Study On Solving Of Integral Equations With Mechanization

Integral equation is an important branch of modern mathematics, which has been one of the principal tools of various areas of scientific research and engineering solving and so on. Integral equation is encountered in a variety of applications in many fields. In the area of integral equation, solving has long been and will continue to be one of the dominant themes. Based on the results of integral equation, especially according to the thoughts and methods of mathematics mechanization due to academician Wentsun Wu, in this paper, several mechanical algorithms in Maple for solving integral equation(s) have been established. The main results are as follows:In chapter 1, we give a survey about the brief history of integral equation, the methods for solving integral equation, the mathematics mechanization with applications and son on. Based on these results, we indicate that it will be useful for solving the integral equation with mechanization.In chapter 2, by using the theories and methods of mathematical analysis and computer algebra, based on the method of Resolvent of a kernel, reliable algorithms for solving the Fredholm integral equation(s), and mechanical Maple algorithms are established, too. The algorithms can provide the continuous solution of the second kind of Fredholm integral equation. Computing examples are presented to illustrate the efficiency and accuracy of the algorithms. Especially, for the sake of reading, we established the form of readable outputting for the results. So, the whole processes of solving the integral equation we obtained are just like we do it with our hands and papers. The form of readable outputting may be a good method for computer-aided proving or solving.In chapter 3, reliable mechanical algorithms for solving the Volterra integral equation(s) with Neumann series, Taylor series and iterated collocation method are established, respectively. In particular, the algorithm can provide the exact solution of the second kind of Volterra integral equation by the methods of Neumann-series and mathematical induction, i.e., the algorithm can also compute the n-th term by finite terms of the iterative kernel. It is shown that the mechanical algorithms are efficient and accurate for solving the Volterra integral equation(s).In chapter 4, a reliable algorithm for solving high-order nonlinear Volterra-Fredholm integro-differential equations was established, and a new Maple algorithm was established too. The results of the examples indicated that the algorithm of Taylor polynomial method is simple and effective, and could provide an accurate approximate solution or exact solution of the high-order nonlinear Volterra-Fredholm integro-differential equation. Furthermore, the mechanical algorithm can be applied to solving high-order ordinary differential equation.And in the last section, we give some discussions and remarks.The results indicate that mathematics mechanization is a valid tool for solving the integral equation(s). It will be useful for solving other equations.