A modified perturbation solution to the one-dimensional Bratu problem

An approximate analytical solution to the one-dimensional Bratu boundary value problem is introduced in this paper. The solution is based on some perturbation expansion methods. The first step is taken from the linearized perturbation technique which was developed to solve initial value problem with a nonlinear term and no small parameter. An artificial small parameter is embedded and the dependent variable can then be expanded in terms of this parameter. However, necessary modifications are introduced to implement some techniques to solve a boundary value problem in order to allow for different nonlinearities. The current solution showed good convergence at any value of the Bratu constant, when compared with the exact lower branch of the solution to the one-dimensional Bratu problem. Also its recursive nature allows for more iterations and adding more correction terms to the final approximate solution which increases the accuracy.