## 6 Numerical Experiments  
In this section, we report some numerical examples of the TFPDEs to demonstrate the accuracy and efficiency of the numerical method. At first, we wanted to show that the approach proposed in this paper properly works, so we chose two examples in such a way that the exact solutions of FPDE could be evaluated analytically. This allowed us to check the accuracy and the order of convergence of the numerical solution. In both the examples, we assumed suitable smooth solutions, and we used the proposed method on the non-uniform grid and on a uniform grid. We compared the numerical results, observing that the finite difference scheme generated more accurate numerical solutions on the non-uniform grid than on the uniform one. In the third test, we solved a TFADE of physical interest with the source term chosen as a linear function of the field variable in order to show that the method is applicable to a wide class of TFADEs. The presented problem is one of the most used mathematical models in the applied sciences. It describes how the concentration of one or more substances, chemical or biological species, distributed in a medium changes under the influence of three processes, namely, advection, diffusion and reaction. With this last test, we illustrate how the changes in the solution behavior arise when the fractional order is varied. **Example 1**. : _We consider the following TFADE_  
\[\frac{\partial^{\alpha}}{\partial t^{\alpha}}u(x,t)+\frac{\partial}{ \partial x}u(x,t)-\frac{\partial^{2}}{\partial x^{2}}u(x,t)=f(x,t),\quad 0<x<1, \quad 0<t\leq T,\] \[u(x,0)=0, 0\leq x\leq 1, \tag{21}\] \[u(0,t)=t^{\beta}, u(1,t)=et^{\beta}, 0<t\leq T,\]  
_where the source term is given by_  
\[f(x,t)=\frac{\Gamma(\beta+1)}{\Gamma(\beta+1-\alpha)}e^{x}t^{ \beta-\alpha}.\]  
_The exact solution is_  
\[u(x,t)=e^{x}t^{\beta}.\]  
In this example, we took \(K_{1}=K_{2}=1\) with \(\alpha=0.5\) and \(\beta=5\). Figure 2 shows the comparison between the numerical solution, \(U_{j}^{n}\), and the exact solution, \(u(x_{j},t^{n})\), at different times computed with \(N=J=20\). From Figure 2, it can be seen that the numerical solution, \(U_{j}^{n}\), is in good agreement with the exact solution \(u(x_{j},t^{n})\). The exact solution is reported with the solid line. We report only the numerical results obtained for the value parameter \(\alpha=0.5\). Analogous results were obtained for \(0.1\leq\alpha\leq 0.9\). In order to investigate the temporal error and the convergence order of the numerical difference method, we defined the maximum error between the exact solution, \(u(x_{j},t^{N})\), and the numerical solution, \(U_{j}^{N}\), at the final time, \(t^{N}\)  
\[e_{\infty}(N,J) = \max_{1\leq j\leq J}|u(x_{j},t^{N})-U_{j}^{N}|, \tag{22}\]  
Figure 2: Comparison of the exact and the numerical solutions of the fractional partial differential equations (FPDE) (21) for \(\alpha=0.5\) with \(N=J=20\) at different time steps. Solid lines: exact solution; star, circle and square points: numerical solutions. and the convergence order, as follows  
\[Order=\log_{2}\left(\frac{e_{\infty}(N,J)}{e_{\infty}(2N,J)}\right). \tag{23}\]  
In order to show the efficiency of the method, we defined \(\bar{U}_{j}^{n}\) as the numerical solution obtained by the implicit difference method on a uniform grid defined by \(\Delta t^{n}=T/N\), for \(n=1,\cdots,N\). We used notations similar to the (22) and (23) to define the maximum error, \(\bar{e}_{\infty}(N,J)\), between the exact solution, \(u(x_{j},t^{N})\), and the numerical solution, \(\bar{U}_{j}^{N}\), at the final time, \(t^{N}\), and the convergence \(Order_{Q}\). In this test, we fixed \(J=100\), a value large enough such that the spatial error is negligible as compared with the temporal error. Table 1 shows the values of \(e_{\infty}\) and \(\bar{e}_{\infty}\) and the corresponding numerical convergence orders for \(\alpha=0.1,0.5\) and \(0.9\). It can be seen that the method is stable and convergent when solving the problem (21) on both the computational grids.