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 conformable fractional calculus and observe its contributions to real-world problems. In this regard, the conformable bivariate Mittag-Leffler function is first proposed. The images of the conformable bivariate Mittag-Leffler functions under the conformable derivatives and the conformable Laplace transforms are calculated. A representation of an explicit solution to the linear sequential fractional differential system involving two generalized fractional orders in the conformable sense is determined based on the conformable bivariate function with the help of the conformable Laplace transform method. Then, it is shown that the obtained solution satisfies the introduced system. The vibration of springs is presented as an application with many simulations and tables for the described system. The effectiveness of the results is shown by discovering a relation between the system's order and its equilibrium position.