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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*
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]
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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.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Makroglou, A., A block-by-block method for Volterra integro-differential equations with weakly-singular kernel, Math. Comp., 37 (1981), pp. 9599.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. 635654.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. 227276.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. 114.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.157188.CrossRefGoogle Scholar
[7]Xu, C. L. and Guo, B. Y., Laguerre pseudospectral method for nonlinear partial differential equations, J. Comp. Math., 20 (2002), pp. 413428.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. 30113022.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. 621642.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. 381402.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. 957982.CrossRefGoogle Scholar
[14]Brunner, H. and Lambert, J. D., Stability of numerical methods for Volterra integro-differential equations, Computing, 12 (1974), pp. 7589.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. 254265.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. 4961.CrossRefGoogle Scholar
[17]Shen, J., Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), pp. 11131133.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. 113.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. 2734.CrossRefGoogle Scholar
[21]Aguilar, M. and Brunner, H., Collocation methods for second-order volterra integro-differential equations, Appl. Numer. Math., 4 (1988), pp. 455470.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. 113.CrossRefGoogle Scholar
[23]Tarang, M., Stability of the spline collocation method for second order Volterra integro-differential equations, Math. Model. Anal., 9 (2004), pp. 7990.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. 110.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. 6166.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. 825837.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. 315345.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. 938950.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. 147167.CrossRefGoogle Scholar