Analytic approximate solution for the Bratuʹʹs problem by optimal homotopy analysis method

In This paper, we present analytic approximate solutions for Bratuʹs problem with high accuracy for different values of . 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 using homotopy analysis method. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter to guarantee the convergence of the obtained series solution.