Journal of Applied Mathematics

A New Modification of Adomian Decomposition Method for Volterra Integral Equations of the Second Kind

Lie-jun Xie

Full-text: Open access

Abstract

We propose a new modification of the Adomian decomposition method for Volterra integral equations of the second kind. By the Taylor expansion of the components apart from the zeroth term of the Adomian series solution, this new technology overcomes the problems arising from the previous decomposition method. The validity and applicability of the new technique are illustrated through several linear and nonlinear equations by comparing with the standard decomposition method and the modified decomposition method. The results obtained indicate that the new modification is effective and promising.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 795015, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808209

Digital Object Identifier
doi:10.1155/2013/795015

Mathematical Reviews number (MathSciNet)
MR3127466

Zentralblatt MATH identifier
06950875

Citation

Xie, Lie-jun. A New Modification of Adomian Decomposition Method for Volterra Integral Equations of the Second Kind. J. Appl. Math. 2013 (2013), Article ID 795015, 7 pages. doi:10.1155/2013/795015. https://projecteuclid.org/euclid.jam/1394808209


Export citation

References

  • G. Adomian, “A review of the decomposition method and some recent results for nonlinear equations,” Mathematical and Computer Modelling, vol. 13, no. 7, pp. 17–43, 1990.
  • G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic, Dordrecht, The Netherlands, 1994.
  • Y. Cherruault, G. Saccomandi, and B. Some, “New results for convergence of Adomian's method applied to integral equations,” Mathematical and Computer Modelling, vol. 16, no. 2, pp. 85–93, 1992.
  • A.-M. Wazwaz, “A reliable modification of Adomian decomposition method,” Applied Mathematics and Computation, vol. 102, no. 1, pp. 77–86, 1999.
  • A.-M. Wazwaz and S. M. El-Sayed, “A new modification of the Adomian decomposition method for linear and nonlinear operators,” Applied Mathematics and Computation, vol. 122, no. 3, pp. 393–405, 2001.
  • M. M. Hosseini, “Adomian decomposition method with Chebyshev polynomials,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1685–1693, 2006.
  • Y. Liu, “Adomian decomposition method with orthogonal polynomials: Legendre polynomials,” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 1268–1273, 2009.
  • W.-C. Tien and C.-K. Chen, “Adomian decomposition method by Legendre polynomials,” Chaos, Solitons & Fractals, vol. 39, no. 5, pp. 2093–2101, 2009.
  • Y. Çenesiz and A. Kurnaz, “Adomian decomposition method by Gegenbauer and Jacobi polynomials,” International Journal of Computer Mathematics, vol. 88, no. 17, pp. 3666–3676, 2011.
  • A.-M. Wazwaz, “A new algorithm for calculating Adomian polynomials for nonlinear operators,” Applied Mathematics and Computation, vol. 111, no. 1, pp. 33–51, 2000.
  • E. Babolian and Sh. Javadi, “New method for calculating Adomian polynomials,” Applied Mathematics and Computation, vol. 153, no. 1, pp. 253–259, 2004.
  • A.-M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press, Beijing, China, 2011.
  • A.-M. Wazwaz, A First Course in Integral Equations, World Scientific Publishing, River Edge, NJ, USA, 1997.
  • M. Rahman, Integral Equations and Their Applications, WIT Press, Southampton, UK, 2007.
  • A.-M. Wazwaz and A. M. Wazwaz, “Necessary conditions for the appearance of noise terms in decomposition solution series,” Applied Mathematics and Computation, vol. 81, no. 2-3, pp. 265–274, 1997.
  • E. Babolian and A. Davari, “Numerical implementation of Adomian decomposition method,” Applied Mathematics and Computation, vol. 153, no. 1, pp. 301–305, 2004.
  • E. Babolian and A. Davari, “Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind,” Applied Mathematics and Computation, vol. 165, no. 1, pp. 223–227, 2005.