Comparative Study of Wavelet Solutions for Forced Oscillatory Problems Arising with and Without Damping

Oscillatory problems are a class of mathematical and physical phenomena in which the solutions display periodic or quasi-periodic variations with respect to time or space. Oscillatory systems subject to external forcing are central in many physical and engineering contexts, and the presence or absence of damping critically influences their behavior. In this comparative study, we develop and evaluate wavelet-based numerical schemes for forced oscillatory differential equations, considering both damped and undamped regimes. Specifically, we employ second-kind Chebyshev wavelets and Haar wavelets, together with their operational integration matrices, to discretize and approximate the solutions of second-order forced oscillators. The wavelet formulations transform the differential problems into systems of algebraic equations, which we then solve under a variety of forcing frequencies and damping parameters. Our numerical experiments demonstrate that Chebyshev wavelets, due to their higher smoothness and spectral accuracy, are particularly effective in capturing the transient decay and subtle features of damped oscillations. In contrast, Haar wavelets provide a computationally efficient and stable approximation in undamped systems, especially in scenarios prone to resonance. We compare these methods across key metrics such as convergence rate, error behavior, and computational cost. The numerical results confirm that Chebyshev wavelets are particularly effective in capturing the fine decay dynamics of damped oscillations, while Haar wavelets deliver fast, stable approximations for undamped systems, especially under resonant forcing. These findings are supported by several computational experiments, which validate both the accuracy and efficiency of the proposed wavelet schemes. These insights offer practical guidance for selecting suitable wavelet approaches tailored to the physical damping conditions of forced oscillatory problems.