## 6 Numerical Simulations  
In this section, we provide computational experiments to verify the robustness and the applicability of the proposed algorithm. In order to gather measurements, we first solve the direct problem. Then, we use these observations to solve the inverse problem. In this quasi-real framework, we are able to calculate the accuracy of the solution to the inverse problem. We solve the problems using the finite difference method [24]. Let us consider the following counterpart of (1). **Example 1** (One-dimensional advection-diffusion solute transport equation). : \[\frac{\partial C}{\partial t}=a\frac{\partial^{2}C}{\partial x^{2}}+b\frac{ \partial C}{\partial x}-eC+f(t)\delta(x-x_{0}),\ \ x\in\mathds{R},\ \ t>0\] (26)  
_with the initial conditions_  
\[C(x,0)=C_{0}(x),\ \ x\in\mathds{R}, \tag{27}\]  
_where the time-varying concentration source \(f(t)\) is located at \(x_{0}\in\mathds{R}\)._  
Again, by substituting  
\[C(x,t)=\exp(\beta t+\mu x)u(x,t), \tag{28}\]  
where  
\[\mu=-\frac{b}{2a},\ \ \beta=-e-\frac{b^{2}}{4a},\]  
then the problem (26), (27) is transformed to  
\[ \begin{split}\mathcal{L}u&=F(x,t)=\exp(-\beta t- \mu x)f(t)\delta(x-x_{0}),\\ \mathcal{L}u&\equiv\frac{\partial u}{\partial t}-a \frac{\partial^{2}u}{\partial x^{2}},\ \ x\in\mathds{R},\ \ t>0,\end{split}  \tag{29}\]  
where  
\[u(x,0)=u_{0}(x)=\exp(-\mu x)C_{0}(x). \tag{30}\]  
Further, the unbounded problem is localized accordingly to \(\Omega=\{-L\leq x\leq L\}\). Now, we will briefly explain the application of the finite difference method on the model (29). Let us introduce the meshes  
\[ \begin{split}\bar{\omega}_{h}&=\{x_{0}=-L,\ldots,x _{i}=-L+ih,\ldots,x_{2l}=L\},\ I=L/h,\\ \bar{\omega}_{t}&\{t^{0}=0,\ldots,t^{i}=j\triangle t,\ldots,t^{l}=T\},\ J=T/\triangle t,\end{split} \]  
where \(h\) and \(\triangle t\) are the spatial and temporal steps, respectively. Let the approximation of \(u(x_{i},t^{j})\approx u_{i}^{j}\) and \(F(x_{i},t^{j})\approx F_{i}^{j}\), \(i=0,\ldots,2I\), \(j=0,\ldots,J\), then the fully implicit finite difference scheme is written in canonical form as  
\[\omega u_{i-1}^{j+1}+\gamma u_{i}^{j+1}+\alpha u_{i+1}^{j+1}=\triangle tF_{i}^{ j+1}+u_{i}^{j},\]  
where  
\[\alpha=-\frac{\triangle t}{h^{2}}a,\quad\gamma=2\frac{\triangle t}{h^{2}}a+1.\]  
The tridiagonal system can be solved by means of the [PERSON] algorithm [24]. Let \(L=20\) m, \(T=10\) s, \(a=5\) m\({}^{2}\)s\({}^{-1}\), \(b=1\) ms\({}^{-1}\), \(e=0.002\) s\({}^{-1}\), \(x_{0}=5\) m and the concentration source be  
\[f(t)=\left\{ \begin{array}{cc}T,&t\leq\dfrac{T}{2},\\ 1-\dfrac{t}{T},&t>\dfrac{T}{2}.\end{array} \right. \tag{31}\]  
If \(C_{0}(x)=10\Phi(x|0,4)\) mgm\({}^{3}\), where \(\Phi(x|\mu_{x},\sigma_{x})\) is the PDF of the normal distribution with mean \(\mu_{x}\) and standard deviation \(\sigma_{x}\), then the solution to the localized problem (26) is plotted on Figure 1 (left), and the boundary conditions are given in Figure 1 (right). Currently, we are able to start solving the inverse problem. Let us take the measurements  
\[C(x_{1},t)=Y(t),\ \ 0\leq t\leq T, \tag{32}\]  
and, respectively,  
\[u(x_{1},t)=Y(t)\exp(-\beta t-\mu x_{1}) \tag{33}\]  
for \(x_{1}=0\). The recovered function \(f(t)\) (31) is presented in Figure 2. It relatively closely follows the true value (31). Figure 1: Concentration (**left**) and boundary conditions (**right**) for problems (26)–(31). Let us try another experiment with a more fluctuating value of \(f(t)\):  
\[f(t)=\sin(t). \tag{34}\]  
The identified source is given in Figure 3. Although lagging slightly, the recovered function follows the true trend. Figure 3: True and recovered values of the concentration source (34). Figure 2: True and recovered values of the concentration source (31). The same is true for the source, similar to [8]; see Figure 4.  
\[f(t)=\left\{ \begin{array}{cc}\frac{2t}{T},&t\leq\frac{T}{2},\\ 1,&t>\frac{T}{2}.\end{array} \right. \tag{35}\]  
The suggested approach could also be successfully used as follows.