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一类弱奇性Volterra积分微分方程的级数展开数值解法

古振东, 孙丽英   

  1. 广东金融学院应用数学系, 广州 510521
  • 收稿日期:2016-12-22 出版日期:2017-12-15 发布日期:2017-11-13
  • 基金资助:

    广东省自然科学基金项目(2017A030310636),广东省高性能计算学会开放课题基金项目(2017060104),中山大学广东省计算科学重点实验室开放基金项目(2016001,2016006),广东省高等学校优秀青年教师培养计划项目(YQ201403),广东高校省级重点平台和重大科研项目(2015GXJK101)和广东金融学院金融数据挖掘与量化投资创新团队项目资助.

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.

Gu Zhengdo, Sun Liying. SERIES EXPANSION METHOD FOR WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2017, 39(4): 351-362.

SERIES EXPANSION METHOD FOR WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

Gu Zhengdo, Sun Liying   

  1. Guangdong University of Finance Guangzhou 510521, China
  • Received:2016-12-22 Online:2017-12-15 Published:2017-11-13
本文考察了一类弱奇性积分微分方程的级数展开数值解法,并给出了相应的收敛性分析.理论分析结果表明,若用已知函数的谱配置多项式逼近已知函数,那么方程的数值解以谱精度逼近方程的真解.数值实验数据也验证了这一理论分析结果.
In this paper, we investigate the series expansion method for weakly singular Volterra integro-differential equations. The provided convergence analysis shows that if given functions is approximated by their own spectral collocation polynomials, then numerical solution converge to exact solution at the rate of spectral accuracy. Numerical experiments are carried out to confirm this result.

MR(2010)主题分类: 

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[1] Brewer D W, Powers R K. Parameter identification in a volterra equation with weakly singular kernel[J]. Journal of Integral Equations & Applications, 1990, 2(3):353-373.

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

[3] Brunner H, Pedas A, Vainikko G. Piecewise polynomial collocation methods for linear volterra integro-differential equations with weakly singular kernels[J]. Siam Journal on Numerical Analysis, 39(3) (2001) 957-982.

[4] Brunner H, Tang T. Polynomial spline collocation methods for the nonlinear basset equation[J]. Computers & Mathematics with Applications, 1989, 18(5):449-457.

[5] Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral methods (fundamental in single domains), 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] Cerezo G M. Solution representation and identication for singular neutral functional dierential equations, Virginia Tech.

[8] Chen Y, Gu Z. Legendre spectral-collocation method for volterra integral differential equations with nonvanishing delay[J]. Communications in Applied Mathematics and Computational Science, 2013, 8(1):67-98.

[9] Chen Y, Li X, Tang T. A note on jacobi spectral-collocation methods for weakly singular volterra integral equations with smooth solutions[J]. J. Comput. Math, 2013, 31:47-56.

[10] Chen Y, Tang T. Spectral methods for weakly singular volterra integral equations with smooth solutions[J]. Journal of Computational and Applied Mathematics, 2009, 233(4):938-950.

[11] 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.

[12] Clements J C, Smith B R. Parameter estimation in a reaction-diffusion model for synaptic transmission at a neuromuscular junction[J]. Canad. appl. math. quart, 1996, 2:157-173.

[13] Corduneanu A, Morosanu G. A linear integro-differential equation related to a problem from capillarity theory[J]. Communications on Applied Nonlinear Analysis, 1996, 3:51-60.

[14] Gu Z. Multi-step chebyshev spectral collocation method for volterra integro-differential equations, Calcolo, 2015, 1-25.

[15] Gu Z, Chen Y. Chebyshev spectral-collocation method for volterra integral equations[J]. Contemporary Mathematics, 2013, 586:163-170.

[16] 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.

[17] Gu Z, Chen Y. Legendre spectral-collocation method for volterra integral equations with nonvanishing delay[J]. Calcolo, 2014, 51(1):151-174.

[18] Gu Z, Chen Y. Piecewise legendre spectral-collocation method for volterra integro-differential equations[J]. LMS Journal of Computation and Mathematics, 2015, 18(01):231-249.

[19] Gu Z, Guo X, Sun D. Series expansion method for weakly singular volterra integral equations[J]. Applied Numerical Mathematics, 2016, 105:112-123.

[20] Lubich C. Runge-kutta theory for volterra and abel integral equations of the second kind[J]. Mathematics of computation, 1983, 41(163):87-102.

[21] Mustapha K, Brunner H, Mustapha H, Schotzau D. An hp-version discontinuous galerkin method for integro-differential equations of parabolic type[J]. Siam Journal on Numerical Analysis, 2011, 49(4):1369-1396.

[22] Prosperetti A. A method for the solution of a class of singular volterra integro-differential equations[J]. Journal of Computational Physics, 1992, 46(3):462-468.

[23] Tang T, Xu X, Cheng J. On spectral methods for volterra integral equations and the convergence analysis[J]. J. Comput. Math, 2008, 26(6):825-837.

[24] Wei Y, Chen Y. Convergence analysis of the spectral methods for weakly singular volterra integrodifferential equations with smooth solutions[J]. Advances in Applied Mathematics & Mechanics, 2012, 4(1):1-20.
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