The linear sequential fractional differential system involving two generalized fractional orders and its application to the vibration theory

Abstract

The main aim of the current paper is to keep developing the theory of fractional calculus and observe its contributions to real-world problems. In this regard, the linear sequential fractional differential system involving two generalized fractional orders in the conformable sense is introduced. The conformable bivariate Mittag-Leffler function is proposed to solve the introduced system. To do this,  the conformable Laplace transform method is used and it is shown that the obtained solution satisfies the system.  The vibration of springs is presented as an application with simulations for the described system. The effectiveness of the results is shown by discovering a relation between the system's order and its equilibrium position.