**1. Introduction**

In the most general form, Lane-Emden type equations are given as

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$$\mathbf{y}'' + \frac{\gamma}{\mathbf{x}} \mathbf{y}' + f(\mathbf{x}, \mathbf{y}) = \mathbf{g}(\mathbf{x}), \quad \mathbf{x} \in [0, \infty), \quad \mathbf{y}(0) = \alpha\_0, \ y'(0) = \beta\_0 \tag{1}$$

where the prime denotes differentiation with respect to *x*, *f*(*x, y*) is a non-linear function, *g*(*x*) is a prescribed function and *γ*, *α*0, *β*<sup>0</sup> are known constants. In recent years, problems described by this class of differential equations have been widely investigated by many researchers because of their applications in astronomy, mathematical biology, mathematical physics, non-Newtonian fluid mechanics, and other areas of science and engineering. From a solution method viewpoint, it has been observed that, owing to the singularity at *x* = 0, Lane-Emden type equations are not trivial to solve. For this reason, the equations are normally used as benchmark equations for testing the effectiveness and robustness of new analytical and numerical methods of solution.

Analytical approaches that have recently been used in solving the Lane-Emden equations are mostly based on truncated series expansions. Examples include the Adomian decomposition method [1–3], differential transformation method [4, 5], Laplace transform [6, 7], homotopy analysis method [8–10], power series expansions [11–14] and variational iteration method [15– 17]. Being power series based, the above methods have a small region of convergence and are not suitable for generating solutions in very large values of *x*. For this reason, most analytical approaches have only reported solutions of Lane-Emden type equations on small interval 0,1 on *x* axis. Despite this limitation, analytical approaches have been found to be desirable because they easily overcome the difficulty caused by the singularity at *x* = 0.

To overcome the limitations of analytical solution methods, several numerical approaches have been proposed for the solution of Lane-Emden type equations. Numerical methods based on spectral collocation have been found to be particularly effective. Collocation methods that have been reported recently for the solution of Lane-Emden type equations include the Bessel collocation method [18, 19], Jacobi-Gauss collocation method [20], Legendre Tau method [21], Sinc-collocation method [22], Chebyshev spectral methods [23], quasi-linearisation based Chebyshev pseudo-spectral method [24, 25] and a collocation method based on radial basis functions [26]. The discretisation scheme of collocation based methods is only implemented on interior nodes of the discretised domain. This property makes it possible for these colloca‐ tion methods to overcome the difficulty of dealing with the singular point.

In this work, we present a multi-domain spectral collocation method for solving Lane-Emden equations. The method is based on the innovative idea of reducing the governing non-linear differential equations to a system of first-order equations which are solved iteratively using a Gauss-Seidel–like relaxation approach. The domain of the problem is divided into smaller nonoverlapping sub-intervals on which the Chebyshev spectral collocation method is used to solve the iteration scheme. The continuity condition is used to advance the solution across neigh‐ bouring sub-intervals. The advantage of the approach is that it does not use Taylor-series based linearisation methods to simplify the non-linear differential equations. The method is free of errors associated with series truncation. The algorithm is also very easy to develop and yields very accurate results using only a few discretisation nodes. The accuracy of the method is validated against known results from the literature. The aim of the study is to explore the applicability of the multi-domain spectral collocation method to Lane-Emden type equations over semi-infinite domains. The results confirm that the method is suitable for solving all types of Lane-Emden equations.
