Diagonalized Chebyshev Rational Spectral Methods For Problems On Unbounded Domains

In this thesis,we propose the diagonalized Chebyshev rational spectral methods for solving second-order elliptic problems on the half/whole line.The main idea of this work is to convert the Chebyshev polynomials in bounded regions into Chebyshev rational functions in unbounded regions by some rational transformation,and then we use these rational functions to approximate the problems of unbounded domains numerically.Some Sobolev bi-orthogonal rational basis functions are constructed which lead to the diagonalization of discrete systems.Accordingly,both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series.Numerical results demonstrate the effectiveness of the suggested approaches.This thesis is divided into four chapters:The first chapter we briefly describe the research contents and structures of this thesis.In the second chapter,some basic concepts and notations of function space and basic properties of Chebyshev polynomials are introduced.In the third chapter,we study the weighted Chebyshev rational spectral method on the whole line.Firstly,the Chebyshev rational function system on the whole line and its basic properties are introduced.Then we construct a biorthogonal Chebyshev rational function system on the whole line and propose a diagonalized Chebyshev rational spectral method for solving the second order elliptic boundary value problem on the whole line.Finally,we provide some numerical results to show the efficiency and accuracy of the suggested approach.In the fourth chapter,we study the weighted Chebyshev rational spectral method on the half line.The Chebyshev rational function system on the half line is introduced.Then we construct a biorthogonal Chebyshev rational function system on the half line and propose a diagonalized Chebyshev rational spectral method for solving the second order elliptic boundary value problem on the half line.In addition,we give a detailed analysis of the numerical results and show that they are in good agreement with the theoretical results.