Exponential fitting Runge–Kutta methods for the delayed recruitment/renewal equation

The so-called delayed recruitment/renewal equation provides the mathematical model in a diverse spectrum of practical applications and may become singularly perturbed when the time-lag is large relative to the reciprocal of the decay rate. In order to accurately capture its solution features numerically, we design a family of exponential fitting Runge–Kutta methods of collocation type to obtain the numerical approximation. The exponential fitting approximations are proved to have higher order of uniform accuracy. We demonstrate the efficiency of this family of exponential fitting Runge–Kutta methods for the delayed recruitment/renewal equation via application to some important problems.