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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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The Lane - Emden Fractional Homogeneous Differential Equation

Journal of Applied Nonlinear Dynamics 6(2) (2017) 237--242 | DOI:10.5890/JAND.2017.06.008

Constantin Milici$^{1}$; Gheorghe Drăgănescu$^{2}$

$^{1}$ Department of Mathematics, Polytechnic University of Timisoara, Timi¸soara, RO 300222, Romania

$^{2}$ Department of Mechanics, Polytechnic University of Timisoara, Timi¸soara, RO 300222, Romania

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In this paper we introduce a nonlinear fractional differential equation of Lane-Emden type. We establish a solution which satisfies the Müntz-Szász theorem conditions in terms of power series. Particular solutions are established for different values of the parameters. A validation of our method is based on a case verified with the aid of a Maple program.


  1. [1]  Lane, J.H. (1870), 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, 2, 57-74.
  2. [2]  Emden, R. (1907), Gaskugeln, Teubner Verlag: Leipzig and Berlin.
  3. [3]  Chandrasekhar, S. (1967), Introduction to the Study of Stellar Structure, Dover: New York.
  4. [4]  Mirza, B.M. (2009), Approximate analytical solutions of the Lane - Emden equation for a self-gravitating isothermal gas sphere, Mon. Not. R. Astron. Soc., 395, 2288 - 2291.
  5. [5]  Hilfer, R. (2000), Fractional calculus and regular variation in thermodynamics. In: Applications of Fractional Calculus in Physics, World Scientific, River Edge, 429-463.
  6. [6]  Mechee, M.S. and Senu, N. (2012), Numerical Study of Fractional Differential Equation of Lane-Emde Type by Least Square Method, International Journal of Differential Equations and Applications, 11(3), 157-168.
  7. [7]  Mechee, M.S., Senu, N. (2012), Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation, Applied Mathematics, 3, 851-856.
  8. [8]  Parand, K., Dehghan, M., Rezaeia, A., Ghaderi, S. (2010), 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 (6) 1096-1108.
  9. [9]  Turkyilmazoglu, M. (2013), Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type, Appl. Math. Model., 37, 7539-7548.
  10. [10]  Bidaut-Véron, M.-F., and Hung, N. Q., Véron, L. (2014), Quasilinear Lane - Emden equations with absorption and measure data, J. Math. Pures et Appl., 102 (2), 315-337.
  11. [11]  El-Nabulsi, R.A. (2013), Non-standard fractional Lagrangians, Nonlinear Dynamics, 74, 381-394.
  12. [12]  Parand, K., Rezaei, A.R., and Taghavi, A. (2010), Lagrangian method for solving Lane‘mden type equation arising in astrophysics on semi-infinite domains, Acta Astronautica, 67 (7-8), 673-680.
  13. [13]  Dehghan, M. and Shakeri, F. (2008), Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy, 13(1), 53-59.
  14. [14]  He, J.-H. (1999), Variational iteration method - a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34, 699-708.
  15. [15]  He, J.-H. (2000), A Review of Some New Recently Developed nonlinear Analytical Techniques, International Journal Nonlinear Science and Numerical Simulation, 1, 51-70.
  16. [16]  Milici, A. and Drăgănescu, G. (2014), A Method for Solve the Nonlinear Fractional Differential Equations, Lambert Academic Publishing: Saarbrücken.
  17. [17]  Milici, A. and Drăgănescu, G. (2015), New Methods and Problems in Fractional Calculus, Lambert Academic Publishing: Saarbrücken.
  18. [18]  Farina, A.(2007), On the classification of solutions of the Lane-Emden equation on unbounded domains of RN, J. Math. Pures Appl., 87(5) 537-561.
  19. [19]  Bozhkov, Y.(2015), The group classification of Lane - Emden systems, Journal of Mathematical Analysis and Applications, 426(1), 89-104.
  20. [20]  Ortigueira, M. D. (2010), On the Fractional Linear Scale Invariant Systems, IEEE Trans. On Signal Process- ing, 58(12), 6406-6410.
  21. [21]  Rudin, W. (1966), Complex Analysis , McGraw - Hill: New York.