• 论文 •

### 一类弱奇性Volterra积分微分方程的级数展开数值解法

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.

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

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:

()
 [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.
 [1] 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. [2] 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. [3] 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. [4] 刘丽华, 马昌凤, 唐嘉. 求解广义鞍点问题的一个新的类SOR算法[J]. 计算数学, 2016, 38(1): 83-95. [5] 黄娜, 马昌凤, 谢亚君. 求解非对称代数Riccati 方程几个新的预估-校正法[J]. 计算数学, 2013, 35(4): 401-418. [6] 陈绍春, 梁冠男, 陈红如. Zienkiewicz元插值的非各向异性估计[J]. 计算数学, 2013, 35(3): 271-274. [7] 任志茹. 三阶线性常微分方程Sinc方程组的结构预处理方法[J]. 计算数学, 2013, 35(3): 305-322. [8] 张亚东, 石东洋. 各向异性网格下抛物方程一个新的非协调混合元收敛性分析[J]. 计算数学, 2013, 35(2): 171-180. [9] 陈争, 马昌凤. 求解非线性互补问题一个新的 Jacobian 光滑化方法[J]. 计算数学, 2010, 32(4): 361-372. [10] 来翔, 袁益让. 一类三维拟线性双曲型方程交替方向有限元法[J]. 计算数学, 2010, 32(1): 15-36. [11] 蔚喜军. 非线性波动方程的交替显-隐差分方法[J]. 计算数学, 1998, 20(3): 225-238.