Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 19 (2024), 361 -- 383

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

ADVANCE IN THE ADOMIAN DECOMPOSITION METHOD FOR SOLVING NEW EMDEN-FOWLER TYPE EQUATIONS

Zainab Ali Abdu AL-Rabahi and Yahya Qaid Hasan

Abstract. In this manuscript, we developed the technique of the Adomian method for solving many Lane-Emden-Fowler type equations. The use of the Adomian Decomposition method to effectively solve these newly developed singular 4th and 5th order equations. The efficacy of our approach is further validated by analyzing various fourth and fifth orders Emden-Fowler type examples encompassing nonlinear cases.

2020 Mathematics Subject Classification: 65L05; 65L11;
Keywords: Emden–Fowler Type Equations; Adomian Method; Adomian Polynomials; Singular Point; Initial Value Problem

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References

  1. G. Adomian, Solving frontier problems of physics: the decomposition method, Kluwer Academic, Dordrecht. 1994. MR1282283. Zbl 0802.65122.

  2. G. Adomian, R. Rach, Inversion of nonlinear stochastic operators, J. Math. Anal. Appl., 91 (1983), 39-46. MR0688530. Zbl 0504.60066.

  3. G. Adomian, R. C. Rach, and R. E. Meyers, Numerical integration, analytic continuation, and decomposition, Appl. Math. Comput. 88 (1997), 95-116. MR1479268. Zbl 0905.65079.

  4. G. Adomian, R. Rach, N. T. Shawagfeh, On the Analytic Solution of the Lane-Emden Equation, Found. Phys. Lett., 8 (1995), 161-181.

  5. Z.A. AL-Rabahi, Y.Q. Hasan, The Solution of Higher-Order Ordinary Differential Equation with Boundary by a New Strategy of Adomian Method, Advances in Mathematics: Scientific Journal 9 (3) (2020), 991 -999.

  6. J. Biazar, F. Goldoust, Wavelet-galerkin method and some numerical method for Lane-Emden type differential equation, Am. J. Appl. Math. Stat. 1 (2013), 83-86.

  7. S. Chandrasekhar, An Introduction to the Study of Stellar Structure, DoverPublications, New York, 1967.

  8. N. M. Dabwan, Y. Q. Hasan, An efficient method to find approximate solutions for Emden-Fowler equations of nth order, Asian Journal of Probability and statistics, 8 (2020), 9-27.

  9. N. M. Dabwan, Y. Q. Hasan, Adomian method for solving Emden-Fowler Equation of higher order, Advances in Mathematics: Scientific Journal. 9 (2020), 231-240. Zbl 1476.34053.

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

  11. J. S. Duan, Recurrence triangle for Adomian polynomials, Appl. Math. Comput. 216 (2010), 1235-1241. MR2607232. Zbl 1190.65031.

  12. J. S. Duan, An efficient algorithm for the multivariable Adomian polynomials, Appl. Math. Comput., 217 (2010), 2456-2467. MR2733688. Zbl 1204.65022.

  13. J. S. Duan, Convenient analytic recurrence algorithms for the Adomian polynomials, Appl. Math. Comput., 217 (2011), 6337-6348. MR2773378. Zbl 1214.65064.

  14. R. Emden, Gaskugeln. Anwendung der mechanischen Wärmetheorie auf kosmologische und meteorologische Probleme(German Edition), Druck and Verlag Von B. G. Teubner, Leipzig, 1907.

  15. R. H. Fowler, Further studies of Emden’s and similar differential equations, Q. J. Math. 2(1) (1931), 259-288. Zbl 0003.23502.

  16. Y. Q. Hasan, A. A. Olalekan, Solving Emden Fowler type equations by adjusted Adomian decomposition strategy, International Journal of Innovative Science and Research Technology. 3 (2018), 831-850.

  17. Y. Q. Hasan, L. M, Zhu, Modified Adomian decomposition method for singular initial value problems in the second-order ordinary differential equations, Surveys in Mathematics and its Applications 3(2008), 183-193. MR2470208. Zbl 1176.65078.

