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March  2020, 13(3): 467-484. doi: 10.3934/dcdss.2020026

Models of fluid flowing in non-conventional media: New numerical analysis

1. 

Institute for Groundwater Studies, Faculty for Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa

2. 

Department of Mathematics, Faculty of Science Madhav Institue of Technology & Sciences, Gwalior - 474005, Madhya Pradesh, India

* Corresponding author: Sonal Jain

Received  March 2018 Revised  April 2018 Published  March 2019

The concept of differentiation with power law reset has not been investigated much in the literature, as some researchers believe the concept was wrongly introduced, it is unable to describe fractal sharps. It is important to note that, this concept of differentiation is not to describe or display fractal sharps but to describe a flow within a medium with self-similar properties. For instance, the description of flow within a non-conventional media which does not obey the classical Fick's laws of diffusion, Darcy's law and Fourier's law cannot be handle accurately with conventional mechanical law of rate of change. In this paper, we pointed out the use of the non-conventional differential operator with fractal dimension and it possible applicability in several field of sciences, technology and engineering dealing with non-conventional flow. Due to the wider applicability of this concept and the complexities of solving analytically those partial differential equations generated from this operator, we introduced in this paper a new numerical scheme that will be able to handle this class of differential equations. We presented in general the conditions of stability and convergence of the numerical scheme. We applied to some well-known diffusion and subsurface flow models and the stability analysis and numerical simulations for each cases are presented.

Citation: Abdon Atangana, Sonal Jain. Models of fluid flowing in non-conventional media: New numerical analysis. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 467-484. doi: 10.3934/dcdss.2020026
References:
[1]

A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernal, Thermal Sci., 20 (2016), 757-763. 

[2]

A. Atangana and S. Jain, A new numerical approximation of the fractal ordinary differential equation, Eur. Phys. J. Plus, 133 (2018), p37. doi: 10.1140/epjp/i2018-11895-1.

[3]

A. Atangana and K. M. Owolabi, Analysis and application of new fractional Adams-Bashforth scheme with Caputo Fabrizio derivative, Chaos, Solitons and Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.

[4]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton Fract., 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.

[5]

W. ChenX. D. Zhang and D. Korosak, Investigation on fractional and fractal derivative relaxation-oscillation models, Int. J. Nonlin. Sci. Num., 11 (2010), 3-9. 

[6]

L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Third edition, Chapman and Hall (CRC Press), Taylor and Francis Group, London and New York, 2014.

[7]

R. Gnitchogna and A. Atangana, New two step laplace adam-bashforth method for integer an non integer order partial differential equations, Numerical Methods for Partial Differential Equations, 34 (2018), 1739-1758.  doi: 10.1002/num.22216.

[8]

S. Jain, Numerical analysis for the fractional diffusion and fractional Buckmaster equation by two step Laplace Adam–Bashforth method, Eur. Phys. J. Plus, 133 (2018), p19. doi: 10.1140/epjp/i2018-11854-x.

[9]

S. KumarX. B. Yin and D. Kumar, A modified homotopy analysis method for solution of fractional wave equations, Adv. Mech. Eng., 7 (2015), 1-8. 

[10]

K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos, Solitons and Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.

[11]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.

[12]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 133 (2018), p15. doi: 10.1140/epjp/i2018-11863-9.

[13]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), p444. doi: 10.1140/epjp/i2017-11717-0.

show all references

References:
[1]

A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernal, Thermal Sci., 20 (2016), 757-763. 

[2]

A. Atangana and S. Jain, A new numerical approximation of the fractal ordinary differential equation, Eur. Phys. J. Plus, 133 (2018), p37. doi: 10.1140/epjp/i2018-11895-1.

[3]

A. Atangana and K. M. Owolabi, Analysis and application of new fractional Adams-Bashforth scheme with Caputo Fabrizio derivative, Chaos, Solitons and Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.

[4]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton Fract., 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.

[5]

W. ChenX. D. Zhang and D. Korosak, Investigation on fractional and fractal derivative relaxation-oscillation models, Int. J. Nonlin. Sci. Num., 11 (2010), 3-9. 

[6]

L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Third edition, Chapman and Hall (CRC Press), Taylor and Francis Group, London and New York, 2014.

[7]

R. Gnitchogna and A. Atangana, New two step laplace adam-bashforth method for integer an non integer order partial differential equations, Numerical Methods for Partial Differential Equations, 34 (2018), 1739-1758.  doi: 10.1002/num.22216.

[8]

S. Jain, Numerical analysis for the fractional diffusion and fractional Buckmaster equation by two step Laplace Adam–Bashforth method, Eur. Phys. J. Plus, 133 (2018), p19. doi: 10.1140/epjp/i2018-11854-x.

[9]

S. KumarX. B. Yin and D. Kumar, A modified homotopy analysis method for solution of fractional wave equations, Adv. Mech. Eng., 7 (2015), 1-8. 

[10]

K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos, Solitons and Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.

[11]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.

[12]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 133 (2018), p15. doi: 10.1140/epjp/i2018-11863-9.

[13]

M. Toufik and A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132 (2017), p444. doi: 10.1140/epjp/i2017-11717-0.

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