Numerical solution of Bratu’s and related problems using a third derivative hybrid block method

A new one-step hybrid block method with third derivatives and optimized features aimed at solving the classical one-dimensional Bratu’s and Reactor design problems is developed. The development of the new method considers three intermediate points that are properly chosen through the optimization of the local truncation errors corresponding to the main formulas to approximate the solution and the first derivative at the end point of the block, and another approximation of the solution at an intermediate point. The convergence analysis and the order of the proposed method are analyzed. Some specific problems are solved to demonstrate the efficiency and feasibility of the technique adopted. The numerical results provided through the implementation of the scheme are very much closer to the exact solutions and are found favorably compared with different methods in the available literature.