Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T16:12:42.931Z Has data issue: false hasContentIssue false
  • English
  • Français

The Global Behavior of Finite Difference-Spatial Spectral Collocation Methods for a Partial Integro-differential Equation with a Weakly Singular Kernel

Published online by Cambridge University Press:  28 May 2015

Jie Tang  and
Da Xu
Jie Tang*
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, Hunan, P.R. China College of Science, Hunan University of Technology, Zhuzhou 412008, Hunan, P.R. China
Da Xu*
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, Hunan, P.R. China
*
Corresponding author.Email address:tj049@163.com
Corresponding author.Email address:daxu@hunnu.edu.cn
Get access

Abstract

The z-transform is introduced to analyze a full discretization method for a partial integro-differential equation (PIDE) with a weakly singular kernel. In this method, spectral collocation is used for the spatial discretization, and, for the time stepping, the finite difference method combined with the convolution quadrature rule is considered. The global stability and convergence properties of complete discretization are derived and numerical experiments are reported.

Keywords

65M1265M1565M7045K05Partial integro-differential equationweakly singular kernelspectral collocation methodsz-transformconvolution quadrature
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 3 , August 2013 , pp. 556 - 570
Copyright
Copyright © Global Science Press Limited 2013

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] Bernardi, C. and Maday, Y., Spectral Methods, in Handbook of Numerical Analysis, edited by Ciarlet, P.G. and Lions, J.L., North-Holland, Amsterdam, 1997.Google Scholar
[2] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.CrossRefGoogle Scholar
[3] Chen, C., Thömee, V. and Wahlbin, L.B., Finte element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comput., 58 (1992), pp. 587–602.Google Scholar
[4] Gottlieb, D. and Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia, 1977.Google Scholar
[5] Guo, B. Y., Spectral Methods and Their Applications, World Scientific, Singapore, 1998.Google Scholar
[6] Guo, B. Y. and Wang, L. L., Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math., 14 (2001), pp. 227–276.Google Scholar
[7] Guo, B. Y. and Wang, L. L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev space, J. Approx. Theory, 1 (2004), pp. 1–41.Google Scholar
[8] Guo, B. Y. and Zhang, K. J., Non-isotropic Jacobi pseudospetral method, J. Comput. Math., 26 (2008), pp. 511–535.Google Scholar
[9] Kim, C. H. and Choi, U. J., Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, J. Austral. Math. Soc. Ser. B, 39 (1998), pp. 408–430.Google Scholar
[10] Li, X. J. and Xu, C. J., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108–2131.Google Scholar
[11] Lin, Y. M. and Xu, C. J., Finite difference/spectral approximation for the time fractional diffusion equations, J. Comput. Phys., 2 (2007), pp. 1533–1552.Google Scholar
[12] Lin, Y. M., Li, X. J. and Xu, C. J., Finite difference/spectral approximations for the fractional cable equation, Math. Comput., 80 (2011), pp. 1369–1396.Google Scholar
[13] Lions, J. L. and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications I, Springer Verlag, Berlin, Heidelberg, New York, 1972.Google Scholar
[14] López-Marcos, J.C., A difference scheme for nonlinear partial integro-differential equation, SIAM J. Numer. Anal., 27 (1990), pp. 20–31.CrossRefGoogle Scholar
[15] Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), pp. 704–719.Google Scholar
[16] Lubich, C., Sloan, I. H. and Thomé, V., Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comput., 65 (1996), pp. 1–17.Google Scholar
[17] Mclean, W. and Thómee, V., Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser. B., 35 (1993), pp. 23–70.Google Scholar
[18] Mclean, W., Thómee, V. and Wahlbin, L.B., Discretization with variable time steps of an evolution equation with a positive type memory term, J. Comput. Appl. Math., 69 (1996), pp. 49–69.Google Scholar
[19] Mustapha, K. and Mclean, W., Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comput., 78 (2009), pp. 1975–1995.CrossRefGoogle Scholar
[20] Mustapha, K. and Mustapha, H., A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel, IMA J. Numer. Anal., 30 (2010), pp. 555–578.Google Scholar
[21] Mustapha, K., Brunner, H., Mustapha, H. and Schótzau, D., An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type, SIAM J. Numer. Anal., 49 (2011), pp. 1369–1396.Google Scholar
[22] Nohel, J. A. and Shea, D. F., Frequency domain methods for Volterra equations, Advances in Math., 22 (1976), pp. 278–304.Google Scholar
[23] Pani, A. K., Fairweather, G. and Fernandes, R. I., Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term, SIAM J. Numer. Anal., 46 (2008), pp. 344–364.Google Scholar
[24] Pani, A. K., Fairweather, G. and Fernandes, R. I., ADI orthogonal spline collocation methods for parabolic partial integro-differential equations, IMA J. Numer. Anal., 30 (2010), pp. 248–276.CrossRefGoogle Scholar
[25] Sanz-Serna, J. M., A numerical method for a partial integrodifferential equation, SIAM J. Numer. Anal., 25 (1988), pp. 319–327.Google Scholar
[26] Shen, J. and Tang, T., Spectral and High-Order Methods with Application, Science Press, Beijing, 2006.Google Scholar
[27] Tang, J., Xu, D. and Liu, J., The global behavior of spacial spectral collocation methods for a partial integro-differential equation with a weakly singular kernel (in Chinese), J. Sys. Sci. and Math. Scis., 29 (2009), pp. 646–656.Google Scholar
[28] Tang, T., A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11 (1993), pp. 309–319.CrossRefGoogle Scholar
[29] Trefethen, L. N., Spectral Methods in Matlab, SIAM, 2000.CrossRefGoogle Scholar
[30] Xu, D., The global behavior of time discretization for an abstract Volterra equation in Hilbert space, CALCOLO, 34 (1997), pp. 71–104.Google Scholar
[31] Xu, D., Uniform l1 behaviour for time discretization of a volterra equation with completely monotonic kernel: I. Stability, IMA J. Numer. Anal., 22 (2002), pp. 133–151.Google Scholar
[32] Xu, D., Uniform l1 behaviour in a second-order difference-type method for a linear Volterra equation with completely monotonic kernel I: Stability, IMA J. Numer. Anal., 31 (2011), pp. 1154–1180.CrossRefGoogle Scholar
[33] Xu, D., Stability of the difference type methods for linear Volterra equations in Hilbert spaces, Numer. Math., 109 (2008), pp. 571–595.Google Scholar