Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-12T08:38:03.937Z Has data issue: false hasContentIssue false
  • English
  • Français

Second-order time discretization with finite-element method for partial integro-differential equations 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:
Department of Applied Mathematics ofKon-Kuk University, Danwol 322, Chungju 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 the second-order time discretization scheme with the finite-element approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the finite element method and the time discretization is based on the Crank-Nicolson scheme with a graded mesh. We show the stability of the scheme and obtain the second-order convergence result for the fully discretized scheme.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 4 , April 1999 , pp. 513 - 524
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Brunner, H., “Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels”, IMA J. Numer Anal. 6 (1986) 221–239.CrossRefGoogle Scholar
[2]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
[3]Kaneko, H. and Xu, Y., “Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind”, Math. Comp. 62 (1994) 725–738.CrossRefGoogle Scholar
[4]Kim, C. H. and Choi, U. J., “Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel”, J. Aust. Math. Soc. B 39 (1998) 408–430.CrossRefGoogle Scholar
[5]Kim, C. H. and Choi, U. J., “Time discretization with collocation methods for the parabolic partial integro-differential equation with weakly singular kernel”, IMA J. Numer. Anal., submitted for publication.Google Scholar
[6]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”, Centre for Mathematics and its Application, The Australian National University, 1991, CM A Report CMA-MR3–91.Google Scholar
[7]Pani, A. K., Thomée, V. and Wahlbin, L. B., “Numerical methods for hyperbolic and parabolic integro-differential equations”, Journal of Integral Equations and Applications 4 (1992) 533–583.CrossRefGoogle Scholar
[8]Renardy, M., Hrusa, W. J. and Nohel, J. A., Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics (Willey, New York, 1987).Google Scholar
[9]Sloan, I. H. and Thomée, V., “Time discretization of an integro-differential equation of parabolic type”, SIAM J. Numer. Anal. 23 (1986) 1052–1061.CrossRefGoogle Scholar
[10]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
[11]Yanik, E. G. and Fairweather, G., “Finite element methods for parabolic and hyperbolic partial integro-differential equation”, Nonlinear Anal. 12 (1988) 785–809.CrossRefGoogle Scholar