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

Richard Olu Awonusika; Oluwaseun Akinlo  Mogbojuri

doi:10.46481/jnsps.2022.687

Authors

Richard Olu Awonusika
richard.awonusika@aaua.edu.ng
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

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