Two highly accurate and efficient numerical methods for solving the fractional Liénard’s equation arising in oscillating circuits

In this paper, we investigate the fractional model of the Liénard and Duffing equations with the Liouville-Caputo fractional derivative. These equations grow with the evolution of radio and vacuum tube technology, which describe oscillating circuits and generalize the spring–mass device equation. We compare two numerical approaches, namely Jacobi and Haar wavelet collocation methods. The given approaches are used to discretize and transform the equation into a system of algebraic equations, and the Broyden-Quasi Newton algorithm is applied to solve the resulting nonlinear system of equations. A complete error analysis and convergence rates for different grid sizes are derived for both methods, which are used to compare the accuracy and efficiency of the two approaches. While both approaches produce correct solutions, according to the numerical findings, the Jacobi collocation method is more efficient and accurate than the Haar wavelet collocation method.