# Approximate Analytical Solution of Fractional Lane-Emden Equation by Mittag-Leffler Function Method

## Authors

• Richard Olu Awonusika Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B. 001, Akungba Akoko, Ondo State, Nigeria
• Oluwaseun Akinlo Mogbojuri Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B. 001, Akungba Akoko, Ondo State, Nigeria

## Keywords:

Fractional differential equation, Lane-Emden differential equation, Mittag-Leffler function, Elementary functions, Series solution, Eigenfunction expansion method

## Abstract

The classical Lane-Emden differential equation, a nonlinear second-order differential equation, models the structure of an isothermal gas sphere in equilibrium under its own gravitation. In this paper, the Mittag-Leffler function expansion method is used to solve a class of fractional LaneEmden differential equation. In the proposed differential equation, the polytropic term f(y(x)) = ym(x) (where m = 0,1,2,... is the polytropic index; 0 < x <=1) is replaced with a linear combination f(y(x)) = a0 + a1y(x) + a2y2(x) + ··· + amym(x) + ··· + aNyN(x),0 <=m <=N,N <= N_0. Explicit solutions of the fractional equation, when f(y) are elementary functions are presented. In particular, we consider the special cases of the trigonometric, hyperbolic and exponential functions. Several examples are given to illustrate the method. Comparison of the Mittag-Leffler function method with other methods indicates that the method gives accurate and reliable approximate solutions of the fractional Lane-Emden differential equation.

Dimensions

R.Garrappa, E.Kaslik &M.Popolizio, “Evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial”, Mathematics 7, (2019) 407. DOI: https://doi.org/10.3390/math7050407

G. Jumarie, “A Fokker-Planck equation of fractional order with respect to time”, Journal of Math. Physics 33 (1992) 3536. DOI: https://doi.org/10.1063/1.529903

G. Jumarie, “Fractional Fokker-Planck equation, solutions and applications”, Physical Review, 63 (2001) 1. DOI: https://doi.org/10.1103/PhysRevE.63.046118

G. Jumarie, “Schr¨odinger equation for quantum-fractal space-time of order n via the complex-valued fractional Brownian motion”, Intern. Y. of Modern Physics A 16 (2001) 5061. DOI: https://doi.org/10.1142/S0217751X01005468

A.A.Kilbas, Theory and applications of fractional differential equations, Elsevier, (2006). DOI: https://doi.org/10.3182/20060719-3-PT-4902.00008

F. Mainardi &R.Gorenflo, “Time-fractional derivatives in relaxation processes: A tutorial survey”, Fractional Calculus and Applied Analysis 10 (2007) 269.

M. D. Ortigueira, Fractional calculus for scientists and engineers Springer (2011). DOI: https://doi.org/10.1007/978-94-007-0747-4

I. Podlubny, Fractional differential equations, Academic Press, San Diego, (1999).

M. ?Zecov´a & J. Terp´ak, “Heat conduction modeling by using fractionalorder derivatives”, Applied Mathematics and Computation 257 (2015) 365. DOI: https://doi.org/10.1016/j.amc.2014.12.136

K. Diethelm, The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, Springer-Verlag Berlin Heidelberg (2010).

A. M. Wazwaz, “A new algorithm for solving differential equations of Lane-Emden type”, Appl. Math. Comput. 118 (2001) 287. DOI: https://doi.org/10.1016/S0096-3003(99)00223-4

S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover, New York, (1967).

O.U.Richardson, TheEmissionofElectricity from HotBodies, Longman, Green and Co., London, New York, (1921).

C. Mohan&A.R.Al-Bayaty, “Powerseries solutions of the Lane-Emden equation”, Astrophysics and Space Science 73 (1980) 227. DOI: https://doi.org/10.1007/BF00642378

J. I. Ramos, “Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method”, Chaos, Solitons & Fractals 38 (2008) 400. DOI: https://doi.org/10.1016/j.chaos.2006.11.018

S. K. Vanani & A. Aminataei, “On the numerical solutions of differential equations of Lane-Emden type”, Computers and Mathematics with Applications 59 (2010) 2815. DOI: https://doi.org/10.1016/j.camwa.2010.01.052

M. O. Ogunniran, O. A. Tayo, Y. Haruna & A. F. Adebisi, “Linear stability analysis of Runge-Kutta methods for singular Lane-Emden equations”, Journal of the Nigerian Society of Physical Sciences 2 (2020) 134. DOI: https://doi.org/10.46481/jnsps.2020.87

E. O. Adeyefa &O.S.Esan, “Exponentially fitted Chebyshev based algorithm as second order initial value solver”, Journal of the Nigerian Society of Physical Sciences 2 (2020) 51. DOI: https://doi.org/10.46481/jnsps.2020.45

S. E. Fadugba, S.N. Ogunyebi & B.O. Falodun, “An examination of a second order numerical method for solving initial value problems”, Journal of the Nigerian Society of Physical Sciences 2 (2020) 120. DOI: https://doi.org/10.46481/jnsps.2020.92