  18. J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114 (2000), 115-123. MR1778658. Zbl 1027.34009.

  19. C. M. Khalique, F. M. Mahomed, B. Muatjetjeja, Lagrangian formulation of a generalized Lane–Emden equation and double reduction, J. Nonlinear Math. Phys., 15 (2008), 152-161. MR2430716. Zbl 1169.34033.

  20. H. J. Lane, 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, Am. J. Sci. (Series 2), 50 (1870), 57-74.

  21. M. K. Mak, Ch. S. Leung, T. Harko, Solving the Nonlinear Biharmonic Equation by the Laplace-Adomian and Adomian Decomposition Methods, Surveys in Mathematics and its Applications, 13 (2018), 183-213. MR3867961. Zbl 1424.35150.

  22. E. Momoniat, C. Harley, Approximate Implicit Solution of a Lane–Emden Equation, New Astronomy, 11 (2006), 520-526.

  23. B. Muatjetjeja, C. M. Khanlique, Exact solutions of the generalized Lane–Emden equations of the first and second Kind, Pramana, 77 (2011), 545-554.

  24. K. Parand, M. Dehghan, A. R. Rezaeia, S. M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun. 181 (2010), 1096-1108. MR2644949. Zbl 1216.65098.

  25. K. Parand, M. Shahini, M. Dehghan, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, J. Comput. Phys. 228 (2009), 8830-8840. MR2558779. Zbl 1177.65100.

  26. R. Rach, A bibliography of the theory and applications of the Adomian decomposition method, 1961–2011. (English) Kybernetes. 41 (2012), 1087-1148. MR3050085. Zbl 1511.65112.

  27. O. W. Richardson, The emission of electricity from hot bodies, Longmans Green and Co., London, 1921.

  28. S. Somali and G. Gokmen, Adomian Decomposition Method for Nonlinear Sturm- Liouville Problems, Surveys in Mathematics and its Applications. 2 (2007), 11-20. MR2350066. Zbl 1132.34333.

  29. A. M. Wazwaz, A New Algorithm for Solving Differential Equations of Lane-Emden Type, Appl. Math. Comput., 118 (2001), 287-310. MR1812960. Zbl 1023.65067.

  30. A. M. Wazwaz, Adomian Decomposition Method for a Reliable Treatment of the Emden-Fowler Equation, Appl. Math. Comput. 161 (2005), 543-560. MR2112423. Zbl 1061.65064.

  31. A. M. Wazwaz, Analytical Solution for the Time-dependent Emden-Fowler Type of Equations by Adomian Decomposition Method, Appl. Math. Comput. 166 (2005), 638-651. MR2150495. Zbl 1073.65105.

  32. A. M. Wazwaz, R. Rach, L. Bougoffa, J. S. Duan, Solving the Lane-Emden–Fowler-type equations of higher orders by the Adomian decomposition method, CMES-Comput. Model. Eng. Sci. 100 (2014), 507-529. MR3308399. Zbl 1356.65214.

  33. A. M. Wazwaz, R. Rach, J. S. Duan, Solving New Fourth–Order Emden–Fowler-Type Equations by the Adomian Decomposition Method, International Journal for Computational Methods in Engineering Science and Mechanics. 16 (2015), 121-131. MR3339358. Zbl 07871406.

  34. A. Yildirim, T. Ozis, Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method, Phys. Lett. A, 369 (2007), 70-76. Zbl 1209.65120.



Zainab Ali Abdu AL-Rabahi (corresponding author)
Department of Mathematics, Faculty of Education,
Aden University, Aden, Yemen.
e-mail: zainabaliabdu99@gmail.com

Yahya Qaid Hasan
Department of Mathematics, Faculty of Education and Sciences,
University of Saba Region, Marib, Yemen
e-mail: ya217hya@gmail.com

https://www.utgjiu.ro/math/sma