  We note a fact that stiff systems or differential equations that have highly
oscillatory solutions cannot be solved efficiently using conventional methods.
In this paper, we study two new classes of exponential Runge-Kutta (ERK)
integrators for efficiently solving stiff systems or highly oscillatory
problems. We first present a novel class of explicit modified version of
exponential Runge-Kutta (MVERK) methods based on the order conditions.
Furthermore, we consider a class of explicit simplified version of exponential
Runge-Kutta (SVERK) methods. Numerical results demonstrate the high efficiency
of the explicit MVERK integrators and SVERK methods derived in this paper
compared with the well-known explicit ERK integrators for stiff systems or
highly oscillatory problems in the literature.
