Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays

Abstract

Delay differential equations (DDEs) are infinitedimensional systems, therefore analyzing their stability is a difficult task. The delays can be discrete or distributed in nature. DDEs with distributed delays are referred to as delay integrodifferential equations (DIDEs) in the literature. In this work, we propose a method to convert the DIDEs into a system of ordinary differential equations (ODEs). The stability of the DIDEs can then be easily studied from the obtained system of ODEs. By using a spacetime transformation, we convert the DIDEs into a partial differential equation (PDE) with a timedependent boundary condition. Then, by using the Galerkin method, we obtain a finitedimensional approximation to the PDE. The boundary condition is incorporated into the Galerkin approximation using the Tau method. The resulting system of ODEs will have timeperiodic coefficients, provided the coefficients of the DIDEs are time periodic. Thus, we use Floquet theory to analyze the stability of the resulting ODE systems. We study several numerical examples of DIDEs with different kernel functions. We show that the results obtained using our method are in close agreement with those existing in the literature. The theory developed in this work can also be used for the integration of DIDEs. The computational complexity of our numerical integration method is O(t), whereas the direct bruteforce integration of DIDE has a computational complexity of O(t2).