The Study Of Numerical Algorithms For Fredholm Integral Equations Of The First Kind

This thesis would research into to solve problems of the first Fredholm integral equation in two-dimensional, which are special cases of inverse problems. Inverse problems commonly are ill-posed, in order to obtain stable approximate solutions, some regularization techniques are required.The main objectives of this work are to describe the theoretical use of the regularization method to solve integral equations of the first kind and to discrete the integral equations of the first kind as to develop a solution method for it. The integral equations of the first kind are ill-posed, in order to obtain stable approximate solutions, some regularization techniques are required. To solve such equations, the kernels are represented by two-dimensional matrices and singular value decomposition is used to get the solutions. The regularization parameter is computed by using the L-curve method and the Discrepancy Principle.Exploration work is focused on the special structure of the coefficient matrix, the study of discrete points, as well as solving the integral equations of the first kind. Since traditional regularization methods only can deal with the continuous problems. The difficulty and main primary content in the thesis is how to construct regularization methods to solve discontinuous problems.Firstly, the math model and methodologies for Fredholm integral equations are introduced. The difficulties of soloving the Fredholm integral equations are discussed. Then, the discretization of the discontinuous exact solution of Fredholm integral equations in one dimension and how to Fredholm integral equations are described in detail,and to solove Fredholm integral equations by total variation regularation. As discreting two-dimensional Fredholm integral equation of the first problem has not been resolved, the first Fredholm integral equation in two-dimensional are described in this paper by numerical integration formula discreting, singular value decomposition, selecting the regularized parameters, and the method is used to solve. There, we describe some numerical methods and some examples are presented. Due to the ill-posedness of integral equations, it is not easy to deal with them. Considering these types of equations in dimensions other than one is a little bit harder because of the complexity of multiple integrals. This can be seen in the type of programs developed which demand a huge quantity of memory in order to achieve acceptable results.Numerical experiments and the analysis of the results shown that, total variation of the feasibility and the effectiveness to solve integral equations of the first kind in one dimensions, and solution method for integral equations of the first kind in two dimensions with no singularities in its kernel was implemented. It gave good results in the two-dimensional case. In each case the quality of the solutions depended on the rate of change the kernels and the smottness of the exact solution. Also, the solutions were influenced by the error in the data and the size of the grid.