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

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.