Numerical Investigation of the Time Fractional Mobile-Immobile Advection-Dispersion Model Arising from Solute Transport in Porous Media

  • Ahmad GolbabaiEmail author
  • Omid Nikan
  • Touraj Nikazad
Original Paper


Evolution equations containing fractional derivatives can offer efficient mathematical models for determination of anomalous diffusion and transport dynamics in multi-faceted systems that cannot be precisely modeled by using normal integer order equations. In recent times, researches have found out that lots of physical processes illustrate fractional order characteristics that alters with time or space. The continuum of order in the fractional calculus permits the order of the fractional operator be accounted for as a variable. In the current research work, radial basis functions (RBFs) approximation is utilized for solving fractional mobile-immobile advection-dispersion (TF-MIM-AD) model in a bounded domain which is applied for explaining solute transport in both porous and fractured media. In this approach, firstly, the discretization process of the aforesaid equation with of convergence order \(\mathcal {O}(\delta t^{})\) in the t-direction is described via the finite difference scheme for \( 0< \alpha <1\). Afterwards, by help of the meshless methods based on RBFs, we will illustrate how to obtain the approximated solution. The stability and convergence of time-discretized scheme are also theoretically discussed in detail throughout the paper. Finally, two numerical instances are included to clarify effectiveness and accuracy of our proposed concepts which is investigated in the current research work.


TF-MIM-AD model Radial basis functions RBF-PS Collocation methods Stability Convergence 

Mathematics Subject Classification

35R11 65M70 91G60 



We express our sincere thanks to Associate Editor and anonymous reviewers for carefully reading this paper and their comments and suggestions which greatly improved the presentation of this paper. The authors are also very much grateful to the Editor-in-Chief, Santanu Saha Ray for his pursuances.


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© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.School of MathematicsIran University of Science and TechnologyNarmak, TehranIran

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