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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (4): 29-39.doi: 10.6040/j.issn.1671-9352.0.2022.352

• • 上一篇    

求解非线性伪抛物方程的重心Lagrange插值配点法

屈金铮1,李金1,2*,苏晓宁1   

  1. 1.华北理工大学理学院, 河北 唐山 063210;2.河北省数据科学与应用重点实验室, 河北 唐山 063210
  • 发布日期:2023-03-27
  • 作者简介:屈金铮(1998— ),男,硕士研究生,研究方向为重心插值配点法理论、网络与信息安全. E-mail:qjz1579043951@163.com*通信作者简介:李金(1980— ),男,博士, 教授,硕士生导师,研究方向为边界元理论及其数值解、超奇异积分的数值计算和重心插值配点法理论. E-mail:lijin@ncst.edu.cn
  • 基金资助:
    河北省自然科学基金资助项目(A2019209533)

Barycentric Lagrange interpolation collocation method for solving nonlinear pseudo-parabolic equations

QU Jin-zheng1, LI Jin1,2*, SU Xiao-ning1   

  1. 1. College of Science, North China University of Science and Technology, Tangshan 063210, Hebei, China;
    2. Hebei Key Laboratory of Data Science and Application, Tangshan 063210, Hebei, China
  • Published:2023-03-27

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摘要： 提出了重心Lagrange插值配点法求解一类非线性伪抛物方程。首先,介绍了重心Lagrange插值并给出了微分矩阵表达式。其次,构造了求解非线性伪抛物方程的直接线性化迭代格式、部分线性化迭代格式、Newton线性化迭代格式。再次,未知函数和初边值条件利用重心Lagrange插值函数来近似,利用配点法得到离散方程,获得了方程的矩阵表达式。最后,数值算例表明,重心Lagrange插值配点法具有高精度和高效率的优点。

关键词: 非线性伪抛物方程, 重心Lagrange插值, 配点法, 迭代格式

Abstract: Barycentric Lagrange interpolation collocation method for solving a class of nonlinear pseudo-parabolic equations is proposed. Firstly, barycentric Lagrange interpolation is introduced and the expression of differential matrix is given. Secondly, direct linearized iterative scheme, partial linearized iterative scheme, Newton linearized iterative scheme for solving nonlinear pseudo-parabolic equation are constructed. Thirdly, unknown functions and initial-boundary value conditions are approximated by barycentric Lagrange interpolation function, discrete equation is obtained by using collocation method, then the matrix equation is obtained. Finally, numerical examples show that the barycentric Lagrange interpolation collocation method has the advantages of high precision and high efficiency.

Key words: nonlinear pseudo-parabolic equation, barycentric Lagrange interpolation, collocation method, iterative scheme

中图分类号: 

