A Truncated Chebyshev Spectral Method For Solving Numerical Differentiation Problem

Based on the truncated spectral method of the first kind Chebyshev polyno-mials, we propose a new method to solve the problem of numerical differentiation in this thesis. After selecting a suitable number of the truncated term by using the Morozov discrepancy principle, we obtain a convergence estimate for the ap-proximate solution. The results for some numerical experiments show that the proposed method is effective and stable.This thesis is divided into four chapters. In Chapter 1 we briefly introduce the concept of inverse problems. As some preliminaries, we give some important results in Chapter 2 for Chebyshev polynomials of the first kind and its property. In Chapter 3, we describe the numerical differentiation problem and propose a regularization method, i.e., by using Chebyshev polynomials of the first kind to get an approximate function in the weighted Sobolev space,we use its derivative to approximate the derivative of exact function. The convergence estimate is also given in Chapter 3. Some numerical examples are tested to verify the efficiency and stability of our proposed method in Chapter 3. In Chapter 4, we give a conclusion.