Kravchenko

This article considers the Cauchy problem for the second order automodeling nonlinear ordinary differential equation in the problem of two-fluid plasma motion. An algorithm for solving this problem with the use of Haar wavelet method was described. A numerical experiment showed that Haar wavelet method is more accurate than the fourth order Runge-Kutta method for few collocation points in problem 1 with a known analytical solution. In problem 2, where the exact solution is unknown, relative error between numerical solutions by Haar wavelet method and Runge-Kutta method tends to decrease with the growth of the number of collocation points.

A simple model of lightning appearance based on hydrodynamic equations of charged gas flow was introduced in paper [3]. It is possible to transform a system of partial differential equations into equation of type (4). The question about analytical solution in general question is still open to this model. Because of the nonlinearity of a problem a question of the effect of initial conditions on the type of the type of solutions is crucial. In this paper an analytical solution of equation (4) is introduced. An ordinary differential equation was investigated by phase plane methods in stationary case. A different kind of motions realized in the model were established.