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

Ritu Agarwal; Mahaveer Prasad Yadav; Dumitru Baleanu; S. D. Purohit; Ritu Agarwal; Mahaveer Prasad Yadav; Dumitru Baleanu; S. D. Purohit

doi:10.3934/math.2020074

AIMS Mathematics

2020, Volume 5, Issue 2: 1062-1073. doi: 10.3934/math.2020074

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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

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

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Abstract

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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