An optimal iterative algorithm for solving Bratu-type problems

Very Recently, Das et al. (J Math Chem 54:527-551, 2016) proposed a method based on variational iteration method for solving Bratu-type problems. In this work, we design a fast iterative scheme based on the optimal homotopy analysis method to obtain series solution for such problem. The proposed method contains a convergence control parameter which adjusts the interval of convergence of the series solution. This parameter is computed by minimizing the sum of the squared residual of the governing equation and with the help of which we obtain an optimal series solution to the problem. Further, our method avoids the usual time-consuming procedure to find the root of a nonlinear algebraic or transcendental equation for the undetermined coefficients. Besides the design of the method, convergence analysis and error estimate of the proposed method are supplemented. Two Bratu type problems are solved to demonstrate the efficiency and accuracy of the proposed method. Numerical results reveal the superiority of the proposed iterative scheme over a newly developed iterative method based on the variational iteration method (Das et al. 2016).