Fractional mass-spring system with damping and driving force for modified non-singular kernel derivatives

The aim of the present work is to discuss the fractional mass-spring system with damping and driving force, considering a simple modification to the fractional derivatives with a non-singular kernel of the Atangana–Baleanu and Caputo–Fabrizio types. We introduce two novel modified fractional derivatives that offer advantages when the fractional differential equations involve higher-order fractional derivatives of order \(1+\alpha \) or \(\alpha +1\), with \(0<\alpha <1\). Previous definitions of fractional derivatives with non-singular kernel do not have a unique definition, leading to significant inconsistencies. One of the main results of the present work is that the proposed modifications provide a unique result for the fractional-order derivatives \(1+\alpha \) and \(\alpha +1\). Additionally, we apply these two novel fractional derivatives to the fractional mass-spring system with damping and driving force. In the case of the modified Caputo–Fabrizio fractional derivative, novel analytical solutions have been constructed, showing interesting oscillating time evolution with a transient term not previously reported. This transient term features an initial nonzero oscillating return away from the equilibrium position. For the modified Atangana–Baleanu fractional derivative, the numerical solutions also exhibit this nonzero oscillating return away from the equilibrium position. These results are not present when using the Caputo singular kernel derivative, as demonstrated in the comparison figures reported here.