# A numerical process of the mobile-immobile advection-dispersion model arising in solute transport

Document Type : Original Article

Authors

1 Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University

2 Department of Mathematics, Qom University of Technology, Qom, Iran

Abstract

In the present article‎, ‎to find the answer to the mobile-immobile advection-dispersion model of temporal fractional order $0< \beta \leq 1$ (MI-ADM-TF)‎, ‎which can be applied to model the solute forwarding in watershed catchment and flood‎, ‎the effective high-order numerical process is gonna be built‎.
‎To do this‎, ‎the temporal-fractional derivative of the MI-ADM-TF is discretized by using the linear interpolation‎, ‎and the temporal-first derivative by applying the first-order precision of the finite-difference method‎.
‎On the other hand‎, ‎After obtaining a semi-discrete form‎, ‎to obtain the full-discrete technique‎, ‎the space derivative is approximated utilizing a collocation approach based on the Legendre basis‎.
‎The convergence order of the implicit numerical design for MI-ADM-TF is discussed in that is linear‎.
‎Moreover‎, ‎the temporal-discretized structure of stability is also discussed theoretically in general in the article‎.
‎Eventually‎, ‎two models are offered to demonstrate the quality and authenticity of the established process‎.

Keywords

#### References

[1] M. Abdelkawy, M.A. Zaky, A.H. Bhrawy, D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys, 67(3) 2015, 773-791.
[2] B. Berkowitz, Characterizing flow and transport in fractured geological media: A review, Advances in water resources, 25(8-12) 2002, 861-884.
[3] A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics, Vol. 378, Springer, 2014.
[4] Z. Chen, J. Qian, H. Zhan, L. Chen, S. Luo, Mobile-immobile model of solute transport through porous and fractured media, Proceedings of ModelCARE, 2009 2010, 274.
[5] K. Coats, B. Smith, et al., Dead-end pore volume and dispersion in porous media, Society of petroleum engineers journal, 4(1) 1964, 73-84.
[6] Z. Deng, L. Bengtsson, V.P. Singh, Parameter estimation for fractional dispersion model for rivers, Environmental Fluid Mechanics, 6(5) 2006, 451-475.
[7] M.S. Field, F.J. Leij, Solute transport in solution conduits exhibiting multi-peaked breakthrough curves, Journal of hydrology, 440 2012, 26-35.
[8] H.R. Ghehsareh, A. Zaghian, M. Raei, A local weak form meshless method to simulate a variable order time-fractional mobile-immobile transport model, Engineering Analysis with Boundary Elements, 90 2018, 63-75.
[9] A. Golbabai, O. Nikan, T. Nikazad, Numerical investigation of the time fractional mobile-immobile advection-dispersion model arising from solute transport in porous media, International Journal of Applied and Computational Mathematics, 5(3) 2019, 50.
[10] S. Kim, M.L. Kavvas, Generalized Ficks law and fractional ADE for pollution transport in a river: Detailed derivation, Journal of Hydrologic Engineering, 11(1) 2006, 80-83.
[11] K. Kumar, R.K. Pandey, S. Sharma, Comparative study of three numerical schemes for fractional integro-differential equations, Journal of Computational and Applied Mathematics, 315 2017, 287-302.
[12] Q. Liu, F. Liu, I. Turner, V. Anh, Y. Gu, A RBF meshless approach for modeling a fractal mobile/immobile transport model, Applied Mathematics and Computation, 226 2014, 336-347.
[13] F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection dispersion models, Computers & Mathematics with Applications, 64(10) 2012, 2990-3007.
[14] H. Mesgarani, A. Beiranvand, Y.E. Aghdam, The impact of the Chebyshev collocation method on solutions of the time-fractional Black-Scholes, Mathematical Sciences, 2020, 1-7.
[15] H. Pourbashash, Application of high-order spectral method for the time fractional mobile/immobile equation, Computational Methods for Differential Equations, 4(4) 2016, 309-322.
[16] H. Pourbashash, D. Baleanu, M.M. Al Qurashi, On solving fractional mobile/immobile equation, Advances in Mechanical Engineering, 9(1) 2017, 168-178.
[17] H. Safdari, Y. E. Aghdam, J. G´ omez-Aguilar, Shifted Chebyshev collocation of the fourth kind with convergence analysis for the space-time fractional advection-diffusion equation, Engineering with Computers, 2020, 1-12.
[18] H. Safdari, H. Mesgarani, M. Javidi, Y.E. Aghdam, Convergence analysis of the space fractional order diffusion equation based on the compact finite difference scheme, Computational and Applied Mathematics, 39(2) 2020, 1-15.
[19] S. Samuel, V. Gill, On Riesz-Caputo fractional differentiation matrix of radial basis functions via complex step differentiation method, Journal of Fractional Calculus and Applications, 9(2) 2018, 133-140.
[20] H. Scher, M. Lax, Stochastic transport in a disordered solid. I. Theory, Physical Review B, 7(10) 1973, 4491.
[21] H. Zhang, F. Liu, M.S. Phanikumar, M.M. Meerschaert, A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model, Computers & Mathematics with Applications, 66(5) 2013, 693-701.

### History

• Receive Date: 26 May 2022
• Revise Date: 14 August 2022
• Accept Date: 16 August 2022
• First Publish Date: 01 September 2022