## 6 Numerical Experiments
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{\alpha}\) & \(\mathbf{N}\) & \(\mathbf{e_{\infty}}\) & \(Order\) & \(\bar{e}_{\infty}\) & \(Order_{0}\) \\ \hline \multirow{4}{*}{0.1} & \(10\) & \(3.7836\times 10^{-5}\) & & \(9.4723\times 10^{-5}\) & \\  & \(20\) & \(9.6542\times 10^{-6}\) & \(1.9705\) & \(2.8745\times 10^{-5}\) & \(1.7204\) \\  & \(40\) & \(2.4674\times 10^{-6}\) & \(1.9682\) & \(8.5307\times 10^{-6}\) & \(1.7526\) \\  & \(80\) & \(6.2843\times 10^{-7}\) & \(1.9732\) & \(2.4911\times 10^{-6}\) & \(1.7759\) \\ \hline \multirow{4}{*}{0.5} & \(10\) & \(4.4597\times 10^{-4}\) & & \(1.3182\times 10^{-3}\) & \\  & \(20\) & \(1.3436\times 10^{-4}\) & \(1.7309\) & \(4.8998\times 10^{-4}\) & \(1.4278\) \\  & \(40\) & \(4.1920\times 10^{-5}\) & \(1.6803\) & \(1.7898\times 10^{-4}\) & \(1.4529\) \\ \cline{1-1}  & \(80\) & \(1.3435\times 10^{-5}\) & \(1.6416\) & \(6.4671\times 10^{-5}\) & \(1.4686\) \\ \hline \multirow{4}{*}{0.9} & \(10\) & \(3.3229\times 10^{-3}\) & & \(6.9875\times 10^{-3}\) & \\  & \(20\) & \(1.4844\times 10^{-3}\) & \(1.1625\) & \(3.3290\times 10^{-3}\) & \(1.0682\) \\ \cline{1-1}  & \(40\) & \(6.7617\times 10^{-4}\) & \(1.1345\) & \(1.5524\times 10^{-3}\) & \(1.0837\) \\ \cline{1-1}  & \(80\) & \(3.1119\times 10^{-4}\) & \(1.1196\) & \(7.2847\times 10^{-4}\) & \(1.0915\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(e_{\infty}\), \(\bar{e}_{\infty}\) and convergence orders, related to the numerical solutions, \(U_{j}^{m}\) and \(\bar{U}_{j}^{m}\), respectively, for different values of \(N\) and \(\alpha\), with \(J=100\). Figure 3: Comparison of the exact and the numerical solutions of the FPDE (24) for \(\alpha=0.1\) with \(N=J\)\(=20\) at different time steps. Solid lines: exact solution; star, circle and square points: numerical solutions. _where the source term is chosen as a linear function of the field variable_  
\[f(u)=\beta u(x,t).\]  
The model describes the one-dimensional transport problem of a concentration, \(u(x,t)\), of a chemical or biological species in a flowing medium, such as air or water. The species concentration is assumed to be horizontally and vertically well mixed, such that it varies only in the longitudinal or downstream direction. Moreover, a steady and uniform flow field is imposed and the effects of the dispersion are constant in time and space. A reaction where the transformation rate \(\beta\) is proportional to the species concentration was considered; according to the sign of rate, decay effects may or may not have occurred. This model has been used by several authors, because it is able to model a wide variety of physical and biological phenomena, see, for example, [53; 54; 55]. In this example, we used \(a=0\), \(b=5\), \(K_{1}=1\), \(K_{2}=1\) and \(\beta=0.2\). Figure 4 shows the solution behavior at different times obtained for \(\alpha=0.1\) and \(\alpha=0.9\) and with \(N=J=100\). The height of the solution profile decreased as the time increased. Analogous results were obtained for other values of \(\alpha\). Lower profiles were obtained for \(\alpha=0.1\), and higher ones were obtained for \(\alpha=0.9\) at the final time, \(t^{N}=1\), that the behavior of solutions were comparable. Figure 5 shows the solution behavior with different values of \(\alpha\) between \(0\) and \(1\) at the final time, \(t^{N}=1\). Figure 5 also shows that the solution exhibited an anomalous diffusion behavior. The height of the solution profile decreased when \(0.1\leq\alpha\leq 0.5\) and increased when \(0.6\leq\alpha\leq 0.9\). Thus, the solution continuously depends on the \(\alpha\)-order of the time-fractional derivative. Figure 4: Numerical solutions of the FPDE (25) with \(N=J=100\) at different time steps. Left frame: \(\alpha=0.1\). Right frame \(\alpha=0.9\). Figure 5: Numerical solutions of the FPDE (25) for different values of \(\alpha\) with \(N=J=100\).