M. S. Mechee & N. Senu, “Numerical study of fractional differential equations of Lane-Emden type by method of collocation”, Applied Mathematics 3 (2012) 851. DOI: https://doi.org/10.4236/am.2012.38126

A. Akg¨ul, M. Inc, E. Karatas & D. Baleanu, “Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique”, Advances in Difference Equations 2015 (2015) 220. DOI: https://doi.org/10.1186/s13662-015-0558-8

A. Saadatmandi, A. Ghasemi-Nasrabady & A. Eftekhari, “Numerical study of singular fractional Lane-Emden type equations arising in astrophysics”, J. Astrophys. Astr. (2019) 40 27. DOI: https://doi.org/10.1007/s12036-019-9587-0

P. K. Sahu & B. Mallick, “Approximate solution of fractional order Lane–Emden type differential equation by orthonormal Bernoulli’s polynomials”, Int. J. Appl. Comput. Math (2019) 5 89. DOI: https://doi.org/10.1007/s40819-019-0677-0

M. I. Nouh & E. A.-B. Abdel-Salam, “Approximate Solution to the Fractional Lane–Emden Type Equations”, Iran J. Sci. Tech. Trans. Sci. 42, (2018) 2199. DOI: https://doi.org/10.1007/s40995-017-0246-5

A. M. Malik & O. H .Mohammed, “Two efficient methods for solving fractional Lane–Emden equations with conformable fractional derivative”, Journal of the Egyptian Mathematical Society (2020) 28. DOI: https://doi.org/10.1186/s42787-020-00099-z

B. C?aruntu, C. Bota, M. L?ap?adat & M. S. Pas¸ca, “Polynomial least squares method for fractional Lane–Emden equations”, Symmetry 11 (2019) 479. DOI: https://doi.org/10.3390/sym11040479

J. Davila, L. Dupaigne & J. Wei, “On the fractional Lane-Emden equation”, Trans. Am. Math. Soc. 369 (2017) 6087. DOI: https://doi.org/10.1090/tran/6872

C. Milici, G. Dr?ag?anescu & J.T. Machado, “Introduction to fractional differential equations”, Nonlinear Systems and Complexity, 25 (2019). DOI: https://doi.org/10.1007/978-3-030-00895-6

U. Saeed, “Haar Adomian method for the solution of fractional nonlinear Lane-Emden type equations arising in astrophysics”, Taiwanese Journal of Mathematics 21 (2017) 1175. DOI: https://doi.org/10.11650/tjm/7969

O. A. Uwaheren, A. F. Adebisi & O. A. Taiwo, “Perturbed collocation method for solving singular multi-order fractional differential equations of Lane-Emden type”, Journal of the Nigerian Society of Physical Sciences 2 (2020) 141. DOI: https://doi.org/10.46481/jnsps.2020.69

A. A. M. Arafa & S. Z. Rida, A. A. Mohammadein, H. M. Ali, “Solving nonlinear fractional differential equation by generalized Mittag-Leffler function method”, Communications in Theoretical Physics 59 (2013) 661. DOI: https://doi.org/10.1088/0253-6102/59/6/01

A. Atangana & A. Secer, “A note on fractional order derivatives and table of fractional derivatives of some special functions”, Abstract and Applied Analysis 2013 (2013) 1. DOI: https://doi.org/10.1155/2013/279681

M.Caputo &M.Fabrizio, “A new definition of fractional derivative without singular kernel”, Progr. Fract. Differ. Appl. 1 (2015) 73.

S. Das, “Functional Fractional Calculus”, Springer (2011). DOI: https://doi.org/10.1007/978-3-642-20545-3

M. Davison & C. Essex, “Fractional differential equations and initial value problems”, The Mathematical Scientist 23 (1998) 108.

S.E.Fadugba, “Solution of fractional order equations in the domain of the Mellin transform”, Journal of the Nigerian Society of Physical Sciences 1 (2019) 138. DOI: https://doi.org/10.46481/jnsps.2019.31

R. Herrmann, Fractional calculus, an introduction for Phycists, World Scientific, (2011). DOI: https://doi.org/10.1142/8072

J. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results”, Computers and Mathematics with Applications 51 (2006) 1367. DOI: https://doi.org/10.1016/j.camwa.2006.02.001

J. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions”, Applied Mathematics Letters 22 (2009) 378. DOI: https://doi.org/10.1016/j.aml.2008.06.003

E. C. de Oliveira & J. A. T. Machado, “A review of definitions for fractional derivatives and integral”, Mathematical Problems in Engineering 2014 (2014) 1. DOI: https://doi.org/10.1155/2014/238459

F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press (2010). DOI: https://doi.org/10.1142/p614

H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, (1962). 2022-05-29

## How to Cite

Awonusika, R. O., & Mogbojuri, . O. A. (2022). Approximate Analytical Solution of Fractional Lane-Emden Equation by Mittag-Leffler Function Method. Journal of the Nigerian Society of Physical Sciences, 4(2), 265–280. https://doi.org/10.46481/jnsps.2022.687

## Section

Original Research  