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Several theoretical studies and numerical methods were used for the study of the Bratu problem. It is a two point boundary value problem of a second order non-linear ordinary differential equation. The Bratu equation models thermal explosion in an enclosure, a non-isothermal zero order reaction in a catalyst particle and appears in the theory of the electric charge around a hot wire, in certain problems of solid mechanics, and the electrospinning of nanofibres. The initial value problem analogue to Bratu equation is called Lane Emden equation of the second kind. The Lane Emden equation is a basic equation in astrophysics that describes dimensionless density distribution in an isothermal gas sphere. We present analytical approximation methods for its solution in case of solid and hollow slab, cylinder and sphere. We use perturbation and collocation methods for its treatment. Perturbation series are recast in the form of rational expressions. We also introduce a rational collocation method. Mathematicians, engineers, astrologists, physicists, and chemists will find the book useful in their work.