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We presented an analysis of evolutions equations generated by three fractional derivatives namely the RiemannLiouville, CaputoFabrizio and the Atangana-aleanu fractional derivatives. For each evolution equation, we presented the exact solution for time variable and studied the semigroup principle. The Riemann-iouville fractional operator verifies the semigroup principle but the associate evolution equation does not. The Caputo-abrizio fractional derivative does not satisfy the semigroup principle but surprisingly, the exact solution satisfies very well all the principle of semigroup. However, the AtanganaBaleanu for small time is the stretched exponential derivative, which does not satisfy the semigroup as operators. For a large time the AtanganaBaleanu derivative is the same with Riemann-iouville fractional derivative, thus satisfies semigroup principle as an operator. The solution of the associated evolution equation does not satisfy the semigroup principle as Riemann-iouville. With the connection between semigroup theory and the Markovian processes, we found out that the Atangana-aleanu fractional derivative has at the same time Markovian and non-Markovian processes. We concluded that, the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects, which are not always Markovian. We presented the exact solutions of some evolutions equation using the Laplace transform. In addition to this, we presented the numerical solution of a nonlinear equation and show that, the model with the AtanganaBaleanu fractional derivative has random walk for small time. We also observed that, the Mittag-Leffler function is a good filter than the exponential and power law functions, which makes the Atangana-Baleanu fractional derivatives powerful mathematical tools to model complex real world problems.
 

Fractional-derivative models are promising tools for characterizing non-Fickian transport in heterogeneous media. Most fractional models utilize an infinite domain, although realistic problems occur on bounded domains. To quantify the impact of a finite or semi-infinite boundary on non-Fickian transport in natural geological media, this study evaluates three representative fractional advection-dispersion equations (FADEs) with absorbing or reflective boundaries. Results show that the temporal FADE (t-FADE) with absorbing/reflective boundaries has analytical solutions, the one-sided spatial FADE (s-FADE) in bounded-domains can be simulated using an Eulerian solver, and the tempered spatiotemporal FADE (st-FADE) can be efficiently solved using a fully Lagrangian approach. Further simulations reveal important impacts of absorbing/reflective boundaries on non-Fickian diffusion. First, the “local” reflective boundary mainly affects the solute dynamics near the boundary for non-local super-diffusion, while the “nonlocal” reflective boundary changes the overall pattern of non-Fickian transport in the whole domain. Second, the total mass for solutes in absorbing boundaries declines non-linearly with respect to time. Third, the mobile and immobile phase plumes tend to respond differently to the boundary because of their different transport mechanisms. Fourth, a field application shows that both the s-FADE with a negative skewness and the t-FADE can be used to quantify bounded-domain sub-diffusion for fluorescein dye transport in the Red Cedar River with a large Péclet number, although the determination of the upstream boundary position contains high uncertainty. Evaluation of the boundary impact on sub-diffusion, super-diffusion, and their mixture may improve our understanding of the nature of non-Fickian transport in bounded domains.