[1]单锐,魏金侠,张雁.Bernstein算子矩阵法求高阶弱奇异积分微分方程数值解[J].华侨大学学报(自然科学版),2012,33(5):595-600.[doi:10.11830/ISSN.1000-5013.2012.05.0595]
 SHAN Rui,WEI Jinxia,ZHANG Yan.Bernstein Operational Matrix Method for Solving the Numerical Solution of High Order Integro-Differential Equation with Weakly Singular[J].Journal of Huaqiao University(Natural Science),2012,33(5):595-600.[doi:10.11830/ISSN.1000-5013.2012.05.0595]
点击复制

Bernstein算子矩阵法求高阶弱奇异积分微分方程数值解()
分享到:

《华侨大学学报(自然科学版)》[ISSN:1000-5013/CN:35-1079/N]

卷:
第33卷
期数:
2012年第5期
页码:
595-600
栏目:
出版日期:
2012-09-20

文章信息/Info

Title:
Bernstein Operational Matrix Method for Solving the Numerical Solution of High Order Integro-Differential Equation with Weakly Singular
文章编号:
1000-5013(2012)05-0595-06
作者:
单锐 魏金侠 张雁
燕山大学 理学院, 河北 秦皇岛 066004
Author(s):
SHAN Rui WEI Jinxia ZHANG Yan
College of Sciences, Yanshan University, Qinhuangdao 066004, China
关键词:
高阶变系数 弱奇异 积分微分方程 Bernstein多项式 算子矩阵 数值解
Keywords:
high order variable coefficients weakly singular integro-differential equation Bernstein polynomial operational matrix numerical solution
分类号:
O241.8
DOI:
10.11830/ISSN.1000-5013.2012.05.0595
文献标志码:
A
摘要:
为了求高阶变系数且带有弱奇异积分核Volterra-Fredholm积分微分方程的数值解,提出了Bernstein算子矩阵法.利用Bernstein多项式的定义及其性质给出任意阶弱奇异积分的近似求积公式,同时也给出Bernstein多项式的微分算子矩阵.通过化简所求方程及离散化简后的方程,可将原问题转换为求代数方程组的解.最后,通过收敛性分析说明该方法是收敛的,并用数值算例验证了方法的有效性.
Abstract:
In order to obtain the numerical solution for high order variable coefficients Volterra- Fredholm integro-differential equation with weakly singular kernels, we present a Bernstein operational matrix method in this paper. A approximate formula which solves solution for any arbitrary order weakly singular integral is given by using the definition of Bernstein polynomial and some properties, and a operational matrix of derivative of Bernstein polynomial is also obtained. By translating the original problem through simplifying and descreting the equation, the problem can be transferred into a system of algebraic equations. Convergence analysis shows that the method is convergent. The numerical example shows that the method is effective.

参考文献/References:

[1] MALEKNEJAD K.A new approach to the numerical solution of Volterra integral equations by using bernstein’s approximation[J].Commun Nonlinear Sci Numer Simul,2011,16(2):647-655.
[2] YOUSEFI S A,BEHROOZIFAR M.Operational matrices of bernstein polynomials and their applications[J].Internat J Systems Sci,2010,41(6):709-716.
[3] MALEKNEJAD K,HASHEMIZADEH E,BASIRAT B.Computational method based on bernstein operational matrices for nonlinear Volterra-Fredholm-hammerstein integral equations[J].Commun Nonlinear Sci Numer Simul,2011,17(1):52-61.
[4] DELVES L M,MOHAMED J L.Computational methods for integral equations[M].Cambridge:Cambridge University Press,1985.
[5] SCHIAVANE P,CONSTANDA C,MIODUCHOWSKI A.Integral methods in science and engineering[M].Boston:Birkhäuser Boston,2002.
[6] RAZZAGHI M.The legendre wavelets operational matrix of integration[J].Int J Syst Sci,2001,32(4):495-502.
[7] MALEKNEJA K.An efficient numerical approximation for the linear class of Fredholm integro-differential equations based on Cattani’s method[J].Commun Nonlinear Sci Numer Simulat,2011,16(7):2672-2679.
[8] MALEKNEJAD K. A Bernstein operational matrix approach for solving a system of high order linear Volterra- Fredholm integro-differential equations[J].Mathematical and Computer Modelling,2012,55(3/4):1363-1372.
[9] PHILLIPS G M.Interpolation and approximation by polynomials[M].New York:Springerr,2003.

相似文献/References:

[1]牛红玲,郝玲,余志先.算子矩阵法求高阶弱奇异积分微分方程数值解[J].华侨大学学报(自然科学版),2013,34(5):581.[doi:10.11830/ISSN.1000-5013.2013.05.0581]
 NIU Hong-ling,HAO ling,YU Zhi-xian.Operational Matrix Method for Solving the Numerical Solution of High Order Integro-Differential Equation with Weakly Singular[J].Journal of Huaqiao University(Natural Science),2013,34(5):581.[doi:10.11830/ISSN.1000-5013.2013.05.0581]

备注/Memo

备注/Memo:
收稿日期: 2012-03-01
通信作者: 单锐(1961-),女,教授,主要从事偏微分方程、积分微分方程数值解和最优化理论的研究.E-mail:weijinxiaymx201366@163.com.
基金项目: 河北省教育厅科学研究计划项目(2009159)
更新日期/Last Update: 2012-09-20