Some Researches On Spline Functions And Radial Basis Functions

Spline functions and radial basis functions are important tools in approximation theo-ry, numerical analysis, computational geometry, and data processing. In this dissertation, some problems in approximation theory with spline functions and radial basis functions (RBFs) are studied. The contents are summarized as follows:In Chapter 1, we presents some preliminaries for the dissertation, including definitions of spline functions and radial basis functions, and some basic facts about them.In Chapter 2, an improved numerical solution of Burgers'equation is presented based on the cubic B-spline quasi-interpolation and the compact finite difference method. For this cubic B-spline quasi-interpolation is introduced. Moreover, the numerical scheme is presented, by using the derivative of the B-spline quasi-interpolation to approximate the spatial derivative and a two-order compact finite difference scheme to approximate the time derivative. Numerical examples show that this scheme has higher accuracy, and it is easy to implement.In Chapter 3, we represent the relationship between network resource utilization efficiency and word length mentioned in telegraphy. The network resource utilization efficiency is a basic problem in telegraphy. If network environment (Access Rate R and Data Header Length Lo) and the quality of service (Max Delay Tmax and Loss Rate Ploss) are given, we can construct the computational formula by using spline interpolation, and compute the optimal word length (Lopt). If the communication services are provided by this word length, we can get maximal network resource utilization efficiency avoiding fussy graphic calculation. Numerical examples show that we can get good result by using cubic natural spline interpolation.In Chapter 4, by using Pade approximation method, we construct two classes of RBFs satisfying that the corresponding interpolation are well-posed. The classical radial basis functions such as Gaussian function and Hardy's multiquadric function (MQ) are respectively in the exponential and irrational forms with higher computational require-ments. In this chapter, we construct two RBFs by using Pade approximation to modify some radial bases, and these new rational RBFs can ensure the non-singularity of the interpolating matrices. Additionally, we give a relationship between Gaussian function and Inverse multiquadric (IMQ) functions.In Chapter 5, we construct a high accuracy MQ quasi-interpolation operator. MQ interpolation yields good approximation order but ill-conditioned linear systems must be solved, so the MQ quasi-interpolation has been widely studied. In this chapter, we propose a new multi-level univariate MQ quasi-interpolation approach with higher approximation order by using IMQ interpolation. Our quasi-interpolation operator has the advantages of the IMQ-RBF interpolation and the MQ quasi-interpolation. By using this operator to solve Sine-Gordon equation numerically, we can get a numerical scheme with good accuracy.