Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T08:55:12.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations

Published online by Cambridge University Press:  28 May 2015

Yunxia Wei  and
Yanping Chen
Yunxia Wei*
Affiliation:
School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

A class of numerical methods is developed for second order Volterra integro-differential equations by using a Legendre spectral approach. We provide a rigorous error analysis for the proposed methods, which shows that the numerical errors decay exponentially in the L∞-norm and L2-norm. Numerical examples illustrate the convergence and effectiveness of the numerical methods.

Keywords

45L1065R2065D15Second order Volterra integro-differential equationsGauss quadrature formulaLegendre-collocation methodsconvergence analysis
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 3 , August 2011 , pp. 419 - 438
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Makroglou, A., A block-by-block method for Volterra integro-differential equations with weakly-singular kernel, Math. Comp., 37 (1981), pp. 95–99.CrossRefGoogle Scholar
[2]Guo, B. Y. and Shen, J., Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86 (2000), pp. 635–654.CrossRefGoogle Scholar
[3]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
[4]Guo, B. Y. and Wang, L. L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory, 128 (2004), pp. 1–14.CrossRefGoogle Scholar
[5]Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods Fundamentals in Single Domains, Springer-Verlag, 2006.CrossRefGoogle Scholar
[6]Qu, C. K. and Wong, R., Szego’s conjecture on Lebesgue constants for Legendre series, Pacific J. Math., 135 (1988), pp.157–188.CrossRefGoogle Scholar
[7]Xu, C. L. and Guo, B. Y., Laguerre pseudospectral method for nonlinear partial differential equations, J. Comp. Math., 20 (2002), pp. 413–428.Google Scholar
[8]Henry, D., Geometric theory of semilinear parabolic equations, Springer-Verlag, 1989.Google Scholar
[9]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
[10]Mastroianni, G. and Monegato, G., Nystrom interpolants based on zeros of Laguerre polynomials for some Weiner-Hopf equations, IMA J. Numer. Anal., 17 (1997), pp. 621–642.CrossRefGoogle Scholar
[11]Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press 2004.CrossRefGoogle Scholar
[12]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.CrossRefGoogle Scholar
[13]Brunner, H., Pedas, A. and Vainikko, G., Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39 (2001), pp. 957–982.CrossRefGoogle Scholar
[14]Brunner, H. and Lambert, J. D., Stability of numerical methods for Volterra integro-differential equations, Computing, 12 (1974), pp. 75–89.CrossRefGoogle Scholar
[15]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
[16]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.CrossRefGoogle Scholar
[17]Shen, J., Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), pp. 1113–1133.CrossRefGoogle Scholar
[18]Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press Beijing 2006.Google Scholar
[19]Balachandran, K., Park, J. Y. and Marshal Anthoni, S., Controllability of second order semilin-ear Volterra integrodifferential systems in banach spaces, Bull. Korean Math. Soc, 36 (1999), pp. 1–13.Google Scholar
[20]Garey, L. E. and Shaw, R. E., Algorithms for the solution of second order Volterra integrodifferential equations, Comput. Math. Appl., 22 (1991), pp. 27–34.CrossRefGoogle Scholar
[21]Aguilar, M. and Brunner, H., Collocation methods for second-order volterra integro-differential equations, Appl. Numer. Math., 4 (1988), pp. 455–470.CrossRefGoogle Scholar
[22]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.CrossRefGoogle Scholar
[23]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
[24]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
[25]Shawa, R. E. and Garey, L. E., A parallel shooting method for second order volterra integro-differential equations with two point boundary conditions, Int. J. Comput. Math., 49 (1993), pp. 61–66.CrossRefGoogle Scholar
[26]Bochkanov, S. and Bystritsky, V, Computation of Gauss-Jacobi quadrature rule nodes and weights, http://www.alglib.net/integral/gq/gjacobi.phpGoogle Scholar
[27]Tang, T., Xu, X. and Chen, J., On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math., 26 (2008), pp. 825–837.Google Scholar
[28]Liu, W. B. and Tang, T., Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems, Appl. Numer. Math., 38 (2001), pp. 315–345.CrossRefGoogle Scholar
[29]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
[30]Chen, Y. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comp., 79 (2010), pp. 147–167.CrossRefGoogle Scholar