On The Study Of Algorithm Of Numerical Solution For 2-D And The Second Kind Fredholm Integral Equation

The source of the integral equation has always been very wide and its practical applications in many engineering problems is very wide. Such as the financial problems in real life, the calculation problem of electromagnetism, physical problems of fluid mechanics and elastic mechanics and some problems in mathematical biology. These areas are involved in solving integral equations, so the integral equation has been the attention of many researchers. However, for a integral equation, we have difficulty in getting its analytic solution. So the numerical method for solving the integral equation has been the attention of many scholars.For the processing method of numerical solution of integral equation, the traditional methods are collocation method, degradation nuclear method and Galerkin method. This thesis mainly studies the linear two-dimensional integral equation of the second kind. Firstly, we introduce some basic theories of Galerkin algorithm. The traditional Galerkin method usually chooses the triangle function or the linear function as its basic function. Secondly, by modifying the traditional Galerkin method, we chose the constant function as the basic function to replace the traditional basic function, so we call this method as a constant function Galerkin method. The advantage of this approach is to reduce the amount calculation. After that, we also make the numerical solution modified and iterated, and we can eventually prove the feasibility and the validity of this method by comparing the numerical solution that we got with the analytic solution.Secondly, this thesis also makes the traditional Nystrom method improved, and a new Nystrom method is put forward. The new Nystrom method does not make integral calculation and only uses the integral mean value theorem to approximate the integral term simply, and it greatly reduces the amount of calculation, especially for high dimension problems. Although the calculation accuracy of this method is slightly lower,if in a rang of allowable error, we chose this method can achieve the goal effectively,and we also give the estimates of the error of the method ultimately. At last, we give the numerical example and get the numerical solution. By comparing the numerical solution with the analytic solution, we can illustrate the effectiveness of this method.Even though the above two methods do some only simple modifies or innovations with the previous method, they make our calculation become much easier. Not only do they provide new methods for our research on the simple equations, but also lay a theoretical foundation for our future research on more general or more complex integral equations for the follow-up work.