Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative
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 havin...
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| Published in | AIMS mathematics Vol. 5; no. 2; pp. 1062 - 1073 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
AIMS Press
01.01.2020
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| Subjects | |
| Online Access | Get full text |
| ISSN | 2473-6988 2473-6988 |
| 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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| AbstractList | 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. |
| Author | Prasad Yadav, Mahaveer Baleanu, Dumitru D. Purohit, S. Agarwal, Ritu |
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| Cites_doi | 10.1002/mma.5822 10.1029/TR039i001p00067 10.1029/WR013i004p00743 10.1017/S0022112059000672 10.5890/DNC.2019.09.009 |
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| CorporateAuthor | 4 Department of HEAS(Mathematics), Rajasthan Technical University, Kota-324010, India 3 Institute of Space Sciences, Magurele-Bucharest-R 76900, Romania 1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, India 2 Department of Mathematics, Cankaya University, Ankara-06430, Turkey |
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| References | 11 12 13 26 16 17 18 19 R. Agarwal, Kritika, S. D. Purohit (1) R. Agarwal, M. P. Yadav, R. P. Agarwal (5) 2 3 4 6 7 P. G. Saffman (23) 8 9 G. De Josselin de Jong (15) 20 21 F. W. Schwartz (25) |
| References_xml | – ident: 1 article-title: i>A mathematical fractional model with non-singular kernel for</i> thrombin receptor activation in calcium signalling</i publication-title: Math. Meth. Appl. Sci. doi: 10.1002/mma.5822 – ident: 16 article-title: F. A. Dullien, Porous – ident: 15 article-title: i>Longitudinal and transverse diffusion in granular deposits</i publication-title: Trans. Am. Geophys. Union doi: 10.1029/TR039i001p00067 – ident: 8 article-title: i>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</i – ident: 3 article-title: t al., Analytic solution of fractional advection dispersion equation with decay for contaminant transport in porous media</i – ident: 2 article-title: 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 – ident: 6 article-title: i>Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer</i – ident: 18 article-title: i>Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models</i – ident: 19 article-title: i>A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence</i – ident: 9 article-title: i>Calculus of variations involving Caputo-Fabrizio fractional differentiation</i – ident: 7 article-title: t al., On high order fractional integro-differential equations including the Caputo-Fabrizio derivative</i – ident: 17 article-title: i>Steady flow through porous media</i – ident: 25 article-title: i>Macroscopic dispersion in porous media: The controlling factors</i publication-title: Water Resour. Res. doi: 10.1029/WR013i004p00743 – ident: 4 article-title: t al., Analytic solution of space time fractional advection dispersion equation with retardation for contaminant transport in porous media</i – ident: 12 article-title: i>The fractional model of spring pendulum: New features within different kernels</i – ident: 13 article-title: i>A new definition of fractional derivative without singular kernel</i – ident: 23 article-title: i>A theory of dispersion in a porous medium</i publication-title: J. Fluid Mech. doi: 10.1017/S0022112059000672 – ident: 20 article-title: i>A new fractional modelling and control strategy for the outbreak of dengue fever</i – ident: 21 article-title: i>Properties of a new fractional derivative without singular kernel</i – ident: 5 article-title: i>Analytic solution of time fractional Boussinesq equation for groundwater flow in unconfined aquifer</i publication-title: J. Discontinuity, Nonlinearity Complexity doi: 10.5890/DNC.2019.09.009 – ident: 11 article-title: t al., New features of the fractional Euler-Lagrange equations for a physical system within non-singular derivative operator</i – ident: 26 article-title: i>Numerical investigation of fractional-fractal Boussinesq equation</i |
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| SubjectTerms | caputo-fabrizio fractional derivative operator fixed point theorem iterative method laplace transform miscible flow |
| Title | Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative |
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