## 2 Materials and Methods  
### Statement of Direct and Inverse Problems  
The mathematical model of the gas lift well operation is described using the following linear system of differential equations [1]:  
\[\frac{\partial P}{\partial t}=-\frac{c^{2}}{\overline{F}}\cdot\frac{\partial Q} {\partial x},\qquad t\geq 0,\ x\in(0;\ 2l), \tag{1}\]  
\[\frac{\partial Q}{\partial t}=-\overline{F}\cdot\frac{\partial P}{\partial x }-2a\cdot Q,\qquad\quad t\geq 0,x\in(0;\ 2l), \tag{2}\]  
where \(t\) is time; \(x\) is a coordinate along the well depth; \(P\) is the pressure; \(Q\) is the volume flow rate of gas; \(\overline{F}\) is the cross-sectional area of the well; \(c\) is the speed of sound in liquid; \(1\) is the well depth; and a is a coefficient that depends on the input parameters. \[c= \begin{cases}c_{1},&x\in(0;\ l)\\ c_{2},&x\in(l;\ 2l)\end{cases} ,\ \overline{F}= \begin{cases}\overline{F}_{1},&x\in(0;\ l)\\ \overline{F}_{2},&x\in(l;\ 2l)\end{cases} ,\ a= \begin{cases}a_{1},&x\in(0;\ l)\\ a_{2},&x\in(l;\ 2l)\end{cases} ,\]  
Parameters \(a_{1}\) and \(a_{2}\) characterize the combined effect of gravitational acceleration and hydraulic resistance on gas movement in the annular space and lift, respectively. These coefficients reflect energy losses due to friction and lifting along the well. Variables \(c_{c}\) and \(c_{2}\) denote the speed of sound in a gas-liquid medium in different zones of the well: \(c_{1}\)--in the annular space, where gas predominantly moves, and \(c_{c}\)--inside the lifting pipe, where a mixture of gas and oil moves. The difference between \(c_{1}\) and \(c_{2}\) is due to differences in the composition and density of the media, which is critical for the correct modeling of wave processes. The initial conditions are as follows:  
\[P(0,x)=P^{0}(x),\ \ \ \ \ Q(0,x)=Q^{0}(x), \tag{3}\]  
and the boundary conditions are as follows:  
\[P(t,0)=P_{0}(t),\ \ \ \ \ Q(t,0)=Q_{0}(t),\ \ \ \ \ for\ \ \ x=0, \tag{4}\]  
\[P(t,l+0)=P(t,l-0)+P_{pl}(t),Q(t,l+0)=Q_{0}(t,l-0)+Q_{pl}(t), \tag{5}\]  
\[for\ x=l,\]  
\[P(t,2l)=P_{out}(t),\ \ \ \ P(t,2l)=Q_{out}(t),\ \ \ x=2l. \tag{6}\]  
where \(Q_{pl}\) is the volumetric flow rate from the reservoir; \(P_{pl}\) is the reservoir pressure; \(P^{0}(x)\) is the initial gas pressure; \(Q^{0}(x)\) is the initial gas volume; \(P_{out}(t)\) is the discharge pressure; and \(Q_{out}(t)\) is the discharge volume. We will reduce the system of hyperbolic Equations (1) and (2) to one second-order equation. To accomplish this, we must find the derivative with respect to \(x\) from Equation (1). \[\frac{\partial^{2}P}{\partial t\partial x}=\frac{\partial}{\partial x}\left( \frac{\partial P}{\partial t}\right)=\frac{\partial}{\partial x}\left(- \frac{c^{2}}{\overline{F}}\cdot\frac{\partial Q}{\partial x}\right)=-\frac{ c^{2}}{\overline{F}}\cdot\frac{\partial^{2}Q}{\partial x^{2}}. \tag{7}\]  
Using Equation (2), we can determine the derivative with respect to \(t\). \[\frac{\partial^{2}Q}{\partial t^{2}}=-\overline{F}\cdot\frac{\partial^{2}P}{ \partial t\partial x}-2a\cdot\frac{\partial Q}{\partial t}. \tag{8}\]Let us substitute (7) into Equation (8). Then, Equation (8) assumes the following form:  
\[\frac{\partial^{2}Q}{\partial t^{2}}+2a\frac{\partial Q}{\partial t}=c^{2}\frac{ \partial^{2}Q}{\partial x^{2}},\qquad x\in(0,2l),\ \ \ t\in(0,T), \tag{9}\]  
The initial conditions are as follows:  
\[Q(0,x)=Q^{0}(x),\quad\frac{\partial Q}{\partial t}(0,x)=G^{0}(x),\ \ \ x\in[0,2l], \tag{10}\]  
and the boundary conditions are as follows:  
\[Q(t,0)=Q_{0}(t),\ \ \ Q(t,2l)=Q_{out}(t),\ \ t\in[0,T], \tag{11}\]  
\[P(t,0)=P_{0}(t),\ \ \ \ \ Q(t,0)=Q_{0}(t),\ \ \ \ \ for\ \ \ x=0, \tag{4}\]0  
where \(G^{0}(x)=-\overline{F}\cdot\frac{\partial^{0}(x)}{\partial x}-2 aQ^{0}(x)\) is the initial rate of gas displacement. In the direct problem, it is necessary to find \(Q(t,x)\) based on the given functions \(Q^{0}(x)\), \(G^{0}(x)\), \(Q_{0}(t)\), \(Q_{pl}(t)\), and \(Q_{out}(t)\).