Analytical solution of a class of Lane–Emden equations: Adomian decomposition method

Abstract

In this paper, the analytical solution of a class of Lane–Emden equation is considered using the Adomian decomposition method. The nonlinear term of the proposed equation is given by normalised Jacobi functions ${\mathscr {P}}_{\gamma }^{(\alpha , \beta )}(y(x))$ ($\gamma \in {\mathbb {C}};\alpha ,\beta >-1$). The Adomian polynomials for the Jacobi functions ${\mathscr {P}}_{\gamma }^{(\alpha , \beta )}(y(x))$ are constructed and the power series solutions are presented. For the special cases $\gamma =0,1$; closed form solutions are obtained. Interestingly, the functions ${\mathscr {P}}_{\gamma }^{(\alpha , \beta )}$ are the spherical functions (normalised eigenfunctions) of the Laplacian on rank one symmetric spaces. In order to present several examples of Lane-Emden type equations and their solutions, we specialise to the spherical functions on the real hyperbolic space and sphere ($\alpha =\beta =(n-2)/2$), the complex hyperbolic space ($\alpha =n-1,\beta =0$), the quaternionic hyperbolic space ($\alpha =2n-1,\beta =1$), and the Cayley hyperbolic plane ($\alpha =7,\beta =3$), as well as their corresponding projective spaces. Comparisons of the results from the present method with other published results show that the Adomian decomposition method gives accurate and reliable approximate solutions of Lane–Emden equations involving Jacobi functions.