A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

doi:10.1007/s10473-022-0121-0

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (1): 387-402.doi: 10.1007/s10473-022-0121-0

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

郑伟珊¹, 陈艳萍²

1. College of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China;
2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

收稿日期:2020-05-27 修回日期:2021-06-01 出版日期:2022-02-25 发布日期:2022-02-24
通讯作者: Yanping CHEN,E-mail:yanpingchen@scnu.edu.cn E-mail:yanpingchen@scnu.edu.cn
作者简介:Weishan ZHENG,E-mail:weishanzheng@yeah.net
基金资助:
This work was supported by the State Key Program of National Natural Science Foundation of China (11931003) and the National Natural Science Foundation of China (41974133, 11671157).

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

Weishan ZHENG¹, Yanping CHEN²

1. College of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China;
2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

Received:2020-05-27 Revised:2021-06-01 Online:2022-02-25 Published:2022-02-24
Contact: Yanping CHEN,E-mail:yanpingchen@scnu.edu.cn E-mail:yanpingchen@scnu.edu.cn
Supported by:
This work was supported by the State Key Program of National Natural Science Foundation of China (11931003) and the National Natural Science Foundation of China (41974133, 11671157).

摘要/Abstract

摘要： In this paper, a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay, which contains a weakly singular kernel. We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules. In the end, we provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both the L^∞-norm and the weighted L²-norm.

关键词: Volterra integro-differential equation, pantograph delay, weakly singular kernel, Jacobi-collocation spectral methods, error analysis, convergence analysis

Abstract: In this paper, a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay, which contains a weakly singular kernel. We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules. In the end, we provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both the L^∞-norm and the weighted L²-norm.

Key words: Volterra integro-differential equation, pantograph delay, weakly singular kernel, Jacobi-collocation spectral methods, error analysis, convergence analysis

中图分类号:

65R20

郑伟珊, 陈艳萍. A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY[J]. 数学物理学报(英文版), 2022, 42(1): 387-402.

Weishan ZHENG, Yanping CHEN. A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY[J]. Acta mathematica scientia,Series B, 2022, 42(1): 387-402.

参考文献

[1] Ali I, Brunner H, Tang T. Spectral methods for pantograph-type differential and integral equations with multiple delays. Front Math China, 2009, 4:49-61
[2] Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods Fundamentals in Single Domains. SpringerVerlag, 2006
[3] Chen Y P, Tang T. Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math, 2009, 233:938-950
[4] Chen Y P, Tang T. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comp, 2010, 79:147-167
[5] Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences 93. 2nd ed. Heidelberg:Springer-Verlag, 1998
[6] Enright W H, Hu M. Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay. Appl Numer Math, 1997, 24:175-190
[7] Gan S Q. Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations. J Comput Appl Math, 2007, 206:898-907
[8] Guo B Y, Wang L L. Jacobi interpolation approximations and their applications to singular differential equations. Adv Comput Math, 2001, 14:227-276
[9] Guo B Y, Shen J, Wang L L. Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J Sci Comput, 2006, 27:305-322
[10] Ishiwata E, Muroya Y. On collocation methods for delay differential and Volterra integral equations with proportional delay. Front Math China, 2009, 4(1):89-111
[11] Kufner A, Persson L E. Weighted Inequalities of Hardy Type. New York:World Scientific, 2003
[12] Mastroianni G, Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey. J Comput Appl Math, 2001, 134:325-341
[13] Nevai P. Mean convergence of Lagrange interpolation. III. Trans Amer Math Soc, 1984, 282:669-698
[14] Ragozin D L. Polynomial approximation on compact manifolds and homogeneous spaces. Trans Amer Math Soc, 1970, 150:41-53
[15] Ragozin D L. Constructive polynomial approximation on spheres and projective spaces. Trans Amer Math Soc, 1971, 162:157-170
[16] Shen J, Tang T. Spectral and High-Order Methods with Applications. Beijing:Science Press, 2006
[17] Wang W S, Li S F. Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations. Front Math China, 2009, 4:195-216
[18] Wei Y X, Chen Y P. Legendre spectral collocation methods for pantograph Volterra delay integro-differential equations. J Sci Comput, 2012, 53:672-688
[19] Wei Y X, Chen Y P. Convergence analysis of the spectral methods for weakly singular Volterra integrodifferential equations with smooth solutions. Adv Appl Math Mech, 2012, 4:1-20
[20] Wei Y X, Chen Y P. Legendre spectral collocation method for Volterra-Hammerstein integral equation of the second kind. Acta Math Sci, 2017, 37B(4):1105-1114
[21] Wu S F, Gan S Q. Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput Math Appl, 2008, 55:2426-2443
[22] Yang Y, Chen Y P, Huang Y Q. Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations. Acta Math Sci, 2014, 34B(3):673-690
[23] Zhang C J, Vandewalle S. Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integrodifferential equations. IMA J Numer Anal, 2004, 24:193-214
[24] Zhao J J, Xu Y, Liu M Z. Stability analysis of numerical methods for linear neutral Volterra delay-integrodifferential system. Appl Math Comput, 2005, 167:1062-1079
[25] Zheng W S, Chen Y P. Numerical analysis for Volterra integral equation with two kinds of delay. Acta Math Sci, 2019, 39B(2):607-617

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

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