Numerical Solutions of Oscillatory Dynamics via Haar Wavelets

Abstract

An efficient numerical approach has been developed for solving second-order differential equations using the Haar Wavelet Collocation Method (HWCM) in dynamical systems that change over time, including applications in physics and engineering.  From a physics perspective, we systematically investigate three sets of differential equations representing simple and damped harmonic motion using the Haar Wavelet Collocation Method (HWCM) and compare the results with the Taylor series method. Numerical results indicate that HWCM provides more accurate approximations than the Taylor series approach. Further, the Haar solutions and exact solutions with absolute errors are calculated. Interestingly, the analysis indicates that error drops exponentially as the resolution level increases. These results provides the accurate tools for predicting displacement, velocity, and acceleration in oscillatory systems. The main key features of the HWCM are that it is simple and effective in solving a wide variety of Initial Value Problems (IVPs).