Exponential fitting of the delayed recruitment/renewal equation

From respiratory physiology to laser-based optical devices, the so-called delayed recruitment/renewal equation (1)εdc(t)dt=−c(t)+f(c(t−1)),provides the mathematical model in a diverse spectrum of practical applications. Here, ε is inversely proportional to the product of the time-delay inherent in the physical system and its rate of decay. When this time-lag is large relative to the reciprocal of the decay rate, ε is small and this delay differential equation (DDE) is singularly perturbed. When this situation obtains, c(t) can exhibit initial layers and chaotic oscillations. In order to accurately capture such solution features numerically, one must use an approximation technique tailored to singular perturbation problems. In this work, we develop such a family of exponentially fitted schemes for the numerical approximation of this fundamental DDE. Application of this new technique is then made to a variety of interesting and important problems, not the least of which is the subject of dynamical diseases.