On The Refinement of an Approximate Solution of a One-Dimensional Singularly Perturbed Model Problem with Discontinuous Nonlinearity

An ordinary differential equation with a small parameter with a derivative, discontinuous phase variable nonlinearity and two-sided initial conditions is considered, which simulates the motion of a material point with an acceleration jump. The features of the problem under study include the fact that there is no solution to the degenerate equation, and that the initial conditions depend on a small parameter and determine a non-smooth function. The problem in this formulation has already been solved by the authors earlier, when, firstly, its exact solution was written down, and secondly, an approximate model was given, a fairly smooth solution was constructed for it, and its analysis was carried out. However, the previous approximate solution had the disadvantage that, although its behavior was regular when moving away from the starting point, it remained unsatisfactory in a small neighborhood near it, based on the requirements of causal constraints. In this paper, an approximation of the equation for a smooth approximate solution is performed in order to eliminate this problem. The constructed smooth solution will obey the left initial condition at the transition point.