analytic approximate solution for the bratu's problem by optimal homotopy analysis method

In This paper, we present analytic approximate solutions for Bratu's problem with high accuracy for different values of $lambda$. We solve this nonlinear problem without any approximations or transformation in the problem and we successfully obtain the two branches of solutions for different values $lambda$ using homotopy analysis method. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter $hbar$ to guarantee the convergence of the obtained series solution.