Galerkin approximations for fractional-order delay-differential equations with constant or time-periodic coefficients

Owing to the versatility of the Galerkin approximations, they have been widely used in developing finite-dimensional approximations of infinite-dimensional systems, such as fractional-order differential equations (FODEs) and delay-differential equations (DDEs). Fractional-order delay-differential equations (FODDEs), which arise in various physical scenarios, are even more complex than FODEs and DDEs due to the presence of two infinite-dimensional terms: the fractional-order derivative and the delay. In this work, we develop a method based on Galerkin approximations to compute the time response and determine the linear stability of time-periodic FODDEs, including those with constant coefficients. Our approach involves two successive Galerkin approximations, first reducing the FODDE to a DDE system and subsequently to an ordinary-differential (ODE) equation system, which is integrated to compute the time response. Furthermore, due to the time-periodic nature of the ODE system, it is amenable to the application of the Floquet theory for stability determination. For demonstration, we determine the stability of five FODDEs of differing nature and compute their time responses, obtaining excellent match with those obtained from direct numerical integration of the FODDEs. Crucially, our method is significantly faster than direct integration, showcasing its utility.