The solution of the one dimensional planar Bratu equation is characterized by having two branches. The two branches meet at a limit point that can be obtained through the solution of an eigen-relation. In this paper a transformation is introduced to transform this highly non-linear problem to a linear one. Rational expressions are given for the solution of the Bratu equation and the limit point. This is done using collocation and perturbation techniques.
Soliman, Moustafa A., "Rational Approximation for the One-Dimensional Bratu Equation" (2013). Chemical Engineering. 6.