemes,and give specific numerical results.