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Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Volterra integral equations Integral equations, integral transforms

Published online by Cambridge University Press:  27 May 2016

Xiulian Shi  and
Yanping Chen
Xiulian Shi*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
*
*Corresponding author. Email:[email protected] (X. L. Shi), [email protected] (Y. P. Chen)
*Corresponding author. Email:[email protected] (X. L. Shi), [email protected] (Y. P. Chen)
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Abstract

A spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially in L∞ norm and weighted L2 norm. Finally, two numerical examples are presented to demonstrate our error analysis.

Keywords

Volterra integro-differential equationsspectral Jacobi-collocation methodpantograph delayweakly singular kernel

MSC classification

Secondary: 65R20: Integral equations 65M70: Spectral, collocation and related methods 45D05: Volterra integral equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 4 , August 2016 , pp. 648 - 669
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Ali, I., Convergence analysis of spectral methods for integro-differential equations with vanishing proportional delays, J. Comput. Math., 28 (2010), pp. 962–973.Google Scholar
[2]Brunner, H. and Schötzau, D., Hp-discontinuous Galerkin time-stepping for Volterra Integrodifferential equations, SIAM J. Numer. Anal., 44 (2006), pp. 224–245.CrossRefGoogle Scholar
[3]Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T., Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988.CrossRefGoogle Scholar
[4]Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T., Spectral Methods Fundamentals in Single Domains, Springer, Heidelberg, 2006.CrossRefGoogle Scholar
[5]Canuto, C., Hussaini, M., Quarteroni, A. and Zang, T., Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer, Heidelberg, 2007.CrossRefGoogle Scholar
[6]Chen, Y., Li, X. and Tang, T., A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Math., 31 (2013), pp. 47–56.CrossRefGoogle Scholar
[7]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.CrossRefGoogle Scholar
[8]Chen, Y. and Tang, T., Convergence analysis of Jacobi spectral collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput., 79 (2010), pp. 147–167.CrossRefGoogle Scholar
[9]Gu, Z. and Chen, Y., Legendre spectral-collocation method for Volterra integral equations with non-vanishing delay, Calcolo, 51 (2014), pp. 151–174.CrossRefGoogle Scholar
[10]Jiang, Y., On spectral methods for Volterra-type integro-differential equations, J. Comput. Appl. Math., 230 (2009), pp. 333–340.CrossRefGoogle Scholar
[11]Kufner, A. and Persson, L., Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.CrossRefGoogle Scholar
[12]Li, X. and Tang, T., Convergence analysis of Jacobi spectral collocations methods for Alber-Volterra integral equations of second kind, Front. Math. China, 7 (2012), pp. 69–84.CrossRefGoogle Scholar
[13]Mastroianni, G. and Occorsio, D., Optimal systems of nodes for Lagrange interpolation on bounded intervals, A survey, J. Comput. Appl. Math., 134 (2001), pp. 325–341.CrossRefGoogle Scholar
[14]Neval, P., Mean convergence of Lagrange interpolation, III, Trans Amer. Math. Soc., 282 (1984), pp. 669–698.Google Scholar
[15]Ragozin, D., Polynomial approximation on compact manifolds and homogeneous space, Trans. Am. Math. Soc., 150 (1970), pp. 41–53.CrossRefGoogle Scholar
[16]Ragozin, D., Constructive polynomial approximation on spheres and projective spaces, Trans. Am. Math. Soc., 162 (1971), pp. 157–170.Google Scholar
[17]Shen, J. and Tang, T., Pectral and High-Order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[18]Shen, J., Tang, T. and Wang, L., Spectral Methods: Algorithms, Analyses and Applications, Springer, New York, 2011.CrossRefGoogle Scholar
[19]Tang, T., Xu, X. and Cheng, J., On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math., 26 (2008), pp. 825–837.Google Scholar
[20]Tao, X., Xie, Z. and Zhou, X., Spectral Petrov-Galerkin methods for the second kind Volterra type integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 216–236.CrossRefGoogle Scholar
[21]Wan, Z., Chen, Y. and Huang, Y., Legendre spectral Galerkin method for second-kind Volterra integral equations, Front. Math. China, 4 (2009), pp. 181–193.CrossRefGoogle Scholar
[22]Wei, Y. and Chen, Y., Convergence analysis of the Legendre spectral collocations methods for second order Volterra integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 419–438.Google Scholar
[23]Wei, Y. and Chen, Y., Legendre spectral collocation methods for pantograph Volterra delay integro-differential equations, J. Sci. Comput., 53 (2012), pp. 672–688.CrossRefGoogle Scholar
[24]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.CrossRefGoogle Scholar
[25]Wei, Y. and Chen, Y., A spectral method for neutral Volterra integro-differential equation with weakly singular kernel, Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 424–446.Google Scholar
[26]Wei, Y. and Chen, Y., Legendre spectral collocation method for neutral and high-order Volterra integro-differential equation, Appl. Numer. Math., 81 (2014), pp. 15–29.CrossRefGoogle Scholar
[27]Willett, D., A linear generalization of Gronwall's inequality, Pro. Am. Math. Soc., 16 (1965), pp. 774–778.Google Scholar
[28]Xie, Z., Li, X. and Tang, T., Convergence Analysis of Spectral Galerkin Methods for Volterra type integral equations, J. Sci. Comput., 53 (2012), pp. 414–434.CrossRefGoogle Scholar
[29]Yang, Y., Chen, Y. and Huang, Y., Spectral-collocation method for fractional Fredholm integro-differential equations, J. Korean Math. Soc., 51 (2014), pp. 203–224.CrossRefGoogle Scholar
[30]Yang, Y., Chen, Y. and Huang, Y., Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations, Acta Math. Sci., 34B (2014), pp. 673–690.CrossRefGoogle Scholar
[31]Yang, Y., Chen, Y., Huang, Y. and Yang, W., Convergence Analysis of Legendre-collocation methods for nonlinear Volterra type integro equations, Adv. Appl. Math. Mech., 7 (2015), pp. 74–88.CrossRefGoogle Scholar
[32]Zhang, R., Zhu, B. and Xie, H., Spectral methods for weakly singular Volterra integral equations with pantograph delays, Front. Math. China, 8 (2013), pp. 281–299.CrossRefGoogle Scholar