Solutions of neutral delay differential equations using a generalized Lambert W function

The Lambert W function is defined by W(a)eW(a)−a=0. One of the many applications of the Lambert W function is in solving delay differential equations (DDEs). In 2003, Asl and Ulsoy provided a solution of some DDEs in terms of the Lambert W functions Asl et al. (2003)[1]. However, the solutions are limited to differential equations with delay in the state variable. Scott et al. (2006)[2] introduced a generalized Lambert function which was further studied by Mező and Baricz (2017)[3]. In our work, we show that this generalization of the Lambert W function provides an analytical solution to neutral delay differential equations (NDDEs). NDDEs are DDEs with time delay not only in the state variables but also in the derivative terms. This analytical solution is advantageous such that it is similar to the general solutions of linear ODEs. Also, one can identify how the parameters affect the solution of the equation since our proposed solution is written in terms of these parameters. We then propose a new numerical method to solve linear NDDEs using the generalized Lambert W function. We test our method to examples with known solutions. We also provide a real-world application by solving an NDDE model of the population growth of an E. coli culture using our proposed approach.