Studying the solution of the system of differential equations obtained in simulation of gas-lift process by the relaxation method

Using the relation method for the system of first order, two-dimensional hyperbolic type partial differential equations describing the motion of gas liquid gas mixture corresponding to gas-lift process of oil production in an annular and lifting pipe, we consider a boundary condition problem. In this problem the existence a boundary condition problem. In this problem the existence of the solution with respect to the equations of motion is studied and it is shown that when constructing the solution of this problem by means of boundary conditions, it is impossible to determine the coefficients of positive degrees of the parameter ε. For this reason, the solution is sought in the form of a series using negative degrees of the parameter ε.