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

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

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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Leg-endre 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.
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 coeffi-cient,and the Chebyshev-Gauss-Lobatto collocation point is used to the calculation of the variable coefficient and integral term.Fi-nally,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 introduc-ing the least squares function of the Volterra type integro-differential equation,the Legendre spectral Galerkin least-squares numeri-cal 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.

integral differential equationsnumerical integrationChebyshev-Gauss-Lobatto interpolationleast square methodLeg-endre Galerkin

范友康、张克磊、覃永辉

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桂林电子科技大学数学与计算科学学院,广西桂林 541004

桂林电子科技大学广西自动检测技术与仪器重点实验室,广西桂林 541004

桂林电子科技大学广西高校数据分析与计算重点实验室,广西桂林 541004

积分微分方程 数值积分 Chebyshev-Gauss-Lobatto插值 最小二乘法 Legendre Galerkin

国家自然科学基金广西自动检测技术与仪器重点实验室基金广西科技基地和人才专项桂林电子科技大学研究生教育创新计划

12161025YQ22106桂科AD182810252020YCXS086

2024

10.16725/j.1673-808X.2022144

桂林电子科技大学学报
桂林电子科技大学

桂林电子科技大学学报

影响因子：0.247
ISSN：1673-808X
年,卷(期)：2024.44(1)