## 4 Conclusions and Discussions  
The distributed-order fractional diffusion-wave equation for the radial flow to or from a well has been investigated by considering the time-fractional derivative with the Mittag-Leffler function of one parameter. The diffusion-wave equation is featured by two temporal fractional orders \(\alpha\) and \(\beta\) to characterize different diffusion types. It is known that Mittag-Leffler functions are solutions to some fractional differential equations; therefore, they are non-local functions. In consequence, the time-fractional Atangana-Baleanu derivative is a fractional derivative with non-singular and non-local kernel. It is expected that the non-locality of the Mittag-Leffler kernel allows better description of the memory within structure and media with different scales. Also, it is known that Mittag-Leffler functions are suitable for interesting generalizations. The fractional derivatives with Prabhakar kernel are suitable tools to describe complex physical processes, such as the anomalous dielectric relaxation or the renewal processes. The analytical solutions of the initial-boundary value fractional problem have been determined using the Laplace and Dirichlet-Weber integral transforms. The solutions from the general case are suitable for generating the solutions corresponding to the ordinary case, namely for the case \(\alpha=\beta=1\). Recall that [PERSON] et al. [35] studied a similar problem by using Caputo time-fractional derivative. It is important to note that the Caputo fractional derivative cannot be obtained as a particular case of the Atangana-Baleanu derivative, so the results obtained in this paper cannot be identical to those in [PERSON]'s article. However, for the case \(\alpha=\beta=1\), the results obtained by us are equivalent to those obtained by [PERSON] for the same values of the fractional parameters. To highlight the influence of the fractional parameters \(\alpha\) and \(\beta\) on the normalized depth of flow, numerical values of the function \(H(r,t)\) have been determined and plotted with the software Mathcad. In the performed analysis, we used the following values of the parameters: \(D_{f}=0.5,\ Q=-0.5,\ T_{f}=20,\ R=0.5\). In Figure 1 are the presented profiles of the normalized depth \(H(r,t)\), versus \(r\) for \(a=b=0.5\) and for different values of the time \(t\) and the fractional parameters \(\alpha\) and \(\beta\). Note that the variation with radial distance of the normalized depth is not significant. The values of the fractional parameters \(\alpha\) and \(\beta\) have a significant influence on the values of the function \(H(r,t)\). For \(\alpha>\beta\), the values of the normalized depth are higher than for \(\alpha<\beta\). This is due to the different mode of the diffusion process in areas with different pores. The influence of the fractional parameters \(\alpha\) and \(\beta\) on the normalized depth \(H(r,t)\) is analyzed in Figure 2 for \(a\) = 0.4 and \(b\) = 0.6 for different values of the time \(t\). It is observed in Figure 2 that for a constant value of the fractional parameter \(\alpha\), the function \(H(r,t)\) is increasing with the fractional parameter \(\beta\). A similar behavior is observed for \(\beta\) constant and increasing values of the parameter \(\alpha\). The increasing of values of a fractional parameter means that the porosity of the medium is larger; therefore the diffusion process will be enhanced. The influence of the parameter \(a\) on the normalized depth \(H(r,t)\) is presented in Figure 3 for \(t\) = 6 and \(\alpha\) = 0.885 and different values of the fractional parameter \(\beta\). As expected, the normalized depth decreases with the parameter \(a\). This result is due to a larger region of the medium with big pores where the flow is faster. Figure 1: The profiles of normalized depth \(H(r,t)\), versus the radial coordinate \(r\), for small time values (**a**), and for large time values (**b**).