Jacobi Spectral Methods And Their Applications To Singular Problems, Unbounded Domains And Axisymmetric Domains

In this dissertation, the theory of the Jacobi spectral methods and their applications to singular problems, unbounded domains and axisymmetric domains are studied. Some efficient algorithms to implement the Jacobi spectral methods are constructed. The main context of this dissertation consists of three parts.Firstly, we establish the Jacobi interpolation approximations in weighted Sobolev spaces and certain Hilbert spaces. They are the theoretical foundation of the Jacobi pseudospectral methods. As indispensable tools, some weighted inverse inequalities and imbedding inequalities are given. Furthermore, various unusual Jacobi orthogonal projections are also included. The Jacobi pseudospectral methods are applied to numerical solutions of singular differential equations, differential equations in infinite intervals and in disc. Some numerical are presented to show the efficiency of these new approaches.Secondly, we develop multiple-dimensional Jacobi polynomial approximations in non-isotropic Hilbert spaces. They are used to numerical solutions of multiple-dimensional singular partial differential equations with different singularities, such as degenerating coefficients, singular boundary values and singular source terms. They are also applied to unbounded domains, such as the whole plane, the half plane and some infinite straps. The Jacobi spectral methods are also available to problems in axisymmetric domains. We propose some efficient algorithms to implement the Jacobi spectral method. It makes this new approach more preferable.On the other hand, we investigate multiple-dimensional, non-isotropic Jacobi interpolation approximations. In particular, we consider three hybrid interpolations, which are useful for problems in different unbounded domains. We take the nonlinear Klein-Gordon equation in an infinite strap and in a cylinder as examples to illustrate how to deal with nonlinear problems by using Jacobi pseudospectral methods.Most of the ideas and techniques in this dissertation can be used to explore other new spectral methods.