This paper aims at the application of an optimized two-step hybrid block method for solving boundary value problems with different types of boundary conditions. The proposed approach produces simultaneously approximations at all the grid points after solving an algebraic system of equations. The final approximate solution is obtained through a homotopy-type strategy which is used in order to get starting values for Newton’s method. The convergence analysis shows that the proposed method has at least fifth order of convergence. Some numerical experiments such as Bratu’s problem, singularly perturbed, and nonlinear system of BVPs are presented to illustrate the better performance of the proposed approach in comparison with other methods available in the recent literature.

本文针对优化的两步混合块法在解决不同类型边界条件下的边值问题中的应用。所提出的方法在求解方程的代数系统后在所有网格点同时产生近似值。最终的近似解是通过同伦类型的策略获得的，该策略用于获得牛顿方法的初始值。收敛性分析表明，所提出的方法至少具有五阶收敛性。提出了一些数值实验，例如Br​​atu问题，奇异摄动和BVP的非线性系统，以说明与最近文献中可用的其他方法相比，该方法具有更好的性能。