变系数Volterra型积分微分方程的2种Legendre谱Galerkin数值积分方法

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

为了进一步提高求解Volterra型积分微分的数值精度,针对一种变系数Volterra型积分微分方程,提出了2种Legendre谱Galerkin数值积分法。采用Galerkin Legendre数值积分对Volterra型积分微分方程的积分项进行预处理,对其构造Legendre tau格式,同时用Chebyshev-Gauss-Lobatto配置点对变系数和积分项部分进行计算,并通过对方程的定义区间进行分解,提出了一种多区间Legendre谱Galerkin数值积分法。该方法的格式对于奇数阶模型具有对称结构。此外,通过引入Volterra型积分微分方程的最小二乘函数,构造了Legendre谱Galerkin最小二乘数值积分法。该方法对应的代数方程系数矩阵是对称正定的。数值算例验证了这2种Legendre谱Galerkin数值积分方法的高阶精度和有效性。

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.