Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T22:18:11.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel

Published online by Cambridge University Press:  17 February 2009

Chang Ho Kim  and
U Jin Choi
Chang Ho Kim
Affiliation:
Current address: Dept. of Applied Mathematics, Konkuk University, Chungju, Chungbuk 380–701, Korea.
U Jin Choi
Affiliation:
Department of Mathematics of KAIST, Kusong-Dong 373–1, Yousong-Gu, Taejon 305–701, Korea.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We propose and analyze the spectral collocation approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 39 , Issue 3 , January 1998 , pp. 408 - 430
Copyright
Copyright © Australian Mathematical Society 1998

References

[1] Bernardi, C. and Maday, Y., “Properties of some weighted Sobolev spaces and application to spectral approximations”, SIAM J. Numer. Anal. 26 (1989) 769–829.CrossRefGoogle Scholar
[2] Bressan, N. and Quarteroni, A., “Analysis of Chebyshev collocation methods for parabolic equa-tions”, SIAM J. Numer. Anal. 23 (1986) 1138–1153.CrossRefGoogle Scholar
[3] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral methods in fluid dynamics (Springer, Berlin, 1987).Google Scholar
[4] Canuto, C. and Quarteroni, A., “Approximation results for orthogonal polynomials in Sobolev spaces”, Math. Comp. 38 (1982) 67–86.CrossRefGoogle Scholar
[5] Chen, C., Thomée, V. and Wahlbin, B., “Finite element approximation of parabolic integro-differential equation with a weakly singular kernel”, Math. Comp. 58 (1992) 587–602.CrossRefGoogle Scholar
[6] Gottlieb, D. and Orszag, S. A., Numerical analysis of spectral methods: theory and applications (SIAM Philadelphia, PA, 1977).Google Scholar
[7] Lin, Y., Thomée, V. and Wahlbin, B., “Ritz-Volterra projections to finite-element spaces and applic-ations to integro-differential and related equations”, SIAM J. Numer. Anal. 28 (1991) 1047–1070.Google Scholar
[8] Linz, P., Analytical and numerical methods for Volterra equations (SIAM Philadelphia, PA, 1985).CrossRefGoogle Scholar
[9] Pani, A. K., Chung, S. K. and Anderssen, R. S., “Convergence of a finite difference scheme for a parabolic integro-differential equation with a weakly singular kernel”, CMA Report CMA-MR8–91, Centre for Mathematics and its Application, The Australian National University (1991).Google Scholar
[10] Pani, A. K., Chung, S. K. and Anderssen, R. S., “On convergence of finite difference scheme for a parabolic generalized solutions of parabolic and hyperbolic integro-differential equations”, CMA Report CMA-MR3-91, Centre for Mathematics and its Application, The Australian National University (1991).Google Scholar
[11] Renardy, M., Hrusa, W. J. and Nohel, J. A., Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics No. 35 (Wiley, New York, 1987).Google Scholar
[12] Thomée, V. and Zhang, N. Y., “Error estimates for semidiscrete finite element methods for parabolic integro-differential equations”, Math. Comp. 53 (1989) 121–139.CrossRefGoogle Scholar
[13] Yanik, E. G. and Fairweather, G., “Finite element methods for parabolic and hyperbolic partial integro-differential equation”, Nonlinear Anal. 12 (1988) 785–809.Google Scholar