In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases

翻译：本研究考察了具有多种边界条件的二阶线性-非线性微分方程的数值逼近，并对其首次逼近进行了残差修正。我们首先采用Bernstein多项式的Galerkin加权残量法获得数值结果。由于首次逼近通过数值方法计算，因此产生了残差。为最小化这些残差，我们利用四阶收敛的紧致有限差分格式，根据误差边界条件求解误差微分方程。同时，针对非线性边值问题，我们提出了四阶收敛紧致有限差分法的公式化表述。通过将误差微分方程逼近解算出的误差值叠加到加权残量值上，得到改进后的逼近解。将数值结果与精确解及现有文献中的解进行对比以验证所提方案，在所有情形下均实现了高精度。