It is a new algorithm to find the exact solution of boundary value problems using the closed-form method using Laplace transforms method and a more accurate numerical solution of boundary value problems using fourth-order finite difference method (FOFDM). It is known that the Laplace transforms method gives a closed-form for initial value problems but in the present study, we were able to use it to find the exact solution of boundary value problems which is the novelty of the present study. Also, high-accurate numerical methods to solve linear boundary value problems which approximate to exact solutions. The uniqueness, convergence, and stability of the new technique (FOFDM) are verified and tested by comparisons with a closed-form method.The uniqueness, convergence, and stability of the new technique (FOFDM) are verified and tested by comparisons with a closed-form method.The uniqueness, convergence, and stability of the new technique (FOFDM) are verified and tested by comparisons with a closed-form method.