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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 652631, 8 pages
http://dx.doi.org/10.1155/2014/652631
Research Article

Numerical Solutions of a Class of Nonlinear Volterra Integral Equations

Department of Pure and Applied Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa

Received 10 April 2014; Revised 17 June 2014; Accepted 26 June 2014; Published 9 July 2014

Academic Editor: Chun-Gang Zhu

Copyright © 2014 H. S. Malindzisa and M. Khumalo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Saveljeva, “Quadratic and cubic spline collocation for Volterra integral equations,” 2006, http://dspace.utlib.ee/dspace/handle/10062/1163.
  2. B. G. Sloss and W. F. Blyth, “Corrington's Walsh function method applied to a nonlinear integral equation,” Journal of Integral Equations and Applications, vol. 6, no. 2, pp. 239–256, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  3. H. Brunner, “Iterated collocation methods for Volterra integral equations with delay arguments,” Mathematics of Computation, vol. 62, no. 206, pp. 581–599, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. G. Blom and H. Brunner, “The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods,” Society for Industrial and Applied Mathematics. Journal on Scientific and Statistical Computing, vol. 8, no. 5, pp. 806–830, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. T. Diogo, “Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations,” Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 363–372, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. T. Diogo and P. Lima, “Collocation solutions of weakly singular Volterra integral equation,” Matemática Aplicada e Computacional, vol. 8, pp. 229–238, 2007.
  7. V. Horvat, “On collocation methods for Volterra integral equations with delay arguments,” Mathematical Communications, vol. 4, no. 1, pp. 93–109, 1999. View at Zentralblatt MATH · View at MathSciNet
  8. R. Benitez and V. J. Bolos, “Blow-up collocation solutions of some Volterra integral equations,” 2011, http://arxiv.org/abs/1112.4658.
  9. M. Aigo, “On the numerical approximation of Volterra integral equations of the second kind using quadrature rules,” International Journal of Advanced Scientific and Technological Research, vol. 1, pp. 558–564, 2013.
  10. R. Katani and S. Shahmorad, “Block by block method for the systems of nonlinear Volterra integral equations,” Applied Mathematical Modelling, vol. 34, no. 2, pp. 400–406, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. F. Mirzaee, “A computational method for solving linear Volterra integral equations,” Applied Mathematical Sciences, vol. 6, no. 17–20, pp. 807–814, 2012. View at MathSciNet · View at Scopus
  12. J. Saberi-Nadjafi and M. Heidari, “A quadrature method with variable step for solving linear Volterra integral equations of the second kind,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 549–554, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. H. Brunner, Collocation Methods For Volterra Integral And Related Functional Differential Equations, Cambridge University Press, Cambridge, Mass, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  14. H. Brunner, “Iterated collocation methods and their discretizations for Volterra integral equations,” SIAM Journal on Numerical Analysis, vol. 21, no. 6, pp. 1132–1145, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. M. S. Corrington, “Solution of differential and integral equations with walsh functions,” IEEE Trans Circuit Theory, vol. 20, no. 5, pp. 470–476, 1973. View at Scopus
  16. R. H. Rand, Lecture notes on nonlinear vibrations, 2003, http://www.math.cornell.edu/~rand/randdocs/nlvibe52.pdf.
  17. D. G. Zill, W. S. Wright, and M. R. Cullen, Differential Equations with Boundary Value Problems, Richard Stratton, 2012.