Modified Adomian Decomposition Method for Solving Singular Differential Equation of Lane-Emden Type
About this article
Keywords:
Modified Adomian decomposition method; Numerical method; Singular differential equation; Lane-Emden equationAbstract
This paper presents a novel adaptation of the Adomian Decomposition Method specifically designed to solve Lane-Emden equations more effectively. Unlike traditional approaches, our method eliminates the need for integral operators, a common source of computational complexity in previous versions of the Adomian method. By relying exclusively on differential operators, we simplify the problem-solving process, allowing for direct differentiation without the approximation challenges associated with integral coefficients. This enhancement not only streamlines the computational workload but also significantly improves both the accuracy and efficiency of solutions to Lane-Emdenequations. We provide rigorous demonstrations of our method through various examples, illustrating marked advantages over conventional techniques in terms of convergence rates and computational simplicity. The unique contribution of this work lies in its potential to inspire further research and applications across diverse fields that utilize higher-order differential equations.
References
G. Adomian. Solving Frontier Problem of Physics: The Decomposition Method. Kluwer Academic Publishers, London, 1994.
Y. Q. Hasan. A new development to the Adamian decomposition for solving singular ivps of lane-emden type. United States of America Research Journal (USARJ), 2(3):9–13, 2014.
Y. Q. Hasan, O. J. Adekunle, and O. A. Afeez. Approximate analytical solution of a class of singular differential equation with Dirichlet boundary conditions by the modified Adamian decomposition method. IJMAO: Theory and Applications, 1:433–442, 2019.
G. Adomian and R. Rach. Modified Adamian polynomials. Mathematical and Computer Modelling, 24(11):39–46, 1996.
R. Rach. A new definition of the Adamian polynomials. Kubernetes, 37:910–955, 2008.
View more references (14)
A. M. Wazwaz. A reliable modification of Adamian decomposition method. Applied Mathematics and Computation, 102:77–86, 1999.
D. Zeidan, C. K. Chau, T. Lu, and W. Zheng. Mathematical studies of the solution of burgers’ equations by Adamian decomposition method. Mathematical Methods in the Applied Sciences, 43(5):2171–2188, 2020.
D. Zeidan, C. K. Chau, and T. Lu. On the characteristic Adamian decomposition method for the riemann problem. Mathematical Methods in the Applied Sciences, 44(10):8097–8112, 2021.
D. Zeidan, C. K. Chau, and T. Lu. On the development of Adamian decomposition method for solving pde systems with non-prescribed data. Computational and Applied Mathematics, 41, 2022.
Aishah Alsulami, Mariam Al-Mazmumy, Huda Bakodah, and Nawal Alzaid. On the application of Adamian decomposition method to special equations in physical sciences. American Journal of Computational Mathematics, 13:387–397, 2023.
N. M. Dabwan and Yahya Qaid Hasan. Solving higher order ordinary differential equations in physics: A new modified Adamian decomposition method approach to mhd flows, elastic beam, and sixth-order boundary value problem. An Najah University Journal for Research-A (Natural Sciences),40(1):11, 2025.
N. M. Dabwan and Yahya Qaid Hasan. Modified Adamian decomposition method for third-order emden-fowler equation. Global Journal of Mathematics Analysis, 12(1):7–19, 2026.
M. S. H. Chowdhury and I. Hashim. Solutions of a class of singular second-order ivps by homotopy-perturbation method. Physics Letters A,365:439–447, 2007.
J. I. Ramos. Linearization techniques for singular initial-value problems of ordinary differential equations. Applied Mathematics and Computation,161:525–542, 2005.
A. Yildirim and T. Ozis. Solutions of singular ivps of lane-emden type by the variational iteration method. Nonlinear Analysis, 70:2480–2484, 2009.
K. Parand, M. Dehghan, A. R. Rezaeia, and S. M. Ghaderi. An approximation algorithm for the solution of the nonlinear lane-emden type equations arising in astrophysics using hermite functions collocation method. Computer Physics Communications, 181:1096–1108, 2010.
A. M. Wazwaz. A new method for solving singular initial value problems in the second-order ordinary differential equations. Applied Mathematics and Computation, 128:45–57, 2002.
A. S. V. Ravi Kanth and K. Aruna. Solution of singular two-point boundary value problems using differential transformation method. Physics Letters A,372:4671–4673, 2008.
L. U. Junfeng. Variational iteration method for solving two-point boundary value problems. Journal of Computational and Applied Mathematics,207:92–95, 2007.