# Combining Laplace transform and Variational iteration method for solving singular IVPs and BVPs of Lane–Emden type equation

## DOI:

https://doi.org/10.31185/wjcms.244## Keywords:

Lane-Emden equation, Laplace Transform method, Variational iteration method## Abstract

This study uses the combined Laplace transform and variational iteration method to analytically solve initial value problems (IVPs) and boundary value problems (BVPs) of Lane- Emden- equation with singular behaviour at . Initially, this method requires calculating the lagrange multiplayer, and after its value was calculated, it was found to be equal , this method is applying on several problems and the results were improved using Padé approximations, the results were compared with the exact solution by calculating the absolute error value between them.

## References

. Lane JH, On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, The American Journal of Science and Arts, 2nd series 50 (1870), 57-74.

. Thomson W, Collected Papers, Vol. 5, p. 266. Cambrige University Press, 1991.

. Emden R, Gaskugeln, Anwendungen der mechanischen Warmen-theorie auf Kosmologie und meteorologische Probleme, Leipzig, Teubner, 1907.

. S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover Publications, New York, NY, USA, 1967.

. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York, NY, USA, 1962.

. O. U. Richardson, The Emission of Electricity of Hot Bodies, London, UK, Longmans Green and Company edition, 1921

. P. L. Chambré, “On the solution of the poisson-boltzmann equation with application to the theory of thermal explosions,” The Journal of Chemical Physics, Vol. 20, no. 11, pp. 1795–1797, 1952

. M. Dehghan and F. Shakeri, “Approximate solution of a differential equation arising in astrophysics using the variational iteration method,” New Astronomy, Vol. 13, no. 1, pp. 53–59, 2008

. J. H. Lane, “On theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its internal heat and depending on the laws of gases known to terrestrial experiment,” The American Journal of Science and Arts, Vol. 2, pp. 57–74, 1870

. R. Emden, Gaskugeln, Teubner, Berlin, Germany, 1907.

. N. T. Shawagfeh, “Nonperturbative approximate solution for Lane-Emden equation,” Journal of Mathematical Physics, Vol. 34, no. 9, pp. 4364–4369, 1993.

. G. Adomian, R. Rach, and N. T. Shawagfeh, “On the analytic solution of the lane-emden equation,” Foundations of Physics Letters, Vol. 8, no. 2, pp. 161–181, 1995.

. Liu, F. K. (1996). Polytropic gas spheres: An approximate analytic solution of the Lane-Emden equation. Monthly Notices of the Royal Astronomical Society, 281(4), 1197-1205

. Datta, B. K. (1996). RETRACTED ARTICLE: Analytic solution to the Lane-Emden equation. Il Nuovo Cimento B (1971-1996), 111(11), 1385-1388.

. Goenner, H., & Havas, P. (2000). Exact solutions of the generalized Lane–Emden equation. Journal of Mathematical Physics, 41(10), 7029-7042

. A.M. Wazwaz, A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput. 118 (2001), 287-310.

. A. Yildirim and T. ¨Ozis, Solutions of Singular IVPs of Lane–Emden Type by Homotopy Perturbation Method, Phys. Lett. A, Vol. 369, pp. 70–76,2007.

. J. Ramos, Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method, Chaos Soliton Fract. 38 (2) (2008), 400-408.

. Iqbal, S.; Javed, A. Application of Optimal Homotopy Asymptotic Method for the Analytic Solution of Singular Lane–Emden Type Equation. Appl. Math. Comput. 2011, 217, 7753–7761.

. Abu Arqub, O., El-Ajou, A., Bataineh, A. S. & Hashim, I. A representation of the exact solution of generalized lane-emden equations using a new analytical method. Abstr. Appl. Anal. 2013, e378593 (2013).

. Z. Šmarda and Y khan An effective computational method for resolving singular initial value issues for equations of the Lane-Emden type J. Comput. Appl. Math(2015).

. S. Gh. Hosseini and S. Abbasbandy, Solution of Lane-Emden Type Equations by Combination of the Spectral Method and Adomian Decomposition Method, Mathematical Problems in Engineering, Volume 2015, Article ID 534754, 10 pages. http://dx.doi.org/10.1155/2015/534754.

. Rajnee Tripathi and Hradyesh Kumar Mishra, Homotopy perturbation method with Laplace Transform (LT‑HPM) for solving Lane–Emden type differential equations (LETDEs), Tripathi and Mishra Springer Plus 5, (2016), 1859. DOI 10.1186/s40064-016-3487-4.

. Waleed Al-Hayani, Laheeb Alzubaidy and Ahmed Entesar, Solutions of Singular IVP’s of Lane-Emden type by Homotopy analysis method with Genetic Algorithm, Appl. Math. Inf. Sci. 11, No. 2, 1-10 (2017). http://dx.doi.org/10.18576/amis/paper.

. K. Parand, S. Hashemi, RBF-DQ method for solving non-linear differential equations of Lane-Emden type, Ain Shams Engineering Journal 9 (2018), 615–629. http://dx.doi.org/10.1016/j.asej.2016.03.010.

. Izadi, M. A discontinuous finite element approximation to singular Lane–Emden-type equations. Appl. Math. Comput. 2021, 401, 126115.

. Saadeh, R., Burqan, A., & El-Ajou, A. (2022). Reliable solutions to fractional Lane-Emden equations via Laplace transform and residual error function. Alexandria Engineering Journal, 61(12), 10551-10562.

. AWONUSIKA, R. O., & OLATUNJI, P. O. (2022). Analytical and numerical solutions of a class of generalised Lane-Emden equations. Journal of the Korean Society for Industrial and Applied Mathematics, 26(4), 185-223

. Parand, K., Aghaei, A. A., Kiani, S., Zadeh, T. I., & Khosravi, Z. (2023). A neural network approach for solving nonlinear differential equations of Lane–Emden type. Engineering with Computers, 1-17

. Awonusika, R. O. (2024). Analytical solution of a class of Lane–Emden equations: Adomian decomposition method. The Journal of Analysis, 32(2), 1009-1056.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2024 Mohamed H. Jassim, Waleed Mohammed Al-Hayani

This work is licensed under a Creative Commons Attribution 4.0 International License.

## How to Cite

*WJCMS*, vol. 3, no. 2, pp. 8–18, Jun. 2024, doi: 10.31185/wjcms.244.