**A Multi-Domain Spectral Collocation Approach for Solving Lane-Emden Type Equations**

Motsa Sandile Sydney, Magagula Vusi Mpendulo, Goqo Sicelo Praisegod, Oyelakin Ibukun Sarah and Sibanda Precious

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/63016

#### **Abstract**

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In this work, we explore the application of a novel multi-domain spectral collocation method for solving general non-linear singular initial value differential equations of the Lane-Emden type. The proposed solution approach is a simple iterative approach that does not employ linearisation of the differential equations. Spectral collocation is used to discretise the iterative scheme to form matrix equations that are solved over a sequence of non-overlapping sub-intervals of the domain. Continuity conditions are used to advance the solution across the non-overlapping sub-intervals. Different Lane-Emden equations that have been reported in the literature have been used for numerical experimentation. The results indicate that the method is very effective in solving Lane-Emden type equations. Computational error analysis is presented to demonstrate the fast convergence and high accuracy of the method of solution.

**Keywords:** Multi-domain, Lame-Emden, Spectral Relaxation method, Collocation, As‐ trophysics
