非线性弱奇性Volterra积分方程的谱配置法

doi:10.12286/jssx.j2019-0642

计算数学 ›› 2021, Vol. 43 ›› Issue (4): 426-443.DOI: 10.12286/jssx.j2019-0642

非线性弱奇性Volterra积分方程的谱配置法

古振东

广东金融学院金融数学与统计学院, 广州 510521

收稿日期:2019-11-14 出版日期:2021-11-14 发布日期:2021-11-12
基金资助:
国家自然科学基金（11971123），广东省自然科学基金（2017A030310636，2018A030313236），广东省高性能计算学会开放基金（2017060104），中山大学广东省计算科学重点实验室开放基金（2016001）资助.

PDF ( 281 ) 引用

古振东. 非线性弱奇性Volterra积分方程的谱配置法[J]. 计算数学, 2021, 43(4): 426-443.

Gu Zhendong. SPECTRAL COLLOCATION METHOD FOR NONLINEAR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS[J]. Mathematica Numerica Sinica, 2021, 43(4): 426-443.

SPECTRAL COLLOCATION METHOD FOR NONLINEAR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS

Gu Zhendong

Guangdong University of Finance, Guangzhou 510521, China

Received:2019-11-14 Online:2021-11-14 Published:2021-11-12

基于已有文献的研究成果及前期工作，我们考察了非线性弱奇性Volterra积分方程（VIE）的谱配置法，并对该方法进行了收敛性分析.得到的结论是数值误差呈谱收敛.误差收敛阶与配置点个数及方程解的正则性相关.数值实验也证实了这一结论.本文的方法解决了已有文献中类似数值方法（Allaei（2016），Sohrabi（2017））存在的问题.

关键词： 弱奇性Volterra积分方程, 非线性, 谱配置逼近, 收敛性分析

Based onsome references and our previous works, we investigate the spectral collocation method for nonlinear weakly singular Volterra integral equations. The provided convergence analysis shows that the presented method has spectral convergence. The convergence orderrelates to the number of collocation points and the regularity of the solution to the equations. We carry out numerical experiments to confirm the theoretical results. The presented method is the improvement to the methods proposed by Allaei(2016) and Sohrabi(2017).

Key words: weaklysingular Volterra integral equation, nonlinearity, spectral approximation, convergence analysis

MR(2010)主题分类:

分享此文:

