High Precision Numerical Methods And Convergence Analysis For Several Kinds Of Integral Equations

In the process of natural sciences research and engineering solving, many problems can be reduced into different types of integral equations, differential equations, integro-differential equations and so on. However, it is difficult to obtain their analysis solutions. Therefore, the research on their high precision numerical solution not only has theoretical values, but also has practical significance. In this thesis, we will mainly discuss some highly accurate numerical methods to solve the Volterra-Fredholm integral equations with complex factor and nonlinear Volterra-Fredholm integral equations, such as the collocation method, the Taylor series method, the least squares approximation method, the homotopy perturbation method and the iterative method. The method of solution, the convergence analysis and comparative analysis of these methods are given. In particular, when h(x)=ax+b, the integral equation with complex factor is functional integral equation with proportional delay, thus the problem of numerical solution of integral equations with proportional delay is solved. The thesis is divided into five chapters, and the structure is arranged as follows:In Chapter1, the research background and significance of integral equation, the research situation of highly accurate numerical methods, and our main work are summarized.In Chapter2, numerical methods such as the Taylor collocation method, the Lagrange collocation method and Taylor series method are used to solve the Volterra-Fredholm integral equations type, respectively. The format of high precision numerical solution and the conver-gence analysis of these methods are obtained. Some numerical examples with different choice of the parameter λi and the functions A(x), B(x) and h(x) are given, and results of the numerical examples verify our convergence analysis.In Chapter3, the least squares approximation method and homotopy perturbation method for solving the Volterra-Fredholm integral equations with complex factor are introduced. The format of the approximation solution and the convergence analysis of these methods are given. Some numerical examples are given to illustrate the accuracy and dependability of the methods.In Chapter4, the iterative method is presented for numerically solving the nonlinear Volterra-Fredholm integral equations. The format of the solution of the iteration method is obtained. The convergence analysis and some numerical examples are given, numerical results verifies our conclusion well.Moreover, at the end of the thesis, that is the Chapter5, some conclusions and discussions about the Volterra-Fredholm integral equations are given again. At the same time, the innovation of the research and the look forward to the future work of subsequent works are introduced.