Abstract
•A new fractional Euler-Lagrange equation for a harmonic oscillator is proposed.•The new equation involves the left and right Caputo-Fabrizio fractional derivatives.•A new numerical method is developed to solve the aforementioned equation effectively.•The fractional model is flexible to exhibit different asymptotic behaviors.•The new approach helps us to extract the hidden aspects of natural phenomena.
The aim of this research is to propose a new fractional Euler-Lagrange equation for a harmonic oscillator. The theoretical analysis is given in order to derive the equation of motion in a fractional framework. The new equation has a complicated structure involving the left and right fractional derivatives of Caputo-Fabrizio type, so a new numerical method is developed in order to solve the above-mentioned equation effectively. As a result, we can see different asymptotic behaviors according to the flexibility provided by the use of the fractional calculus approach, a fact which may be invisible when we use the classical Lagrangian technique. This capability helps us to better understand the complex dynamics associated with natural phenomena.