DISCUSSION OF A NON-LINEAR DIFFERENTIAL EQUATION CONTAINING EXCEPTIONAL POINTS

摘要

A theoretical treatment of spherical plasma probes in the Vlassov approximation leads to a non-linear ordinary differential equation which has somewhat exceptional properties. Certain mathematical phenomena caused by this fact are studied in the present report. Within the region of integration, one encounters certain points for which a Lipchitz condition is not satisfied; fortunately, a weaker condition suffices to establish uniqueness of the initial value problem in the present case. Next, we ask whether the solutions of the variational equation are suited to approximate neighboring solutions for a given solution. Two cases are discussed where the variational equation will yield such an approximation. In a third case, the approximation fails in the vicinity of the exceptional point. Here an auxiliary approximation valid for this vicinity is introduced and the auxiliary solution is matched with the solution of the variational equation. The result is somewhat unexpected. Finally, the actual numerical procedure is discussed. No definite conclusion can be reached, but one recognizes which form of intermediate approximations may cause numerical difficulties.