非线性第二类Volterra积分方程的Chebyshev谱配置法

doi:10.12286/jssx.2020.4.445

计算数学 ›› 2020, Vol. 42 ›› Issue (4): 445-456.DOI: 10.12286/jssx.2020.4.445

非线性第二类Volterra积分方程的Chebyshev谱配置法

古振东¹, 孙丽英²

1 广东金融学院金融数学与统计学院, 广州 5105211;
2 广东金融学院保险学院, 广州 510521

收稿日期:2018-11-05 出版日期:2020-11-15 发布日期:2020-11-15
基金资助:
广东省自然科学基金项目（2017A030310636，2018A030313236），广东省高性能计算学会开放基金项目（2017060104），中山大学广东省计算科学重点实验室开放基金项目（2016001），广东高校省级重点平台和重大科研项目（2017KTSCX131），广东省教育厅科研项目（2017KTSCX130）资助.

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古振东, 孙丽英. 非线性第二类Volterra积分方程的Chebyshev谱配置法[J]. 计算数学, 2020, 42(4): 445-456.

Gu Zhendong, Sun Liying. CHEBYSHEV SPECTRAL COLLOCATION METHOD FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND[J]. Mathematica Numerica Sinica, 2020, 42(4): 445-456.

CHEBYSHEV SPECTRAL COLLOCATION METHOD FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND

Gu Zhendong¹, Sun Liying²

1 School of Financial Mathematics&Statistics, Guangdong University of Finance, Guangzhou 510521, China;
2 School of Insurance, Guangdong University of Finance, Guangzhou 510521, China

Received:2018-11-05 Online:2020-11-15 Published:2020-11-15

我们在参考了相关文献的基础上，考察了一类非线性Volterra积分方程的Chebyshev谱配置法.方法中，我们将该类非线性方程转化为两个方程进行数值逼近.我们选择N阶Chebyshev Gauss-Lobatto点作为配置点，对积分项用N阶高斯数值积分公式逼近.收敛性分析结果表明数值误差的收敛阶为N^（1/2）-m，其中m是已知函数最高连续导数的阶数.我们也开展数值实验证实这一理论分析结果.

关键词： 非线性, Volterra积分方程, 谱配置逼近, 收敛性分析

Base on related references, we investigate the Chebyshev spectral collocation method for nonlinear Volterra integral equations of the second kind. The main idea of the presented method is to approximate numerically the new equations which are tranformedfrom the nonlinear Volterra integral equations of the second kind. The collocation points are Chebyshev Gauss-Lobatto points of order N. The integral terms are approximated by Gauss quadrature formula of order N. The provided convergence analysis shows thatthe convergence rate of the numerical errors is N^(1/2)-m provided that the given functions are m times continuous differentiable. This theoretical result is confirmed by the provided numerical experiments.

Key words: Nonlinear Volterra integral equations, Spectral collocation approximation, Convergence analysis

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摘要

非线性第二类Volterra积分方程的Chebyshev谱配置法

CHEBYSHEV SPECTRAL COLLOCATION METHOD FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND

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