The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the approximate solutions of 2nd order boundary value problems. Here the Bernoulli polynomials over the interval [0, 1] are chosen as trial functions so that care has been taken to satisfy the corresponding homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method. In addition to that the given differential equation over arbitrary finite domain [a, b] and the boundary conditions are converted into its equivalent form over the interval [0, 1]. All the formulas are verified by considering numerical examples. The approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases.

翻译：本文旨在求解带有Dirichlet、Neumann及Robin边界条件的二阶线性和非线性微分方程的数值解。我们采用伯努利多项式的线性组合来逼近二阶边值问题的解。在Galerkin加权残量法中，选取区间[0,1]上的伯努利多项式作为试探函数，以确保满足Dirichlet边界条件的齐次形式。此外，将任意有限域[a,b]上的给定微分方程及边界条件转换为其在区间[0,1]上的等价形式。通过数值算例验证所有公式，并将近似解与精确解及现有方法所得解进行比较。所有情形均获得了可靠的高精度结果。