  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
