Second Order Jacobi Approximations And Jacobi Interpolation Approximations In Non-uniformly Weighted Sobolev Spaces With Their Applications

Spectral method is one of important numerical method for solving differential equations. The basic idea of Fourier spectral methods stems from 19th. But only in the past three decades,vareous spectral methods formed a branch of computational mathematics with strict theortical ananlysis. The fascinating merit of spectral method is its high accuracy. Because of this, spectral method has been applied successfully to computation of many problems arising in science, technology and economy, such as numerical simulations of many problems in heat conduction, fluid dynamics, quantum mechanics and financial mathematics and so on.In this paper, we investigate second-order Jacobi approximations and Jacobi-Gauss-type interpolation approximations in non-uniformaly Jacobi-weighted Sobolev spaces with their applications to fourth-order differential equations.In Chapter Two, second-order Jacobi approximation in non-uniformly Jacobi-weighted Sobolev spaces is investigated. Some non-uniformly Jacobi-weighted Sobolev spaces are introduced. Some approximation results on various orthogonal projections in these spaces are established, which serve as the mathematical foundation of Jacobi spectral methods for differential equations of fourth-order. Jacobi spectral schemes are provided for several model problems. The convergence is proved. Numerical results agree well with theoretical analysis and show the efficiency of this new approach.In Chapter Three,we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly Jacobi-weighted Sobolev spaces, which serve as important tools in analysis of numerical quadratures, and numerical methods of differential and integral equations. Then we propose Jacobi pseudospectral schemes for several singular problems and multiple-dimensional problems of fourth order. Numerical results demonstrate the spectral accuracy of these schemes, and coincide well with theoretical analysis.The main ideas and techniques used in this dissertation are also useful for other fourth-order differential equations.