Solving Generalized Hill’s and Meissner’s Equations by a Semianalytical Technique

This paper presents a comparative study of analytical and numerical approaches for solving generalized forms of second-order Hill’s and Meissner’s ordinary differential equations (ODEs), which model a wide range of periodic and oscillatory phenomena in applied mathematics and physics. Using the Akbari–Ganji Method (AGM), a powerful semianalytical technique, we derive approximate solutions that effectively capture the essential behavior of these equations. To assess the accuracy and reliability of AGM, we compare its solutions with those obtained using the Fehlberg fourth-fifth order Runge–Kutta method with degree four interpolant (RKF45), a robust numerical method. The comparison involves a detailed error analysis, including the computation of absolute error, maximum absolute error, norm error, root mean square error (RMSE), and Average Error. Results demonstrate that AGM offers high precision and computational efficiency while maintaining agreement with RKF45 numerical solutions. This work highlights AGM’s effectiveness in handling periodic coefficients and broadens its scope within periodic and quasi-periodic systems, providing a promising alternative for researchers tackling complex oscillatory systems.