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

  • Received: 04 October 2019 Accepted: 19 December 2019 Published: 10 January 2020
  • MSC : Primary: 34A08; Secondary: 26A33

  • 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.

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

    Related Papers:

  • 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.


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    [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. doi: 10.1002/mma.5822
    [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. doi: 10.5890/DNC.2019.09.009
    [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. doi: 10.1029/TR039i001p00067
    [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. doi: 10.1017/S0022112059000672
    [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. doi: 10.1029/WR013i004p00743
    [26] M. P. Yadav, R. Agarwal, Numerical investigation of fractional-fractal Boussinesq equation, Chaos: An Interdiscip. J. Nonlinear Sci., 29 (2019), 013109.
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