## 4 Numerical Examples: MATLAB Implementation
We considered the following function: \(g(x,y)=ye^{x}+e^{y}+x^{2}\), which in polar coordinates is  
Figure 4: Exact source is shown in red, and recovered source with regularization is in blue. In this case, the regularized source given by (17), which was obtained using Tikhonov regularization, allowed us to handle numerical instability to get a good approximation of the exact source. Figure 3: Exact source is shown in red, and recovered source without regularization is in blue. In this case, the recovered source was obtained using the coefficients given by Equation (11), and it was far from the exact source due to the numerical instability of the inverse source problem. Figure 2: Exact measurement is shown in red, and recovered measurement is in blue. The error in the exact measurement was obtained by adding a random error to the exact measurement using the rand function of MATLAB. given by \(g(\theta)=sin(\theta)e^{cos(\theta)}+e^{sin(\theta)}+cos^{2}(\theta)\). Using Equation (8), we found the solution to the forward problem \(V\), i.e., to find the ideal (exact) measurement \(V\), we had to find the solution to problem (1)-(5) and then restrict it to the boundary \(S_{2}\). The noisy data \(V_{\delta}\) was obtained as in the previous example. Figure 5 displays both the exact measurement \(V\) (in red) and the measurement with error \(V_{\delta}\) (in blue). Figure 6 shows the plot of both the exact source and the recovered source without regularization. As in the previous example, we also observed the necessity of applying regularization methods. Figure 7 displays both the exact and recovered sources when regularization was applied. The relative error between the exact source \(g\) and the recovered regularized source, \(g_{\alpha(\delta),N}\), denoted by \(RE_{S_{1}}(g_{\alpha(\delta),N},g)\), is given by  
\[RE_{S_{1}}(g_{\alpha(\delta),N},g)=0.054.\]  
\[g_{\alpha(\delta),N}(\theta)=\sum_{k=1}^{16}A_{k}(\alpha)\Big{[}V_{k,\delta}^{1 }\cos k\theta+V_{k,\delta}^{2}\sin k\theta\Big{]},\]0 Exact measurement is shown in red, and measurement with error is in blue. The latter was obtained by adding a random error to the exact measurement, which was done using the rand function of MATLAB. \[g_{\alpha(\delta),N}(\theta)=\sum_{k=1}^{16}A_{k}(\alpha)\Big{[}V_{k,\delta}^{1 }\cos k\theta+V_{k,\delta}^{2}\sin k\theta\Big{]},\]1 Exact source is shown in red, and recovered source without regularization is in blue. The difference shows the necessity to apply regularization methods. **Example 3**. : We considered the same values for \(R_{1}\), \(R_{2}\), \(\sigma_{1}\), and \(\sigma_{2}\) as in the previous example. We considered the following square-wave function given in polar coordinates as  
\[g(\theta)=\left\{ \begin{aligned} -1,&\quad\text{if}\ -\pi\leq\theta<0,\\ 1,&\quad\text{if}\quad\ 0\leq\theta<\pi.\end{aligned} \right. \tag{20}\]  
We found the solution \(V\) to the forward problem and the noisy data \(V_{\delta}\) for the boundary \(S_{2}\) as in the previous examples. Figure 8 displays both the exact source \(g\) (red) and its approximation by Fourier series \(g_{N}\) (blue) when truncating the Fourier series (7) to the first N = 16 terms. Figure 9 shows both the exact measurement \(V\) (in red) and the measurement with error \(V_{\delta}\) (in blue). Figure 8: Plot of the exact square-wave function \(g\) (red) and its approximation by Fourier series (blue). We observe the Fourier series oscillations and the presence of the Gibbs phenomenon. This jump function is frequently used in electronic signals. Figure 7: Exact source is shown in red, and the recovered source with regularization is in blue. In this case, the regularized source given by Equation (17) allowed us to obtain a stable approximation of the source. Figure 10 displays both the exact and recovered sources without regularization. As in the previous examples, we observed the importance of regularization methods to obtain stable solutions to the inverse problem. Figure 11 displays both the exact and recovered sources when regularization was applied.