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.

PDF ( 166 ) 引用

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

MR(2010)主题分类:

分享此文:

[1] Brauer F, Castillo-Chávez C. Mathematical models in population biology and epidemiology[M]. Springer New York.

[2] Brunner H. Collocation methods for Volterra integral and related functional differential equations[M]. vol. 15, Cambridge University Press, 2004.

[3] Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral methods (fundamental in single domains)[M]. Springer, 2006.

[4] Elnagar G N, Kazemi M. Chebyshev spectral solution of nonlinear volterra-hammerstein integral equations[J]. Journal of Computational and Applied Mathematics, 1996, 76(1-2):147-158.

[5] Gilding B H. A singular nonlinear volterra integral equation[J]. Journal of Integral Equations & Applications, 1993, 5(4):465-502.

[6] Gouyandeh Z, Allahviranloo T, Armand A. Numerical solution of nonlinear volterra-fredholmhammerstein integral equations via tau-collocation method with convergence analysis[J]. Journal of Computational & Applied Mathematics, 2016, 308:435-446.

[7] Ma Y, Huang J, Wang C. Numerical Solutions of Nonlinear Volterra-Fredholm-Hammerstein Integral Equations Using Sinc Nystrom Method[M]. Springer International Publishing, 2017.

[8] Maleknejad K, Hashemizadeh E, Basirat B. Computational method based on bernstein operational matrices for nonlinear volterra-fredholm-hammerstein integral equations[J]. Communications in Nonlinear Science & Numerical Simulation, 2012, 17(1):52-61.

[9] Marzban H R, Tabrizidooz H R, Razzaghi M. A composite collocation method for the nonlinear mixed volterra-fredholm-hammerstein integral equations[J]. Communications in Nonlinear Science & Numerical Simulation, 2011, 16(3):1186-1194.

[10] Parand K, Rad J A. Numerical solution of nonlinear volterra-fredholm-hammerstein integral equations via collocation method based on radial basis functions[J]. Applied Mathematics & Computation, 2012, 218(9):5292-5309.

[11] Unterreiter A. Volterra integral equation models for semiconductor devices[J]. Mathematical Methods in the Applied Sciences, 1996, 19(6):425-450.

[1]	尚在久, 宋丽娜. 关于辛算法稳定性的若干注记[J]. 计算数学, 2020, 42(4): 405-418.
[2]	洪庆国, 刘春梅, 许进超. 一种抽象的稳定化方法及在非线性不可压缩弹性问题上的应用[J]. 计算数学, 2020, 42(3): 298-309.
[3]	刘子源, 梁家瑞, 钱旭, 宋松和. 带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法[J]. 计算数学, 2019, 41(4): 440-452.
[4]	林霖. 类Hartree-Fock方程的数值方法[J]. 计算数学, 2019, 41(2): 113-125.
[5]	王俊俊, 李庆富, 石东洋. 非线性抛物方程混合有限元方法的高精度分析[J]. 计算数学, 2019, 41(2): 191-211.
[6]	王志强, 文立平, 朱珍民. 时间延迟扩散-波动分数阶微分方程有限差分方法[J]. 计算数学, 2019, 41(1): 82-90.
[7]	李郴良, 田兆鹤, 胡小媚. 一类弱非线性互补问题的广义模系矩阵多分裂多参数加速松弛迭代方法[J]. 计算数学, 2019, 41(1): 91-103.
[8]	陈圣杰, 戴彧虹, 徐凤敏. 稀疏线性规划研究[J]. 计算数学, 2018, 40(4): 339-353.
[9]	袁晓, 肖瑾. 受参考价格影响的变质产品销售最优动态价格和保存技术投资的联合策略研究[J]. 计算数学, 2017, 39(4): 363-377.
[10]	古振东, 孙丽英. 一类弱奇性Volterra积分微分方程的级数展开数值解法[J]. 计算数学, 2017, 39(4): 351-362.
[11]	裕静静, 江平, 刘植. 两类五阶解非线性方程组的迭代算法[J]. 计算数学, 2017, 39(2): 151-166.
[12]	张旭, 檀结庆, 艾列富. 一种求解非线性方程组的3p阶迭代方法[J]. 计算数学, 2017, 39(1): 14-22.
[13]	郭俊, 吴开腾, 张莉, 夏林林. 一种新的求非线性方程组的数值延拓法[J]. 计算数学, 2017, 39(1): 33-41.
[14]	刘金魁. 解凸约束非线性单调方程组的无导数谱PRP投影算法[J]. 计算数学, 2016, 38(2): 113-124.
[15]	王艳芳, 王然, 康彤. 一类带有铁磁材料参数的非线性涡流问题的A-φ有限元法[J]. 计算数学, 2016, 38(2): 125-142.

阅读次数

全文

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

扫码分享