Numerical Methods Of Fredholm Integral Equations Of The First Kind And Volterra Integral Equations With Weakly Singular Kernels

Integral equations are very important for scientific and engineering applications.Fredholm integral equations and Volterra integral equations are two kinds of important and fundamental equations,which have always been important research topics for scholars.Most of the integral equations are difficult to give analytical solutions.So it is necessary to consider numerical methods to obtain the numerical solutions of integral equations.Fredholm integral equations of the first kind are ill-posed and difficult to be solved.The regularization method is one of the most efficient methods to deal with ill-posed problems.In this thesis,Fredholm integral equations of the first kind are numerically solved based on the neural network and regularization method.First,assume that the Hilbert space generated by the kernel function of the integral equation is the space where the solution is.The approximate solution can be represented as a linear combination of finite number of the basis functions.Then,construct a deep residual neural network,and obtain the coefficients in the linear combination by training the residual neural network.Numerical experiments show that this method is effective.For Volterra integral equations with a singular kernel,it is difficult to deal with the improper integral.For a class of Volterra integral equations with weakly singular kernels,this thesis presents a numerical method to solve the integral equations based on the piecewise linear maximum entropy method.This method can eliminate the singularity and make the numerical method more accurate and more efficient.The error and the convergent rate of the numerical method are given in the thesis.Numerical examples show that the convergent rate of this method is consistent with the theoretical analysis.