Two kinds of Legendre spectral Galerkin numerical integration methods for Volterra type integral differential equations with variable coefficient

In order to further improve the numerical accuracy of solving Volterra integro-differential, two kinds of Legendre spectral Galerkin numerical integration methods are investigated for the Volterra-type integro-differential equation with variable coefficients. Firstly, the Galerkin Legendre numerical integration is applied to deal with the integral term of the Volterra-type integro-differential equations. Secondly, the Legendre tau scheme is developed for the Volterra-type integral-differential equations with variable coefficient, and the Chebyshev-Gauss-Lobatto collocation point is used to the calculation of the variable coefficient and integral term. Finally, by decomposing the definition interval of the function, the multi-interval Legendre spectral Galerkin numerical integration method is also designed. Its scheme of the proposed method has symmetric structure for odd-order model. In addition, by introducing the least squares function of the Volterra type integro-differential equation, the Legendre spectral Galerkin least-squares numerical integration method of is constructed. The corresponding coefficient matrix of the algebraic equation is symmetric positive. Some numerical examples are given to test the high-order accuracy and the effectiveness of our methods.