A comparison of numerical methods for solving the bratu and bratu-type problems - Md.Kasmani, Ruhaila (2005)
The Bratu problem ul/(x) + /\eu(x) = 0 with u(O) = u(l) = 0 has two exact solutions for values of 0 < A < Ac, no solutions if A > Ac while unique solution is obtained when A = Ac where Ac = 3.513830719 is the critical value. The First Bratu-Type problem corresponds A = _7[2 while the Second Bratu-Type problem is ul/(x) + 7[ 2e-u(x) = o. The exact solution of the First Bratu-Type problem blows up at x = 0.5 while the Second Bratu-Type problem is continuous. The present work seeks to compare various numerical methods for solving the Bratu and Bratu-Type problems. The numerical methods are the standard Adomian decomposition method, the modified Adomian decomposition method, the shooting method and the finite difference method. These methods are implemented using Maple. Convergence is achieved by applying the four methods when 0 < A ::; 2, however the shooting method is the most effective method as it gives the smallest maximum absolute error. ·When A = Ac, none of these methods give the convergence solutions. Due to the nature of the solution of the First Bratu-Type problem, only the shooting method and the modified Adomian decomposition method can give the convergence values to the exact solution. The finite difference method is proved to be the most effective method for the Second Bratu-Type problem compared to other methods. Keywords: Bratu problem, Bratu-Type problems, standard Adomian decomposition method, modified Adomian decomposition method, shooting method, finite difference method.