Volterra型积分方程的Galerkin Legendre Jacobi数值积分法

doi:10.12288/szjs.s2021-0815

数值计算与计算机应用 ›› 2023, Vol. 44 ›› Issue (1): 81-94.DOI: 10.12288/szjs.s2021-0815

• 论文 • 上一篇

Volterra型积分方程的Galerkin Legendre Jacobi数值积分法

范友康¹, 卢荣伟², 覃永辉^1,2

1. 桂林电子科技大学, 数学与计算科学学院, 广西高校数据分析与计算重点实验室, 广西自动检测技术与仪器重点实验室, 桂林 541004;
2. 桂林电子科技大学, 广西应用数学中心, 桂林 541004

收稿日期:2021-12-29 发布日期:2023-03-16
通讯作者: 覃永辉,yonghui1676@163.com;yonghui@csrc.ac.cn.
基金资助:
国家自然科学基金（12161025，61963011，11961012，11961010）；广西自动检测技术与仪器重点实验室基金（YQ22106）；桂林电子科技大学研究生教育创新计划（2020YCXS086）资助.

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范友康, 卢荣伟, 覃永辉. Volterra型积分方程的Galerkin Legendre Jacobi数值积分法[J]. 数值计算与计算机应用, 2023, 44(1): 81-94.

Fan Youkang, Lu Rongwei, Qin Yonghui. GALERKIN LEGENDRE JACOBI NUMERICAL INTEGRAL FOR VOLTERRA INTEGRAL EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2023, 44(1): 81-94.

GALERKIN LEGENDRE JACOBI NUMERICAL INTEGRAL FOR VOLTERRA INTEGRAL EQUATIONS

Fan Youkang¹, Lu Rongwei², Qin Yonghui^1,2

1. School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, China;
2. Center for Applied Mathematics of Guangxi(GUET), Guilin 541004, China

Received:2021-12-29 Published:2023-03-16

研究Volterra型积分方程的Galerkin Legendre Jacobi数值积分方法.首先，利用Jacobi Galerkin数值积分对方程中的积分项进行离散，从而我们得到一个等价方程.其次，对该等价模型构造Legendre Galerkin方程，且在积分项部分用Chebyshev插值计算.然后，该方法还被推广到非线性Volterra积分方程的计算.最后，对计算区间较大的模型，基于上述方法，构造了两级多区域格式且将其应用于含有一个间断点的Volterra型积分方程的计算.此外，还将其推广到第三类Volterra积分方程.通过数值算例验证该方法的高阶精度与有效性.

关键词： Volterra积分方程, Legendre Galerkin, Jacobi Galerkin数值积分, 间断性

Galerkin Legendre Jacobi numerical integration method is investigated for the Volterratype integral equations. First, the integral term of the equation is approximated by using the Jacobi Galerkin numerical integral method, and an equivalent equation is obtained. Second, the Legendre Galerkin equation is developed for this equivalent model, and the integral term is calculated by Chebyshev interpolation. Then, our method is also extended to solve the nonlinear Volterra integral equations. Finally, by using the above method, we give a two-level multi-interval scheme for the Volterra integral equation with discontinuity. In addition, we also apply our scheme to solve the third class of Volterra integral equations. Some numerical examples are given to test the high-order accuracy and the effectiveness of the proposed method.

Key words: Volterra integral equations, Legendre Galerkin, Jacobi Galerkin numerical integration, discontinuouty

MR(2010)主题分类:

45D05
65N35

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Volterra型积分方程的Galerkin Legendre Jacobi数值积分法

GALERKIN LEGENDRE JACOBI NUMERICAL INTEGRAL FOR VOLTERRA INTEGRAL EQUATIONS

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