AIMS Mathematics, 2020, 5(2): 1062-1073. doi: 10.3934/math.2020074.

Research article Special Issues

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative

1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India
2 Department of Mathematics, Cankaya University, Ankara-06430, Turkey
3 Institute of Space Sciences, Magurele-Bucharest-R 76900, Romania
4 Department of HEAS(Mathematics), Rajasthan Technical University, Kota-324010, India

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

In this paper, we discuss the phenomenon of miscible flow with longitudinal dispersion in porous media. This process simultaneously occur because of molecular diffusion and convection. Here, we analyze the governing differential equation involving Caputo-Fabrizio fractional derivative operator having non singular kernel. Fixed point theorem has been used to prove the uniqueness and existence of the solution of governing differential equation. We apply Laplace transform and use technique of iterative method to obtain the solution. Few applications of the main result are discussed by taking different initial conditions to observe the effect on derivatives of different fractional order on the concentration of miscible fluids.
  Article Metrics

Keywords Caputo-Fabrizio fractional derivative operator; miscible flow; fixed point theorem; Laplace transform; iterative method

Citation: Ritu Agarwal, Mahaveer Prasad Yadav, Dumitru Baleanu, S. D. Purohit. Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative. AIMS Mathematics, 2020, 5(2): 1062-1073. doi: 10.3934/math.2020074


  • 1. R. Agarwal, Kritika, S. D. Purohit, A mathematical fractional model with non-singular kernel for thrombin receptor activation in calcium signalling, Math. Meth. Appl. Sci., 42 (2019), 7160-7171.    
  • 2. R. Agarwal, M. P. Yadav, R. P. Agarwal, Collation analysis of fractional moisture content based model in unsaturated zone using q-homotopy analysis method, Methods of Mathematical Modelling: Fractional Differential Equations, CRC Press, Taylor & Francis, 151, 2019.
  • 3. R. Agarwal, M. P. Yadav, R. P. Agarwal, et al., Analytic solution of fractional advection dispersion equation with decay for contaminant transport in porous media, Matematicki Vesnik, 71 (2019), 5-15.
  • 4. R. Agarwal, M. P. Yadav, R. P. Agarwal, et al., Analytic solution of space time fractional advection dispersion equation with retardation for contaminant transport in porous media, Progress in Fractional Differentiation and Applications, 5 (2019), 283-295.
  • 5. R. Agarwal, M. P. Yadav, R. P. Agarwal, Analytic solution of time fractional Boussinesq equation for groundwater flow in unconfined aquifer, J. Discontinuity, Nonlinearity Complexity, 8 (2019), 341-352.    
  • 6. A. Atangana, D. Baleanu, Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer, J Eng. Phys., 143 (2017), Article Number: D4016005.
  • 7. M. S. Aydogan, D. Baleanu, A. Mousalou, et al., On high order fractional integro-differential equations including the Caputo-Fabrizio derivative, Boundary Value Probl., 2018 (2018), 90.
  • 8. D. Baleanu, A. Mousalou, S. Rezapour, The extended fractional Caputo-Fabrizio derivative of order 0 ≤ σ < 1 on $C_\mathbb{R}[0,1]$ and the existence of solutions for two higher-order series-type differential equations, Adv. Differ. Equations, 2018 (2018), 255.
  • 9. N. R. Bastos, Calculus of variations involving Caputo-Fabrizio fractional differentiation, Statistics, Optimization & Information Computing, 6 (2018), 12-21.
  • 10. J. Bear, Dynamics of fluids in porous media, Courier Corporation, 2013.
  • 11. D. Baleanu, S. S. Sajjadi, A, Jajarmi, et al., New features of the fractional Euler-Lagrange equations for a physical system within non-singular derivative operator, Eur. Phys. J. Plus, 134 (2019), 181.
  • 12. D. Baleanu, A. Jajarmi, J. H. Asad, The fractional model of spring pendulum: New features within different kernels, Proc. Rom. Acad., 19 (2018), 447-454.
  • 13. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fractional Differ. Appl., 2 (2015), 73-85.
  • 14. G. Dagan, Flow and transport in porous formations, Springer Science & Business Media, 2012.
  • 15. G. De Josselin de Jong, Longitudinal and transverse diffusion in granular deposits, Trans. Am. Geophys. Union, 39 (1958), 67-74.    
  • 16. F. A. Dullien, Porous media: Fluid transport and pore structure, Academic press, 2012.
  • 17. R. A. Greenkorn, Steady flow through porous media, AIChE Journal, 27 (1975), 529-545.
  • 18. J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Front. Fract. Calc., 1 (2017), 270-342.
  • 19. A. Jajarmi, B. Ghanbari, D. Baleanu, A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence, Chaos: An Interdiscip. J. Nonlinear Sci., 29 (2019), 093111.
  • 20. A. Jajarmi, S. Arshad, D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A: Stat. Mech. Appl., 535 (2019), 122524.
  • 21. J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fractional Differ. Appl., 1 (2015), 87-92.
  • 22. P. I. Polubarinova-Koch, Theory of ground water movement, Princeton University Press, 2015.
  • 23. P. G. Saffman, A theory of dispersion in a porous medium, J. Fluid Mech., 6 (1959), 321-349.    
  • 24. A. Scheidegger, On the theory of flow of miscible phases in porous media, International Union of Geodesy and Geophysics, 1957.
  • 25. F. W. Schwartz, Macroscopic dispersion in porous media: The controlling factors, Water Resour. Res., 13 (1977), 743-752.    
  • 26. M. P. Yadav, R. Agarwal, Numerical investigation of fractional-fractal Boussinesq equation, Chaos: An Interdiscip. J. Nonlinear Sci., 29 (2019), 013109.


This article has been cited by

  • 1. Dumitru Baleanu, Pshtiwan Othman Mohammed, Shengda Zeng, Inequalities of trapezoidal type involving generalized fractional integrals, Alexandria Engineering Journal, 2020, 10.1016/j.aej.2020.03.039
  • 2. Shahid Khan, Muhammad Adil Khan, Saad Ihsan Butt, Yu-Ming Chu, A new bound for the Jensen gap pertaining twice differentiable functions with applications, Advances in Difference Equations, 2020, 2020, 1, 10.1186/s13662-020-02794-8
  • 3. Hong-Hu Chu, Saima Rashid, Zakia Hammouch, Yu-Ming Chu, New fractional estimates for Hermite-Hadamard-Mercer’s type inequalities, Alexandria Engineering Journal, 2020, 10.1016/j.aej.2020.06.040
  • 4. Agneta M. Balint, Stefan Balint, Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective, Fractal and Fractional, 2020, 4, 3, 36, 10.3390/fractalfract4030036
  • 5. Shrideh Al-Omari, Serkan Araci, Mohammed Al-Smadi, Ghaleb Gumah, Hussam Alrabaiah, Estimates of certain paraxial diffraction integral operator and its generalized properties, Advances in Difference Equations, 2020, 2020, 1, 10.1186/s13662-020-02859-8
  • 6. Ritu Agarwal, Mahaveer Prasad Yadav, Ravi P. Agarwal, Fractional flow equation in fractured aquifer using dual permeability model with non-singular kernel, Arabian Journal of Mathematics, 2020, 10.1007/s40065-020-00293-y

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved