Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2010, Article ID 901587, 21 pages
http://dx.doi.org/10.1155/2010/901587
Research Article

High Accuracy Combination Method for Solving the Systems of Nonlinear Volterra Integral and Integro-Differential Equations with Weakly Singular Kernels of the Second Kind

1College of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2Department of Scientific Computing, The Florida State University, Tallahassee, FL 32310, USA

Received 21 October 2009; Accepted 1 April 2010

Academic Editor: Gradimir V. Milovanović

Copyright © 2010 Lu Pan et al. 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. H. Brunner, “The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes,” Mathematics of Computation, vol. 45, no. 172, pp. 417–437, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. Brunner, “Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels,” IMA Journal of Numerical Analysis, vol. 6, no. 2, pp. 221–239, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, vol. 4 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 1997. View at MathSciNet
  4. W. Hackbusch, Integralgleichungen, Teubner Studienbücher Mathematik, B. G. Teubner, Stuttgart, Germany, 1989. View at MathSciNet
  5. Z. Chen, Y. Xu, and J. Zhao, “The discrete Petrov-Galerkin method for weakly singular integral equations,” Journal of Integral Equations and Applications, vol. 11, no. 1, pp. 1–35, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. T. Lü and H. Yong, “Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kind,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 225–237, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Pedas and E. Tamme, “Spline collocation method for integro-differential equations with weakly singular kernels,” Journal of Computational and Applied Mathematics, vol. 197, no. 1, pp. 253–269, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Pedas and E. Tamme, “Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 111–126, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. Brunner, Y. Lin, and S. Zhang, “Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methods,” Journal of Integral Equations and Applications, vol. 10, no. 4, pp. 375–396, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. Qun and T. Lü, “The combination of approximate solutions for accelerating the convergence,” RAIRO Analyse Numérique, vol. 18, no. 2, pp. 153–160, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Liu and T. Lü, “High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equations,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 441–451, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. N. Lyness and B. W. Ninham, “Numerical quadrature and asymptotic expansions,” Mathematics of Computation, vol. 21, pp. 162–178, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. Lü and Y. Huang, “A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind,” Journal of Mathematical Analysis and Applications, vol. 282, no. 1, pp. 56–62, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Liu and T. Lü, “Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations,” Journal of Computational and Applied Mathematics, vol. 201, no. 1, pp. 300–313, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. Navot, “A further extension of Euler-Maclaurin summation formula,” Journal of Mathematical Physics, vol. 41, pp. 155–184, 1962. View at Google Scholar
  16. T. Lü and Y. Huang, “A high accuracy combinatorial algorithm and a posteriori error estimation for solving second-kind Abel integral equations,” Journal of Systems Science and Mathematical Sciences, vol. 24, no. 1, pp. 110–117, 2004. View at Google Scholar · View at MathSciNet