AM  Vol.3 No.8 , August 2012
Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation
ABSTRACT
Lane-Emden differential equations of order fractional has been studied.Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analytic methods has been introduced.

Cite this paper
M. Mechee and N. Senu, "Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation," Applied Mathematics, Vol. 3 No. 8, 2012, pp. 851-856. doi: 10.4236/am.2012.38126.
References
[1]   M. Yigider, “The Numerical Method for Solving Differential Equations of Lane-Emden Type by Pade Approximation,” Discrete Dynamics in Nature and Society, Vol. 2011, 2011, Article ID: 479396. doi:10.1155/2011/479396

[2]   R. P. Agarwal D. O. Regan and V. Lakshmikanthamr, “Quadratic Forms and Nonlinear Non-Resonant Singular Second Order Boundary Value Problems of Limit Circle Type,” Zeitschrift fur Analysis und ihre Anwendungen, Vol. 20, 2001, pp. 727-737.

[3]   R. P. Agarwal and D. O. Regan, “Existence Theory for Single and Multiple Solutions to Singular Positone Boundary Value Problems,” Journal of Differential Equations, Vol. 175, No. 2, 2001, pp. 393-414. doi:10.1006/jdeq.2001.3975

[4]   R. P. Agarwal and D. O. Regan, “Existence Theory for Singular Initial and Boundary Value Problems: A Fixed Point Approach,” Applicable Analysis: An International Journal, Vol. 81, No. 2, 2002, pp. 391-434. doi:10.1080/0003681021000022023

[5]   M. M. Coclite and G. Palmieri, “On a Singular Nonlinear Dirichlet Problem,” Communications in Partial Differential Equations, Vol. 14, No. 10, 1989, pp. 1315-1327. doi:10.1080/03605308908820656

[6]   J. H. 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,” The American Journal of Science and Arts, Vol. 50, 1870, pp. 57-74.

[7]   R. Emden, “Gaskugeln,” Teubner, Leipzig and Berlin, 1907.

[8]   S. Chandrasekharr, “Introduction to the Study of Stellar Structure,” Dover, New York, 1967.

[9]   M. Chowdhury and I. Hashim, “Solutions of Emden Fowler Equations by Homotopy-Perturbation Method,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 1, 2009, pp. 104-115. doi:10.1016/j.nonrwa.2007.08.017

[10]   A. Yildirim and T. ?zi?, “Solutions of Singular IVPs of Lane-Emden Type by the Variational Iteration Method,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, No. 6, 2009, pp. 2480-2484. doi:10.1016/j.na.2008.03.012

[11]   K. Parand and A. Pirkhedri, “Sinc-Collocation Method for Solving Astrophysics Equations,” New Astronomy, Vol. 15, No. 6, 2010, pp. 533-537. doi:10.1016/j.newast.2010.01.001

[12]   E. Momoniat and C. Harley, “An Implicit Series Solution for a Boundary Value Problem Modelling a Thermal Explosion,” Mathematical and Computer Modelling, Vol. 53, No. 1-2, 2011, pp. 249-260. doi:10.1016/j.mcm.2010.08.013

[13]   K. Parand, M. Dehghan, A. Rezaeia and S. 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, Vol. 181, No. 6, 2010, pp. 1096-1108. doi:10.1016/j.cpc.2010.02.018

[14]   H. Adibi and A. Rismani, “On Using a Modified Legendre-Spectral Method for Solving Singular IVPs of Lane-Emden Type,” Computers & Mathematics with Applications, Vol. 60, No. 7, 2010, pp. 2126-2130. doi:10.1016/j.camwa.2010.07.056

[15]   A. H. Bhrawy and A. S. Alofi, “A JacobiGauss Collocation Method for Solving Nonlinear LaneEmden Type Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 1, 2012, pp. 62-70.

[16]   R. P. Agarwal and D. O. Reganr, “Singular Boundary Value Problems for Superlinear Second Order Ordinary and Delay Differential Equations,” Journal of Differential Equations, Vol. 130, No. 2, 1996, pp. 333-335. doi:10.1006/jdeq.1996.0147

[17]   R. Lewandowski and B. Chorazyczewski, “Identification of the Parameters of the KelvinVoigt and the Maxwell Fractional Models, Used to Modeling of Viscoelastic Dampers,” Computers and Structures, Vol. 88, No. 1-2, 2010, pp. 1-17. doi:10.1016/j.compstruc.2009.09.001

[18]   F. Yu, “Integrable Coupling System of Fractional Soliton Equation Hierarchy,” Physics Letters A, Vol. 373, No. 41, 2009, pp. 3730-3733. doi:10.1016/j.physleta.2009.08.017

[19]   K. Diethelm and N. Ford, “Analysis of Fractional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 265, No. 2, 2002, pp. 229-248. doi:10.1006/jmaa.2000.7194

[20]   R. W. Ibrahim and S. Momanir, “On the Existence and Uniqueness of Solutions of a Class of Fractional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 334, No. 1, 2007, pp. 1-10. doi:10.1016/j.jmaa.2006.12.036

[21]   S. M. Momani and R. W. Ibrahim, “On a Fractional Integral Equation of Periodic Functions Involving Weyl-Riesz Operator in Banach Algebras,” Journal of Mathematical Analysis and Applications, Vol. 339, No. 2, 2008, pp. 1210-1219. doi:10.1016/j.jmaa.2007.08.001

[22]   B. Bonilla, M. Rivero and J. J. Trujillor, “On Systems of Linear Fractional Differential Equations with Constant Coefficients,” Applied Mathematics and Computation, Vol. 187, No. 1, 2007, pp. 68-78. doi:10.1016/j.amc.2006.08.104

[23]   I. Podlubny, “Fractional Differential Equations,” Academic Press, London, 1999.

[24]   S. Zhangr, “The Existence of a Positive Solution for a Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 252, No. 2, 2000, pp. 804-812. doi:10.1006/jmaa.2000.7123

[25]   R. W. Ibrahim and M. Darusr, “Subordination and Superordination for Analytic Functions Involving Fractional Integral Operator,” Complex Variables and Elliptic Equations, Vol. 53, No. 11, 2008, pp. 1021-1031. doi:10.1080/17476930802429131

[26]   R. W. Ibrahim and M. Darusr, “Subordination and Superordination for Univalent Solutions for Fractional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 345, No. 2, 2008, pp. 871-879. doi:10.1016/j.jmaa.2008.05.017

 
 
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