## 3 Results and Discussion  
### Numerical Solution for the Problem  
The initial data for the computational experiment are as follows: \(Q^{0}=0.21\) is the volumetric flow rate of injected gas (m\({}^{3}\)/s); \(P^{0}=5177500\) is the initial pressure distribution (Pa); \(l=1485\) is the well depth (m); \(\lambda_{1}=0.01\) is the hydraulic resistance in the ring; \(\lambda_{2}=0.23\) is the hydraulic resistance in the pipe; \(d_{1}\approx 0.1353\) is the effective diameter of the well annular space (m); and \(d_{2}=0.073\) is the internal diameter of the well (m). \(\rho_{1}=0.75\) is the gas density (kg/m\({}^{3}\)); \(\rho_{2}=700\) is the oil density (kg/m\({}^{3}\)); g = 9.8 is the gravity acceleration (m/s\({}^{2}\)); \(c_{1}=331\) is the speed of sound in the annular space (m/s);\(c_{2}=850\) is the speed of sound in the pipe (m/s); \(\text{r}_{1}=\frac{\text{d}_{1}}{2}\) is the radius of the annular space (m); \(\text{r}_{2}=\frac{\text{d}_{2}}{2}\) is the radius of the inner pipe (m); \(\text{F}_{1}=\pi\text{r}_{1}^{2}\) is the cross-sectional area of the pipe of the well's annular space (m\({}^{2}\)); \(\text{F}_{2}=\pi\text{r}_{2}^{2}\) is the cross-sectional area of the inner pipe of the well (m\({}^{2}\)); \(\text{w}_{1}=\frac{\text{d}_{1}}{\text{F}_{1}\rho_{1}}\) is averaged over the cross-section speed of the mixture in the ring (m/s); and \(\text{w}_{2}=\frac{\text{d}_{0}}{\text{F}_{2}\rho_{2}}\) is the average cross-sectional speed of the mixture in the pipe (m/s). The coefficients in the ring and pipe are described as follows:  
\[a_{1}=\frac{g}{2w_{1}}+\frac{\lambda_{1}w_{1}}{4d_{1}},\;a_{2}=\frac{g}{2w_{2 }}+\frac{\lambda_{2}w_{2}}{4d_{2}}\]  
Calculations of the test problem were performed for the parameters \(T=0.5\), \(l=1\), \(N_{x}=100\), \(N_{t}=100\), \(h=\frac{2l}{N_{x}}\), and \(\tau=\frac{T}{N_{t}}\). The step of gradient descent is \(\alpha=0.01\), the exact solution of which is known, as follows:  
\[G^{0}(x)=sin[(\pi x/2l)]\]  
The initial approximation is as follows:  
\[G^{0}_{0}(x)=0.1.\]  
Figure 1 shows a graph of change in the functional \(J\left(G^{0}_{n}\right)\) during the numerical solution process of the inverse problem. The graph demonstrates a monotonic decrease in functional values during the iteration process. To quantitatively assess the accuracy of the numerical method, an error analysis was performed between the approximate and exact solution of the function \(Q(T,x)\). The Euclidean (discrete \(L_{2}\)) error norm was used as a measure as follows:  
\[\varepsilon=\left(\sum_{i=1}^{N_{x}}\left[Q^{N_{t}}_{i}-Q^{1}(x_{i})\right]^{ 2}h\right)^{1/2}.\]  
When using the gradient descent step, \(\alpha=0.01\), and \(46\) iterations, the error value was \(\varepsilon\approx 0.018\). This confirms the good accuracy of the approximation, especially in the  
Figure 1: Graph of decreasing functional \(J\left(G^{0}_{n}\right)\). central part of the region, while the main discrepancies are concentrated in the interval \(x\in(0.8,\ 1.6)\), where high gradients of the function are observed. The convergence analysis of the functional \(J(G_{n}^{0})\) showed its monotonous decrease from the initial value to the level of \(\approx 0.02\) over 46 iterations. Further increases in the number of iterations did not provide significant improvement, which indicates the achievement of the optimum and the stability of the method. Thus, the proposed numerical method provides a reliable approximation of the desired function and demonstrates stable behavior with respect to computational costs. Figure 2 shows (a) a two-dimensional and (b) a three-dimensional picture of the smooth distribution of the function \(Q(t,x)\), reflecting correct wave and symmetry development. The range of changes in the function is wider, and the changes in gradient are smoother. Figure 3 shows (a) two-dimensional and (b) three-dimensional solutions, which diverge for an approximate value of the initial condition, \(G_{n}^{0}(x)\), in areas close to \(x=l\) and \(t\approx 0.4\).