An approach for solving singular two point boundary value problems: analytical and numerical treatment

C. Chun, A. Ebaid, Mi Lee, Emad Aly

Abstract


The numerical treatment of two point singular boundary value problems has always been a difficult and challenging task due to the singularity behaviour that occurs at a point. Various efficient numerical methods have been proposed to deal with such boundary value problems. We present a new efficient modification of the Adomian decomposition method for solving singular boundary value problems, both linear and nonlinear. Numerical examples illustrate the efficiency and accuracy of the proposed method.

References
  • G. Adomian. A review of the decomposition method and some recent results for nonlinear equation. Math. Comput. Modelling, 3, 1992, 17--43.
  • G. Adomian. Solving frontier problems of physics: the decomposition method. Kluwer Academic Publishers, Boston, 1994.
  • G. Adomian. Solution of the Thomas--Fermi equation. Appl. Math. Lett., 11(3), 1998, 131--133.
  • D. Lesnic. A computational algebraic investigation of the decomposition method for time--dependent problems. Appl. Math. Comput., 119, 2001, 197--206.
  • E. Babolian and J. Biazar. Solving the problem of biological species living together by Adomian decomposition method. Appl. Math. Comput., 129, 2002, 339--343.
  • M. Benabidallah and Y. Cherruault. Application of the Adomian method for solving a class of boundary problems. Kybernetes, 33, 2004, 118--132.
  • E. H. Aly, A. Ebaid and R. Rach. Advances in the Adomian decomposition method for solving two--point nonlinear boundary value problems with Neumann boundary conditions. Compu. Math. Applic., 63, 2012, 1056--1065.
  • B. Jang. Two--point boundary value problems by the extended Adomian decomposition method. J. Comput. Appl. Math., 219, 2008, 253--263.
  • A. M. Wazwaz. Partial differential equations and solitary waves theory. Springer, New York, 2009.
  • M. Kumar and N. Singh. Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems. Comput. Chem. Eng., 34, 2010, 1750--1760.
  • Y. Cherruault, G. Adomian, K. Abbaoui and R. Rach. Further remarks on convergence of decomposition method. Bio--Medical Comput., 38, 1995, 89--93.
  • M. M. Hosseini and H. Nasabzadeh. On the convergence of Adomian decomposition method. Appl. Math. Comput., 182, 2006, 536--543.
  • A. Ebaid. A new analytical and numerical treatment for singular two--point boundary value problems via the Adomian decomposition method. J. Comput. Appl. Math., 235, 2011, 1914--1924.
  • J. Janus and J. Myjak. A generalized Emden--Fowler equation with a negative exponent. Nonlin. Analy., 23, 1994, 953--970.
  • M. K. Kadalbajoo and V. K. Aggarwal. Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline. Appl. Math. Comput., 160, 2005, 851--863.
  • A. S. V. Ravi Kanth, K. Aruna. Solution of singular two--point boundary value problems using differential transformation method. Phys. Lett. A, 372, 2008, 4671--4673.
  • Sami Bataineh, M. S. M. Noorani and I. Hashim. Approximate solutions of singular two--point bvps by modified homotopy analysis method. Phys. Lett. A, 372, 2008, 4062--4066.
  • A. M. Wazwaz. Adomian decomposition method for a reliable treatment of the Emden--Fowler equation. Appl. Math. Comput., 161, 2005, 543--560.
  • M. Inc, M. Ergut, Y. Cherruault. A different approach for solving singular two-point boundary value problems. Kybernetes, 34, 2005, 934--940.
  • C. Chun. A modified Adomian decomposition method for solving higher-order singular boundary value problems. Z. Naturforsch. A, 65, 2010, 1093--1100.
  • A. M. Wazwaz. The modified decomposition method for analytic treatment of differential equations. Appl. Math. Comput., 173, 2006, 165--176.
  • M. Cui and F. Geng. Solving singular two--point boundary value problem in reproducing kernel space. J. Comput. Appl. Math., 205, 2007, 6--15.
  • S. M. El-Sayed. Integral methods for computing solutions of a class of singular two--point boundary value problems. Appl. Math. Comput., 130, 2002, 235--241.
  • A. Ebaid. Exact solutions for a class of nonlinear singular two-point boundary value problems: The decomposition method. Z. Naturforsch. A, 65, 2010, 145--150.

Keywords


Adomian decomposition method; singular two-point boundary value problems

Full Text:

PDF BibTeX


DOI: http://dx.doi.org/10.21914/anziamj.v53i0.4582



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.