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We consider the asymptotic solution of the Tonks-Langmuir integro-different equation with an Emmert kernel, which describes the behavior of the potential both inside the main plasma volume and in a thin boundary layer. Equations of this type are singularly perturbed due to the small coefficient at the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the regular expansion series and in the boundary function expansion. The equation for the first coefficient of the regular series has only a trivial solution. Second-order differential equations are obtained for the first two boundary functions. The equation for the first boundary function is solved numerically on a discrete grid with locally uniform spacing. An approximate analytical expression for the first boundary function is obtained from the linearized equation. This solution adequately describes the behavior of the potential on small distances only.

The asymptotic solution of the integro-differential plasma -sheath equation is considered. This equation is singularly perturbed because of the small coefficient multiplying the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the form of both a regular series expansion and an expansion in boundary functions. The equation for the first coefficient of the regular series has only a trivial solution. A numerical algorithm is considered for the solution of the second-order differential equation describing the behavior of the zeroth-order boundary function. The proposed algorithm efficiently solves the boundary-value problem and produces a wellbehaved solution of the Cauchy problem.

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This is a review of studies on hybrid simulation of low-frequency processes in space plasmas. We discuss the main approximations used in the derivation of the hybrid model: particle representation ~br ions; massless fluid representation for electrons. The main numerical schemes for the implementation of this model are considered: the generalized Ohm law scheme, the predictor-corrector scheme, the scheme using Boris and Runge-Kutta methods to compute the fields. The article reviews the literature on simuIation of collisionless shocks: quasiperpendicular shocks with anisotropic (mirror and ioncyclotron) instabilities; quasiparallel shocks with inclusion of re-formation processes ("periodic" destruction and repeated formation of the shock front), as well as collision of two shocks. Numerical aspects of simulation are discussed in some cases. Initialization of shocks and collisionless discontinuities is examined.