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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]
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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

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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.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