Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T13:56:09.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence Analysis of the Spectral Methods for Weakly Singular Volterra Integro-Differential Equations with Smooth Solutions

Published online by Cambridge University Press:  03 June 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
*
URL: http://202.116.32.252/userinfo.asp?usernamesp=%B3%C2%D1%DE%C6%BC, Email: [email protected]
Corresponding author. Email:[email protected]
Get access

Abstract

The theory of a class of spectral methods is extended to Volterra integro-differential equations which contain a weakly singular kernel (t - s)->* with 0 < μ < 1. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in L°°-norm and weighted L2-norm. The numerical examples are given to illustrate the theoretical results.

Keywords

45J0565R20Volterra integro-differential equationsweakly singular kernelsspectral methodsconvergence analysis
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 1 , February 2012 , pp. 1 - 20
Copyright
Copyright © Global-Science Press 2012

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] 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.CrossRefGoogle Scholar
[3] Bernardi, C. and Maday, Y., Spectral Methods, in: Ciarlet, G.P. and Lions, J. L. (Eds.), Handbook of Numerical Analysis, Vol. 5, Techniques of Scientific Computing, Elsevier, Amsterdam, 1997, 209–486.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.CrossRefGoogle Scholar
[5] Brunner, H., Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 6 (1986), pp. 221–239.CrossRefGoogle Scholar
[6] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004.CrossRefGoogle Scholar
[7] 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
[8] Brunner, H. and Schoötzau, D., hp-discontinuous Galerkin time-stepping for Volterra In-tegrodifferential equations, SIAM J. Numer. Anal., 44 (2006), pp. 224–245.CrossRefGoogle Scholar
[9] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods Fundamentals in Single Domains, Springer-Verlag, 2006.CrossRefGoogle Scholar
[10] 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
[11] Chen, Y. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra Integral equations with a weakly singular kernel, Math. Comput., 79 (2010), pp. 147–167.CrossRefGoogle Scholar
[12] Colton, D. and Kress, R., Inverse Coustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Springer-Verlag, Heidelberg, 2nd Edition, 1998.CrossRefGoogle Scholar
[13] Guo, B. Y., Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl., 243 (2000), pp. 373–408.CrossRefGoogle Scholar
[14] 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
[15] Guo, B. Y. and Wang, L. L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory., 128 (2004), pp. 1–41.CrossRefGoogle Scholar
[16] Jiang, Y. J., On spectral methods for Volterra-type Integro-differential equations, J. Comput. Appl. Math., 230 (2009), pp. 333–340.CrossRefGoogle Scholar
[17] Kufner, A. and Persson, L. E., Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.CrossRefGoogle Scholar
[18] 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
[19] 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
[20] 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
[21] Nevai, P., Mean convergence of Lagrange interpolation: III, Trans. Am. Math. Soc., 282 (1984), pp. 669–698.CrossRefGoogle Scholar
[22] Ragozin, D. L., Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Am. Math. Soc., 150 (1970), pp. 41–53.CrossRefGoogle Scholar
[23] Ragozin, D. L., Constructive polynomial approximation on spheres and projective spaces, Trans. Am. Math. Soc., 162 (1971), pp. 157–170.Google Scholar
[24] 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
[25] Shen, J., Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), pp. 1113–1133.CrossRefGoogle Scholar
[26] Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[27] Tang, T., Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373–382.CrossRefGoogle Scholar
[28] 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
[29] 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
[30] Y. Wei, X. 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
[31] Xu, C. L. and Guo, B. Y., Laguerre pseudospectral method for nonlinear partial differential equations, J. Comput. Math., 20 (2002), pp. 413–428.Google Scholar
[32] 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