Analytical Solutions of a Class of Generalised Lane-Emden Equations: Power Series Method Versus Adomian Decomposition Method

Abstract

In this paper, we obtain highly accurate analytical solutions of a class of strongly nonlinear Lane-Emden equations using a power series method and the Adomian decomposition method. The nonlinear term of the proposed problem involves the integer powers of a continuous real-valued function $\Lambda(y(x))$. In each of the proposed methods, a unified result is presented for the function $\Lambda(y(x))$. The particular cases of the trigonometric functions $\Lambda(y(x))=\tan y(x)$, $\sec y(x)$ and the hyperbolic functions $\Lambda(y(x))=\tanh y(x)$, $\sech y(x)$ are considered explicitly using the proposed methods. Lane-Emden equations involving the first integer powers of these trigonometric and hyperbolic functions are given as examples to illustrate the reliability, efficiency and accuracy of the proposed methods. Numerical comparisons of the results obtained show excellent agreements between the two methods, an indication that both methods are accurate, effective, reliable and convenient in solving singular strongly nonlinear ordinary differential equations with appropriate initial conditions.