Jacobi Spectral Collocation Method For Nonlinear Fredholm Integral Uation With Weakly Singular

Integral equation is a very important mathematical model,which is widely applied in fluid mechanics,theory ofelasticity,electrodynamics,electromagnetic field theory,radiation biology and population problems.As everyone knows,the nonlinear problem is a common problem in real life.Therefore,Nonlinear Fredholm integral equations are studied by many scholars.Due to the unknowns are in the integral,the exact solution is difficult to be given by an exact analytical expression.Therefore,in practical application,numerical methods for the integral equation have received significant.The principle of the spectral method is to take the finite series expansion of the approximate expansion of the solution into an orthogonal polynomial,which is the approximate spectral expansion of the solution.And then,based on the expansion formula and the original equation,the equations of the expansion coefficient are obtained.So,the more the number of series type,the higher the accuracy of the spectral method will be obtained.With the continuous development of fast Fourier transform,the calculation of spectral methods is also decreasing.We can get expected high accuracy without many terms.Therefore,the spectral method is applied to the solving problem of integral equation with high accuracy,stability and convergence.But the Jacobi spectral collocation method of nonlinear Fredholm integral equation don't involved in available literatures.So,this paper uses Jacobi spectral collocation to solve the nonlinear Fredholm integral equation and gets approximation with high accuracy.Firstly,in this paper,we give the research status of Fredholm integral equation at home and abroad and some preparatory knowledge about collocation method,orthogonal polynomial and spectral method,which can make a paving for introducing the Jacobi spectral collocation method.Then,specific algorithm of nonlinear Fredholm integral equation with weakly singular by Jacobi spectral collocation method are introduced.Firstly,We absorb the weakly singular function into the weight function.And then,The Jacobi-Gauss quadrature formula is used to discretizes the integral.Regarding to the processing of nonlinear parts,the Newton iterative method is selected?Finally,the error analysis of the numerical scheme and the numerical solution from Jacobi spectral-collocation methods is given.We show that exponential rates of convergence can be achieved for Jacobi-spectral collocation solution in L?-norm and weighted L2-norm.Numerical experiments results are provided to prove the effectiveness of the given methods.