[1] Allaei S S, Diogo T, Rebelo M. The jacobi collocation method for a class of nonlinear volterra integral equations with weakly singular kernel[J]. Journal of Scientific Computing, 2016, 69(2):1-23.
[2] Bieniasz L K. An adaptive huber method for nonlinear systems of volterra integral equations with weakly singular kernels and solutions[J]. Journal of Computational & Applied Mathematics, 2017, 323:136-146.
[3] Brunner H. Collocation methods for Volterra integral and related functional differential equations[M]. vol. 15, Cambridge University Press, 2004.
[4] Brunner H, Pedas A, Vainikko G. The piecewise polynomial collocation method for nonlinear weakly singular volterra equations[J]. Mathematics of Computation 68(227) (1999) 1079-1096.
[5] Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral methods (fundamental in single domains)[M]. Springer, 2006.
[6] Cao Y, Herdman T, Xu Y. A hybrid collocation method for volterra integral equations with weakly singular kernels[J]. Siam Journal on Numerical Analysis, 2003, 41(1):364-381.
[7] Chen Y, Tang T. Convergence analysis of the jacobi spectral-collocation methods for volterra integral equations with a weakly singular kernel[J]. Mathematics of computation, 2010, 79(269):147-167.
[8] Frankel J I. A nonlinear heat transfer problem:solution of nonlinear, weakly singular volterra integral equations of the second kind[J]. Engineering Analysis with Boundary Elements, 1991, 8(5):231-238.
[9] Gu Z. Spectral collocation method for system of weakly singular volterra integral equations[J]. Advances in Computational Mathematics, 2019, 2019:1-25.
[10] Gu Z. Spectral collocation method for weakly singular volterra integro-differential equations[J]. Applied Numerical Mathematics, 2019, 143:263-275.
[11] Gu Z, Chen Y. Chebyshev spectral-collocation method for a class of weakly singular volterra integral equations with proportional delay[J]. Journal of Numerical Mathematics, 2014, 22(4):311-342.
[12] Han H, He X, Liu Y, Tao L. Extrapolation for solving a system of weakly singular nonlinear volterra integral equations of the second kind[J]. International Journal of Computer Mathematics, 2011, 88(16):3507-3520.
[13] Jumarhon B, Mckee S. Product integration methods for solving a system of nonlinear volterra integral equations[J]. Journal of Computational & Applied Mathematics, 1996, 69(2):285-301.
[14] Mastroianni G, Occorsio D. Optimal systems of nodes for lagrange interpolation on bounded intervals. a survey[J]. Journal of Computational & Applied Mathematics, 2001, 134(1):325-341.
[15] Olmstead W E. A nonlinear integral equation associated with gas absorption in a liquid[J]. Zeitschrift F ü r Angewandte Mathematik Und Physik Zamp, 1977, 28(3):513-523.
[16] Padmavally K. On a non-linear integral equation[J]. Journal of Mathematics & Mechanics, 1958, 7(4):533-555.
[17] Ragozin D L. Polynomial approximation on compact manifolds and homogeneous spaces[J]. Transactions of the American Mathematical Society, 1970, 150(1):41-53.
[18] Ragozin D L. Constructive polynomial approximation on spheres and projective spaces[J]. Transactions of the American Mathematical Society, 1971, 162(NDEC):157-170.
[19] Rebelo M. Extrapolation methods for a nonlinear weakly singular volterra integral equation[C]. in:American Institute of Physics Conference Series, 2010.
[20] Rebelo M, Diogo T. A hybrid collocation method for a nonlinear volterra integral equation with weakly singular kernel[J]. Journal of Computational & Applied Mathematics, 2010, 234(9):2859-2869.
[21] Shen J, Tang T, Wang L L. Spectral methods:algorithms, analysis and applications[M]. vol. 41, Springer Science & Business Media, 2011.
[22] Sohrabi S, Ranjbar H, Saei M. Convergence analysis of the jacobi-collocation method for nonlinear weakly singular volterra integral equations[J]. Applied Mathematics and Computation, 2017, 299(3):141-152.
[23] Zakeri G A, Navab M. Sinc collocation approximation of non-smooth solution of a nonlinear weakly singular volterra integral equation[J]. Journal of Computational Physics, 2010, 229(18):6548-6557.
[24] Zhong C, Wei J. Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind[J]. Applied Mathematics & Computation, 2011, 217(19):7790-7798.

[1]	许志强. 相位恢复:理论、模型与算法[J]. 计算数学, 2022, 44(1): 1-18.
[2]	张法勇, 安晓丽. 带五次项的非线性Schrödinger方程的守恒差分格式[J]. 计算数学, 2022, 44(1): 63-70.
[3]	刘嘉诚, 陈先进, 段雅丽, 李昭祥. 改进的PNC方法及其在非线性偏微分方程多解问题中的应用[J]. 计算数学, 2022, 44(1): 119-136.
[4]	唐耀宗, 杨庆之. 非线性特征值问题平移对称幂法的收敛率估计[J]. 计算数学, 2021, 43(4): 529-538.
[5]	马积瑞, 范金燕. 信赖域方法在Hölderian局部误差界下的收敛性质[J]. 计算数学, 2021, 43(4): 484-492.
[6]	胡雅伶, 彭拯, 章旭, 曾玉华. 一种求解非线性互补问题的多步自适应Levenberg-Marquardt算法[J]. 计算数学, 2021, 43(3): 322-336.
[7]	李旭, 李明翔. 连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速[J]. 计算数学, 2021, 43(3): 354-366.
[8]	刘金魁, 孙悦, 赵永祥. 凸约束伪单调方程组的无导数投影算法[J]. 计算数学, 2021, 43(3): 388-400.
[9]	唐玲艳, 郭嘉, 宋松和. 求解带刚性源项标量双曲型守恒律方程的保有界WCNS格式[J]. 计算数学, 2021, 43(2): 241-252.
[10]	袁光伟. 非正交网格上满足极值原理的扩散格式[J]. 计算数学, 2021, 43(1): 1-16.
[11]	王然, 张怀, 康彤. 求解带有非线性边界条件的涡流方程的A-φ解耦有限元格式[J]. 计算数学, 2021, 43(1): 33-55.
[12]	丁戬, 殷俊锋. 求解一类非线性互补问题的松弛two-sweep模系矩阵分裂迭代法[J]. 计算数学, 2021, 43(1): 118-132.
[13]	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.
[14]	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.
[15]	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.

阅读次数

全文

摘要

非线性弱奇性Volterra积分方程的谱配置法

SPECTRAL COLLOCATION METHOD FOR NONLINEAR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS

扫码分享