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In this study, we examine numerical approximations for 2nd-orderlinear-nonlinear differential equations with diverse boundary conditions,followed by the residual corrections of the first approximations. We firstobtain numerical results using the Galerkin weighted residual approach withBernstein polynomials. The generation of residuals is brought on by the factthat our first approximation is computed using numerical methods. To minimizethese residuals, we use the compact finite difference scheme of 4th-orderconvergence to solve the error differential equations in accordance with theerror boundary conditions. We also introduce the formulation of the compactfinite difference method of fourth-order convergence for the nonlinear BVPs.The improved approximations are produced by adding the error values derivedfrom the approximations of the error differential equation to the weightedresidual values. Numerical results are compared to the exact solutions and tothe solutions available in the published literature to validate the proposedscheme, and high accuracy is achieved in all cases

Numerical Approximations of a Class of Nonlinear Second-Order Boundary Value Problems using Galerkin-Compact Finite Difference Method