Time frequency method of solving one boundary value problem for a hyperbolic system and its application to the oil extraction

We consider the boundary value problem, where the motion of the object is described by the two-dimensional linear system of partial differential equations of hyperbolic type where a discontinuity is at a point within the interval that defines the phase coordinate $x$. Using the method of series and Laplace transformation in time $t$ (time-frequency method), an analytical solution is found for the determination of debit $Q(2l,t)$ and pressure $P(2l,t)$, which can be effective in the calculation of the coefficient of hydraulic resistance in the lift at oil extraction by gas lift method where $l$ is the well depth. For the case where the boundary functions are of exponential form, the formulas for $P(2l,t)$ and $Q(2l,t)$ depending only on $t$ are obtained. It is shown that at constant boundary functions, these formulas allow us to determine the coefficient of hydraulic resistance in the lift of gas lift wells, which determines the change in the dynamics of pollution.