Abstract
In this paper, we propose a new class of multistep collocation methods for solving nonlinear Volterra Integral Equations, based on Hermite interpolation. These methods furnish an approximation of the solution in each subinterval by using approximated values of the solution, as well as its first derivative, in the r previous steps and m collocation points. Convergence order of the new methods is determined and their linear stability is analyzed. Some numerical examples show efficiency of the methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baker, C.T.H., Keech, M.S.: Stability regions in the numerical treatment of Volterra Integral Equations. SIAM. J. Numer. Anal. 15, 394–417 (1978)
Bellen, A., Jackiewicz, Z., Vermiglio, R., Zennaro, M.: Stability analysis of Runge–Kutta method for Volterra Integral Equations of second kind. IMA J. Numer. Anal. 10, 103–118 (1990)
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)
Brunner, H.: On the divergence of collocation solutions in smooth piecewise polynomial spaces for Volterra Integral Equations. BIT Numer. Math. 44, 631–650 (2004)
Brunner, H., Van der Houwen, P.J.: The numerical solution of Volterra equations. In: CWI Monographs, vol. 3. North Holland, Amesterdam (1986)
Cash, J.R.: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J. Numer. Anal. 18, 21–36 (1981)
Conte, D., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for Volterra Integral Equations. Appl. Math. Comput. 204, 839–853 (2008)
Conte, D., Paternoster, B.: A family of multistep collocation methods for Volterra Integral Equations. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) Numerical Analysis and Applied Mathematics. AIP Conference Proceeding, vol. 936, pp. 45–49. Springer, New York (2007)
Conte, D., Paternoster, B.: Multistep collocation methods for Volterra Integral Equations. Appl. Numer. Math. 59, 1721–1736 (2009)
Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1981)
Hojjati, G., Rahimi, M.Y., Hosseini, S.M.: New second derivative multistep methods for stiff systems. Appl. Math. Model. 30, 466–476 (2006)
Ferguson, D.: Some interpolation theorems for polynomials. J. Approx. Theory 9, 327–348 (1973)
Grimaldi, R.P.: Discrete and Combinatorial Mathematics. Pearson (1999)
Liniger, W., Willoughby, R.A.: Efficient numerical integration of stiff systems of ordinary differential equations. Technical report RC-1970, Thomas J. Watson Research Center, Yorktown Heihts, New York (1976)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (2002)
Van Der Houwen, P.J., Te Riele, H.J.J.: Backward differentiation type formulas for Volterra Integral Equations of the second kind. Numer. Math. 37, 205–217 (1981)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fazeli, S., Hojjati, G. & Shahmorad, S. Multistep Hermite collocation methods for solving Volterra Integral Equations. Numer Algor 60, 27–50 (2012). https://doi.org/10.1007/s11075-011-9510-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-011-9510-5
Keywords
- Volterra Integral Equations
- Multistep collocation methods
- Hermite interpolation
- Linear stability
- Convergence