Comparative numerical study of the second-order boundary value problems

In this paper, we present a comprehensive analytical investigation of second-order boundary value problems using the semi-analytical Homotopy Analysis Method (HAM). A key aspect of our approach involves determining the optimal value of the convergence control parameter ℏ by analyzing the residual error associated with the approximate solution. This enables us to enhance both the accuracy and convergence of the method. To illustrate the applicability and effectiveness of HAM, several representative examples are provided, each demonstrating the method’s flexibility and precision. Furthermore, a comparative study is conducted in which the results obtained via HAM are evaluated against those produced by other established numerical techniques, such as the B-spline method and finite difference approaches. The comparison clearly demonstrates the superior accuracy and robustness of HAM in solving second-order boundary value problems, thereby affirming its potential as a reliable tool for a wide range of applications, including higher-order and fractional differential equations.