Singularly perturbed equation with a jump in solutions

The asymptotics of the solution of the model one-dimensional singularly perturbed Lighthill equation is constructed by the parametrization method. A feature of the problem is that there is a singular point at the point x=0. It is proved that at this singular point the solution of the singularly perturbed Lighthill problem changes its value sharply, i.e., jump occurs. The value of this jump is calculated.