Applying the dual operator formalism to derive the zeroth-order boundary function of the plasma-sheath equation (Q2458135)

The author constructs an asymptotic solution of Tonks-Langmuir integro-differential equation. The equation has Emmert kernel which describes the behaviour of potential both in the main plasma volume and in the wall layer. The equation is singularly perturbed because the highest-order (here the second) derivative is multiplied by a small factor. The asymptotic solution is obtained by the boundary function method. The second-order differential equation describing the behaviour of zeroth-order boundary function is investigated using the dual operator formalism, which is an analogon of conjugate operator in the linear theory. The asymptotic solution makes it possible to propose a number of homogeneous discrete three-point schemes for solving the original equation.