Solution of a Subclass of Singular Second Order Differential Equation of Lane Emden type by Taylor Series Method

In this paper, a subclass of second order differential equation of Lane Emden type followed by imposed initial or boundary condition is identified for solving by Taylor's series method. Such types of problems come into existence while modeling upon some of peer physical phenomenon. The proposed method eventually produces an analytic solution in the form of a polynomial function. Again some of the problems available in literature are also considered and tested for solution to justify the suitability and viability of the method. Keywords: Variation iteration method, boundary value problems, analytic solution, Lane-Emden equation, He's Polynomial, Taylor series

Adomian Decomposition Method for Solving Higher Order Boundary Value Problems

In this paper, we present efficient numerical algorithms for the approximate solution of linear and non-linear higher order boundary value problems. Algorithms are, based on Adomian decomposition. Also, the Laplace Transformation with Adomian decomposition technique is proposed to solve the problems when Adomian series diverges. Three examples are given to illustrate the performance of each technique. Keyword: Higher order Singular boundary value problems, Adomian decomposition techniques, Laplace transformations

Solution of a Subclass of Lane-Emden Differential Equation by Variational Iteration Method

In this paper we apply He's variational iteration method to find out an appropriate solution to a class of singular differential equation under imposed conditions by introducing and inducting in a polynomial pro satisfying the given subject to conditions at the outset as selective function to the solution extracting process. As for as application part is concerned, Illustrative examples from the available literature when treated all over reveal and out show that the solution deduced by proposed method is exact and again polynomial. Overall, a successful produce of exact solutions by proposed process itself justify the effectiveness and efficiency of the method so very much. Keywords: He's variational iteration method, Lane-Emden differential equation, exact solution, polynomial, Lagrange multiplier

Solution of a broad Class Singular Boundary Value Problem by Variational Iteration Method

In this paper an iterative method for finding a compatible solution to a class of singular second order differential equation of prescribed boundary values often observed common is considered by constructing a successive sequence of correction functional via variational theory. The analytical convergence of such iteratively generated sequential scheme is analyzed explicitly and duly discussed.  Interestingly, the proposed method when applied on, over hither to widely quote numerical problems turns out to be quite encouraging and renders appropriate solution. May sometimes by this method the limiting value of functional sequence happens to be an exact solution too. Keywords; singular problem, Variational iteration method, Convergence, sequence, smooth function Lagrange multiplier, linearization, discretization, transformation

A Successive Linearization Method Approach to Solve Lane-Emden Type of Equations

We propose a new application of the successive linearization method for solving singular initial and boundary value problems of Lane-Emden type. To demonstrate the reliability of the proposed method, a comparison is made with results from existing methods in the literature and with exact analytical solutions. It was found that the method is easy to implement, yields accurate results, and performs better than some numerical methods

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

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