Further developments to the decomposition method for solving singular initial-value problems

Singular initial-value problems in second-order ordinary differential equations of Lane–Emden type are investigated. In this study, we shall introduce a further development in the Adomian decomposition method to overcome the difficulty at the singular point of non-homogeneous, linear and non-linear Lane–Emden-like equations; especially when the singularity appears on the right-hand side of this type of equations. A more general concept of this type of initial-value problem is introduced with various examples. The numerical results of these examples are compared with exact solutions when available.