It is pointed out that the order reduction of singular perturbed systems with non-Lipschitzian nonlinearities can be performed, using the new concept of positively (negatively) invariant manifolds, if the five assumptions of the Tikhonov theorem are satisfied. Examples when these assumptions are satisfied and not satisfied are presented and discussed. The previously derived singularly perturbed system of three equations (droplet radii, gas temperature and fuel vapour concentration), describing evaporation and ignition of monodisperse fuel sprays, are shown to satisfy all five assumptions of the Tikhonov theorem. This allows us to reduce this system to a system of two equations describing the gas temperature and fuel concentration when the droplet evaporation process has been completed.
- Monodisperse spray
- singular perturbations
- non-Lipschitzian equations
- positively (negatively) invariant manifolds
Sazhin, S., Shchepakina, E., & Sobolev, V. (2010). Order reduction of a non-Lipschitzian model of monodisperse spray ignition. Mathematical and Computer Modelling, 52(3-4), 529-537. https://doi.org/10.1016/j.mcm.2010.03.051