Contents lists available at ScienceDirect International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media O. Nikan a , J.A. Tenreiro Machado b , A. Golbabai a, ⁎ , T. Nikazad a a School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran b Department of Electrical Engineering, ISEP-Institute of Engineering, Polytechnic of Porto, Porto, Portugal ARTICLE INFO Keywords: Riemann-Liouville fractional derivative Fractal mobile/immobile transport model RBF-FD Stability Convergence 2010 MSC: 35R11 65M70 91G60 ABSTRACT The fractal mobile/immobile model of the solute transport is based on the assumption that the waiting times in the immobile region follow a power-law, and this leads to the application of fractional time derivatives. The model covers a wide family of systems that include heat difusion and ocean acoustic propagation. This paper develops an efcient computational technique, stemming from the radial basis function-generated fnite dif- ference (RBF-FD), to solve the fractal mobile-immobile transport model (FMTM). The time fractional derivative of the FMTM is discretized via the shifted Grünwald-Letnikov formula with second-order accuracy. On the other hand, the spatial derivative is approximated using the local RBF-FD method. The main beneft of the local collocation technique is that we only need to consider discretization points present in each of the sub-domains around the collocation point. The stability and convergence analysis of the proposed method are proven via the energy method in the L 2 space. The numerical results for the FMTM on regular and irregular domains confrm the theoretical formulation and efciency of the proposed scheme. 1. Introduction In recent decades fractional calculus (FC) has received considerable attention in many scientifc areas. Indeed, because of the intrinsic nonlocality, fractional derivatives are better than traditional integer- order formulations for modeling real-life problems where the under- lying process has hereditary properties [1–4]. A wide range of such problems can be found in the scientifc literature namely, in physics, engineering, biology, mechanics, thermodynamics, fnance, and others [5–12]. The governing transport model derived by means of the Fick's law is often referred to as the advection-dispersion equation (ADE) [13]. A breakthrough curve (BTC) results from the ADE in the form of a Gaussian distribution function for a solute source that is released in- stantaneously [14]. Nonetheless, several evidences led to diferent fndings. In fact, there are two striking features that cannot be described by the ADE. First, the ADE represents the peak in the BTC at an earlier time (early arrival). Second, the tailing of the BTC lasts much longer than expected [15]. The frst evidence stems from the continuous time random walk (CTRW) that was introduced by Scher and Lax [16] and later addressed in hydrology by Berkowitz et al. [14,15]. The second represents is the mobile/immobile model (MIM) approach [17]. A simple hypothesis is the basis of MIM approach, namely the assumption that the regions in the global fow of geological mediums are mobile and immobile. The MIM approach has been considered by several re- searchers in the feld of hydrology for dealing with the transport pro- blems in saturated and unsaturated regions, both in granular and fractured media, since the leading work by Coats et al. [18]. The MIM describes the reactive solute transport with scale-dependent dispersion in heterogeneous porous media and divides the porous medium's liquid phase into mobile (fowing) and immobile (stagnant) regions. The convective-dispersive transport is assumed to be limited to the mobile liquid region, and the exchange of solvents between these two liquid regions can be explained as a process of frst order [19]. Dentz et al. [20] tried to fnd the equivalence between the La- grangian random walks and the MIM continuum models. They applied the same approach as the one used for showing the equivalence of Brownian motion and the difusion equation. The equivalence between the CTRW and the MIM is important because the limit processes related to the CTRW is known and it is associated with a MIM continuum model. The signifcance of the convergence is given by the possibility of achieving clear predictions in long-term processes. Goltz et al. [21] and Harvey et al. [22] found that the single-rate MIM asymptotically pre- dicts an exponential model where the proportion of mobile and injected https://doi.org/10.1016/j.icheatmasstransfer.2019.104443 ⁎ Corresponding author. E-mail addresses: omid_nikan@mathdep.iust.ac.ir (O. Nikan), jtm@isep.ipp.pt (J.A.T. Machado), golbabai@iust.ac.ir (A. Golbabai), tnikazad@iust.ac.ir (T. Nikazad). International Communications in Heat and Mass Transfer 111 (2020) 104443 0735-1933/ © 2019 Elsevier Ltd. All rights reserved. T