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A Spectral Method for Second Order Volterra Integro-Differential Equation with Pantograph Delay

Published online by Cambridge University Press:  03 June 2015

Weishan Zheng  and
Yanping Chen
Weishan Zheng*
Affiliation:
School of Mathematics sciences, South China Normal University, Guangzhou 510631, Guangdong, China
Yanping Chen*
Affiliation:
School of Mathematics sciences, South China Normal University, Guangzhou 510631, Guangdong, China
*
Email: weishanzheng@yeah.net
Corresponding author. URL: http://math.xtu.edu.cn/myphp/math/ypchen/index.htm, Email: yanpingchen@scnu.edu.cn
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Abstract

In this paper, a Legendre-collocation spectral method is developed for the second order Volterra integro-differential equation with pantograph delay. We provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both L2-norm and L∞-norm.

Keywords

65R2034K28Legendre-spectral methodsecond order Volterra integro-differential equationpantograph delayerror analysis
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 2 , April 2013 , pp. 131 - 145
Copyright
Copyright © Global-Science Press 2013

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References

[1]Ali, I., Brunner, H. and Tang, T., A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math., 27 (2009), pp. 254–265.Google Scholar
[2]Ali, I., Brunner, H. and Tang, T., Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China, 4 (2009), pp. 49–61.Google Scholar
[3]Balachandran, K., Park, J. Y. and Anthoni, S. Marshal, Controllability of second order semilinear Volterra integrodifferential systems in banach spaces, Bull. Korean Math. Soc., 36 (1999), pp. 1–13.Google Scholar
[4]Bologna, M., Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels, J. Phys. A Math. Theor., 43 (2010), pp.1–13.Google Scholar
[5]Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press 2004.CrossRefGoogle Scholar
[6]Brunner, H., Makroglou, A. and Miller, R. K., Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution, Appl. Numer. Math., 23 (1997), pp. 381–402.Google Scholar
[7]Brunner, H. and Lambert, J. D., Stability of numerical methods for Volterra integro-differential equations, Computing, 12 (1974), pp. 75–89.CrossRefGoogle Scholar
[8]Chen, Y. and Tang, T., Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math., 233 (2009), pp. 938–950.Google Scholar
[9]Shen, J., Tang, T. and Wang, L.-L., Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011.Google Scholar
[10]Garey, L. E. and Shaw, R. E., Algorithms for the solution of second order Volterra integro-differential equations, Comput. Math. Appl., 22 (1991), pp. 27–34.CrossRefGoogle Scholar
[11]Guo, B. Y. and Wang, L. L., Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math., 14 (2001), pp. 227–276.Google Scholar
[12]Kato, T. and Mcleod, J.B., The functional-differential equation y′(x)= ay(λx)+by(x), Bull. Amer. Math. Soc., 77 (1971), pp. 891–937.Google Scholar
[13]Makroglou, A., A block-by-block method for Volterra integro-differential equations with weakly-singular kernel, Math. Comput., 37 (1981), pp. 95–99.Google Scholar
[14]Matinfar, M., Saeidy, M. and Vahidi, J., Application of homotopy analysis method for solving systems of Volterra integral equations, Adv. Appl. Math. Mech., 4 (2012), pp. 36–45.Google Scholar
[15]Qu, C. K. and Wong, R., Szego’s conjecture on Lebesgue constants for Legendre series, Pacific J. Math., 135 (1988), pp. 157–188.Google Scholar
[16]Rawashdeh, E., Mcdowell, D. and Rakesh, L., Polynomial spline collocation methods for second-order Volterra integro-differential equations, IJMMS, 56 (2004), pp. 3011–3022.Google Scholar
[17]Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press Beijing 2006.Google Scholar
[18]Tao, X., Xie, Z. Q. and Zhou, X. J., Spectral Petrov-Galerkin methods for the second kind Volterra type integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 216–236.Google Scholar
[19]Tarang, M., Stability of the spline collocation method for second order Volterra integro-differential equations, Math. Model. Anal., 9 (2004), pp. 79–90.CrossRefGoogle Scholar
[20]Wei, Y. and Chen, Y., Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 419–438.Google Scholar
[21]Wei, Y. and Chen, Y., Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv. Appl. Math. Mech., 4 (2012), pp. 1–20.Google Scholar
[22]Zarebnia, M. and Nikpour, Z., Solution of linear Volterra integro-differential equations via Sinc functions, Int. J. Appl. Math. Comput., 2 (2010), pp. 1–10.Google Scholar
[23]Zhang, K. and Li, J., Collocation methods for a class of Volterra integral functional equations with multiple proportional delays, Adv. Appl. Math. Mech., 4 (2012), pp. 575–602.CrossRefGoogle Scholar