Transforming linear FDEs with rational orders into ODEs by modified differential operator multiplication method

Abstract

This paper presents a modified differential operator multiplication (DOM) method for solving a certain class of fractional differential equations (FDEs), with emphasis on linear oscillators subjected to periodic excitation. The main idea of DOM is to transform the considered FDEs with rational order r/m into rth-order ordinary differential equations (ODEs), herein r and m are positive integers. The transformation is realized by differentiating the FDEs stepwise with fractional-order r/m, until the rth-order derivative is reached due to the accumulative property for FDs. In the modified method, differently, first-order ODEs are deduced by transforming the FDEs with order r/m into 1/m. In addition, we introduce auxiliary state space variables to represent the FDs of inhomogeneous terms such as triangle functions. Exact and explicit first-order ODEs are deduced and solved by the Runge–Kutta algorithm. To validate the modified DOM, we solve the fractional Kelvin-Voigt equation and harmonically excited oscillators with visco-elastic dampings. The presented method can provide highly accurate solution, at the expense of linearly increasing computational resources as the solution domain expands. Such high computation accuracy and efficiency would potentially enable the presented method to become a benchmark comparison for solving certain FDEs, especially harmonically excited fractional oscillators.