Path-regularization of linear neutral delay differential equations with several delays

For differential equations with discontinuous right-hand side and,in particular,for neutral delay equations it may happen that classicalsolutions do no exist beyond a certain time instant. In this situation,it is common to consider weak solutions of Utkin (Filippov) type.This article extends the concept of weak solutions and proposes anew regularization which eliminates the discontinuities.Codimension-$1$ and codimension-$2$ weaksolutions are considered. Numerical experiments show theadvantages of the new regularization.