A wavelet approach to the solution of Ermakov-Pinney equation

Abstract

The Ermakov–Pinney (EP) equation is well known for its exact solvability and its connection to time-dependent harmonic oscillators. However, for cases involving complex or time-dependent parameters, numerical approaches become essential. This work employs the Haar wavelet method to solve the EP equation under various forms and conditions. We demonstrate the efficacy of the method by solving a range of initial and boundary value problems, comparing the results with exact solutions and those obtained via the Runge–Kutta method. To the best of our knowledge, this is the first systematic application of the Haar wavelet method to the EP equation with time-dependent frequency and coupled boundary conditions. Our findings indicate that the Haar wavelet method is not only easier to implement but also yields superior accuracy compared to the Runge–Kutta approach.