Legendre Galerkin Chebyshev Collocation Spectral Method For A Class Of Integral Equations

In this paper,the Galerkin Legendre Jacobi numerical integration method of the Volterra integral equation is studied.Further,the method is applied to the calculation of the Volterra integral equation with fixed discontinuities arising from the source term and the third kind of Volterra integral equation.For the Volterra integro-differential equation with variable coefficients,based on the Legendre tau method,the Legendre spectral Galerkin numerical integration method is designed.The Legendre spectral Galerkin least-squares numerical integration method is also given by using the least-squares method.The specific research contents are:The Galerkin Legendre Jacobi numerical integration method is constructed for the Volterra integral equations of the second kind.This method uses Galerkin numerical integration to discretize the integral term in the equation,thereby obtaining an equivalent equation,and then designing the Legendre Galerkin format for the equivalent equation.The interpolation polynomial and the source term are calculated by Chebyshev interpolation.The error estimation of the method in the sense of the L~2-norm is given.For Volterra integral equations with relatively large intervals,combined with the interval division method,a multi-interval format is given.Further,for the problem that the source term contains fixed discontinuities,the two-level multi-interval format is used to calculate the format.In addition,the method is applied to the calculation of the third kind of Volterra integral equations.For the Volterra integro-differential equations with variable coefficients,the Legendre spectral Galerkin numerical integration method is developed.The method adopts the Legendre tau form.The integral terms of the Volterra type integro-differential equations are preprocessed with Galerkin Legendre numerical integration,and the variable coefficients and interpolating polynomials are computed with Chebyshev-Gauss-Lobatto collocation points.We give the error estimate of the method in the sense of the L~2-norm.By introducing the least-squares function of the Volterra type integro-differential equation,the Legendre spectral Galerkin least squares numerical integral method is constructed.By choosing appropriate basis functions,the coefficient matrix of the system of algebraic equations obtained by this method has the characteristics of symmetric positive definite.