## 3 Results and Discussion
### Numerical Solution for the Problem
A distortion in the shape and a narrowing in the range of maximum values of the function \(Q(t,x)\) indicate incomplete convergence of the iterative process. The main discrepancies between the exact and approximate solutions are in the region of \(x\in(0.8,1.6)\), where the values of \(Q(t,x)\) on the right graph are smaller than the exact solution. This discrepancy is caused by insufficient correction of the initial condition in the Landweber method at this stage. The exact solution reflects the correct distribution of wave processes, while the approximate solution shows minor deviations in areas with high gradients. Additional iterations of the Landweber method can improve convergence and reduce discrepancies between the numerical and exact solutions. The graphs clearly show the importance of accurately approximating the initial condition, \(G^{0}(x)\), to achieve high accuracy in modeling gas lift wells. Figure 2: An exact solution, \(Q(t,x)\), using the true initial condition, \(G^{0}(x)\). (**a**) shows the smooth wave propagation and symmetric distribution of the function \(Q(t,x)\) across the domain. (**b**) illustrates the behavior of \(Q(t,x)\) near regions where high gradients appear, highlighting the accuracy of the true initial condition. Both graphs (Figures 4 and 5) show a smooth change in the function along the x-axis for small values of time intervals, t. As the time increases, an increase in the amplitude is observed in the central part of the graph, near \(x\approx 1\). This increase indicates the formation of a localized maximum, which is typical for solutions of the considered system of equations. The function \(Q_{fix}(t,x)\) demonstrates a more pronounced symmetry with respect to the center than the function \(Q_{n}(t,x)\), indicating the high accuracy of the exact solution. In both cases, a distinct peak is visible in the vicinity of \(x\approx 1.2\). With increasing time, this peak becomes stronger and reaches its maximum values at \(t=0.5\). However, in the right graph, which represents an approximate solution, the central peak shows a slight shift and fluctuation, which may indicate the approximation's inaccuracy. At the initial time intervals, the values of the function \(Q(t,x)\) at the boundaries \(x=0\) and \(x=2\) remain practically unchanged. However, with an increase in the time parameter, t, a decrease in the values of the function at the boundaries is observed. This phenomenon  
Figure 4: Graph of the function \(Q_{fix}(t,x)\) for different values of \(t\) with precisely specified initial conditions. Figure 3: An approximate solution, \(Q(t,x)\), using the initial condition, \(G_{n}^{0}(x)\), reconstructed using the Landweber method. (**a**) shows the distribution of the approximate solution of \(Q(t,x)\), showing deviations in waveform and amplitude. (**b**) identifies regions near \(x\approx l\) and \(t\approx 0.4\) where the approximation diverges from the exact solution. is consistent with the physical interpretation of the boundary conditions and reflects the expected behavior of the solution for the system. The proposed method has important practical applications, as it helps optimize the parameters of the gas lift process and minimizes gas consumption for maximum oil production. This optimization leads to a significant reduction in operating costs during oil production, which is especially important for aging fields with low flow rates.