Some Studies On The Projection Method For Fredholm Integral Equations Of The Second Kind

In this paper, we mainly studies extrapolation of discrete Multi-Projection methods, New Projection method, Kantorovich method and iterated Kantorovich method for Fredholm integral equations of the second kind. Firstly analyze the asymptotic error expansion for the approximation solution, under the expansion,the order of convergence can be increased by two further powers of h by Richard-son extrapolation. Secondly compare the accuracy and arithmetic workload a-mong New Projection method, Kantorovich method and iterated Kantorovich method, it proves that the new methods are better than the Kantorovich method in numerical results. The outline of this paper is as follow:In Chapter 1,we review the history and status of the study of the Fredholm integral equation of the second kind, and then introduces the research background and some of the basic concepts and conclusions related to this research. At last the preliminary knowledge of this paper are summarized.In Chapter 2, we first adopt the multiple projection method for the problem of the second kind Fredholm integral equation,then lead to approximation equa-tion, then make use of Galerkin method to obtain asymptotic error expansion of the iterative solution, and carry out Richardson extrapolation on the basis of asymptotic error expansion, thus it improve the precision of the approximate solution and the order of convergence.In Chapter 3, firstly, we adopt multiple projection method for the Fredholm integral equation of the second kind, it leads to the approximation equation,and take advantage of Collocation method to obtain asymptotic error expansion of iteration solution, then carry out Richardson extrapolation, thus the order of convergence can be increased by two further powers by Richardson extrapolation.In Chapter 4, for the solution of the weak singular integral equation, firstly we introduce New Projection method, Kantorovich method and iterated Kantorovich method, respectively. Secondly we compare three methods in term of accuracy and arithmetic workload. Finally, the numerical example shows that two kinds of new methods are better than the Kantorovich method.