Jacobi Spectral Galerkin Method For Nonlinear Fredholm Integral Equation With A Weakly Singular Kernel

The Fredholm integral equation is an important class of mathematical models,which is widely applied.Many problems of engineering,physics,mechanics,electromagnetism and so on can be modeled as Fredholm integral equation.Moreover,the nonlinear Fredholm integral equation with a weakly singular kernel has attracted great attention due to its complexity.As we all know,the spectral method is studied by many scholars due to its good exponential convergence.However,when it comes to the study of the integral equation's numerical algorithm,the spectral method is mostly used to solve the Volterra integral equation and it is rarely used to investigate the Fredholm integral equation.So the main purpose of this paper is to apply the Jacobi spectral Galerkin method to investigate the numerical scheme and convergence analysis of the nonlinear Fredholm integral equation with a weakly singular kernel.This paper firstly introduces the research status of integral equation at home and abroad as well as spectral method,Galerkin method and the basic theory of orthogonal polynomials and so on.Secondly,the nonlinear weak singular Fredholm integral equation is solved by the Jacobi spectral Galerkin method,the numerical scheme and convergence theory of the Jacobi spectral Galerkin method are obtained,and Jacobi pseudospectral Galerkin method and it's matrix form are also given.Besides,we give the numerical scheme which is gained by using the Newton iteration method to deal with the nonlinear part.At last,we offer the convergence analysis of Jacobi spectral Galerkin method and Jacobi pseudospectral Galerkin method respectively.And numerical experiments about numerical examples of nonlinear weakly singular Fredholm integral equations with smooth exact solution and nonsmooth solutions are carried out respectively by using Matlab software.The numerical results tell that the exponential convergence is reached,which is consistent to the convergence analysis of this paper.In other words,the correctness of the theoretical analysis is verified by the numerical experiments.