  • O241.82
[1] 李宁,郭艳,秦卫,等. 一类热流密码体制非线性模型的有限元算法[J]. 南京邮电学院学报(自然科学版),2001,21(3):43-45,57. LI Ning, GUO Yan, QIN Wei, et al. A finite element algorithm used for nonlinear heat flow cryptosystems[J]. Journal of Nanjing University of Posts and Telecommunications(Natural Science), 2001, 21(3):43-45,57.
[2] 高常忠,涂慧,宋惠元. 一类热流密码体制模型及计算机模拟结果分析[J]. 信息工程大学学报,2004,5(4):28-31. GAO Changzhong, TU Hui, SONG Huiyuan. A class of heat flow cryptosystems and analysis of results of computer simulations[J]. Journal of Information Enqineering University, 2004, 5(4):28-31.
[3] 陈雪鸿,江成顺. 热流密码体制的一种非齐次半线性模型的加,解密实现[C] //中国密码学学术会议. 上海:上海交通大学,2004:160-165. CHEN Xuehong, JIANG Chengshun. An realization of encryption and decryption in the semi-linear case of heat flow cryptsystems[C] //Chinese Association for Cryptologic Research. Shanghai:Shanghai Jiao Tong University, 2004: 160-165.
[4] 刘霖雯. 一类Sobolev方程的伪谱解法及其在热流密码体制中的应用[D]. 郑州:解放军信息工程大学,2006. LIU Linwen. Fourier pseudo-spectral method for some Sobolev equation and its applications in heat flow cryptosystem[D]. Zhengzhou: PLA Information Engineering University, 2006.
[5] 曹杨. 一类伪抛物型方程解的渐进行为及其在图像处理中的应用[D]. 长春:吉林大学,2010. CAO Yang. Asymptotic behavior of a class of pseudo-parabolic equations with applications to image processing[D]. Changchun: Jilin University, 2010.
[6] 李伟,宋惠元. 伪抛物型方程组的比较原理及其应用[J]. 信息工程大学学报,2002,3(1):35-39. LI Wei, SONG Huiyuan. The comparative theory of the pseudoparabolic equation system and its application[J]. Jounal of Information Engineering University, 2002, 3(1):35-39.
[7] TING T W. A cooling process according to two-temperature theory of heat conduction[J]. Journal of Mathematical Analysis and Applications, 1974, 45(1):23-31.
[8] SHIVAMOGGI B K. A symmetric regularized long-wave equation for shallow water waves[J]. Physics of Fluids, 1986, 29(3):890-891.
[9] TAYLOR W J. Method of Lagrangian curvilinear interpolation[J]. Journal of Research of the National Bureau of Standards, 1945, 35(2):151-155.
[10] BERRUT J P,TREFETHEN L N. Barycentric Lagrange interpolation[J]. SIAM Review, 2004, 46(3):501-517.
[11] 虎晓燕,韩惠丽. 重心插值配点法求解分数阶Fredholm积分方程[J]. 郑州大学学报(理学版),2017,49(1):17-23. HU Xiaoyan, HAN Huili. Barycentric interpolation collocation method for solving Fredholm integral equation of fractional order[J]. Journal of Zhengzhou University(Natural Science Edition), 2017, 49(1):17-23.
[12] 王兆清,李淑萍,唐炳涛. 一维重心型插值:公式、算法和应用[J]. 山东建筑大学学报,2007,22(5):448-453. WANG Zhaoqing, LI Shuping, TANG Bingtao. Formulations, algorithms and applications on barycentric interpolation in 1D[J]. Journal of Shandong Jianzhu University, 2007, 22(5):448-453.
[13] WANG Zhaoqing, JIANG Jian,TANG Bingtao, et al. Numerical solution of bending problem for elliptical plate using differentiation matrix method based on barycentric Lagrange interpolation[J]. Applied Mechanics and Materials, 2014, 638/639/640:1720-1724.
[14] WANG Zhaoqing, TANG Bingtao, ZHENG Wei. A barycentric interpolation collocation method for Darcy flow in two-dimension[J]. Applied Mechanics and Materials, 2014, 684:3-10.
[15] 李树忱,王兆清,袁超. 极坐标系下弹性问题的重心插值配点法[J]. 中南大学学报(自然科学版),2013,44(5):2031-2040. LI Shuchen, WANG Zhaoqing, YUAN Chao. Barycentric interpolation collocation method for solving elastic problems[J]. Journal of Central South University(Science and Technology), 2013, 44(5):2031-2040.
[16] 王兆清,李淑萍,唐炳涛. 圆环变形及屈曲的重心插值配点法分析[J]. 机械强度,2009,31(2):245-249. WANG Zhaoqing, LI Shuping, TANG Bingtao. Deformation and buckling analysis of ring by barycentric interpolation collocation method[J]. Journal of Mechanical Strength, 2009, 31(2):245-249.
[17] 赵岳月,王兆清,李金. 自由边界问题的重心插值迭代配点法研究[J]. 山东建筑大学学报,2018,33(2):29-32. ZHAO Yueyue, WANG Zhaoqing, LI Jin. Study of barycentric interpolation iteration collocation method for free boundary problem[J]. Journal of Shandong Jianzhu University, 2018, 33(2):29-32.
[18] 李树忱,王兆清. 高精度无网格重心插值配点法:算法、程序及工程应用[M]. 北京:科学出版社,2012:168-179. LI Shuchen, WANG Zhaoqing. Meshless barycentric interpolation collocation method with high precision-algorithms: programs & applications in engineering[M]. Beijing: Science Press, 2012: 168-179.
[19] 刘婷. 求解电报方程的重心Lagrange插值配点法[D]. 银川:宁夏大学,2016. LIU Ting. Barycentric Lagrange interpolation collocation method for solving telegraph equation[D]. Yinchuan: Ningxia University, 2016.
[20] LI Jin, SU Xiaoning, QU Jinzheng. Linear barycentric rational collocation method for solving telegraph equation[J]. Mathematical Methods in the Applied Sciences, 2021, 44(14):11720-11737.
[21] 虎晓燕. 分数阶积分微分方程的无网格重心插值配点法[D]. 银川:宁夏大学,2016. HU Xiaoyan. Meshfree barycentric interpolation collocation method for solving integral-differential equations of fractional order[D]. Yinchuan: Ningxia University, 2016.
[22] LI Jin, CHENG Yongling. Numerical solution of Volterra integro-differential equations with linear barycentric rational method[J]. International Journal of Applied and Computational Mathematics, 2020, 6(5):1-12.
[23] LI Jin, CHENG Yongling. Linear barycentric rational collocation method for solving heat conduction equation[J]. Numerical Methods for Partial Differential Equations, 2020, 37(1):533-545.
[24] LI Jin, SANG Yu. Linear barycentric rational collocation method for beam force vibration equation[J]. Shock and Vibration, 2021, 2021(2):1-11.
[25] 李淑萍. 基于重心型插值的数值计算方法[J]. 山东科学,2010,23(4):13-16. LI Shuping. A survey of numerical method based on barycentric interpolation[J]. Shandong Science, 2010, 23(4):13-16.
[1] 王志刚 秦新强 党发宁 苏李君. 楔形基无网格法解的存在惟一性[J]. J4, 2010, 45(2): 44-49.
[2] 张青洁,王文洽 . 色散方程高阶差分格式的并行迭代法[J]. J4, 2008, 43(2): 8-11 .
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