doi: 10.3934/dcdss.2020021

MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives

1. 

Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan

2. 

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

* Corresponding author: Ilyas Khan (ilyaskhan@tdt.edu.vn)

Received  May 2018 Revised  September 2018 Published  March 2019

Fund Project: The author Kashif Ali Abro is highly thankful and grateful to Mehran University of Engineering and Technology, Jamshoro, Pakistan, for generous support and facilities of this research work.

The novelty of this research is to utilize the modern approach of Atangana-Baleanu fractional derivative to electrically conducting viscous fluid embedded in porous medium. The mathematical modeling of the governing partial differential equations is characterized via non-singular and non-local kernel. The set of governing fractional partial differential equations is solved by employing Laplace transform technique. The analytic solutions are investigated for the velocity field corresponding with shear stress and expressed in term of special function namely Fox-H function, moreover a comparative study with an ordinary and Atangana-Baleanu fractional models is analyzed for viscous flow in presence and absence of magnetic field and porous medium. The Atangana-Baleanu fractional derivative is observed more reliable and appropriate for handling mathematical calculations of obtained solutions. Finally, graphical illustration is depicted via embedded rheological parameters and comparison of models plotted for smaller and larger time on the fluid flow.

Citation: Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020021
References:
[1]

A. Abdon and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20 (2016), 763-769. doi: 10.2298/TSCI160111018A. Google Scholar

[2]

K. A. AbroS. H. SaeedN. MustaphaI. Khan and A. Tassadiq, A Mathematical Study of Magnetohydrodynamic Casson Fluid via Special Functions with Heat and Mass Transfer embedded in Porous Plate, Malaysian Journal of Fundamental and Applied Sciences, 14 (2018), 20-38. Google Scholar

[3]

K. A. AbroM. Hussain and M. M. Baig, An analytic study of molybdenum disulfide nanofluids using the modern approach of Atangana-Baleanu fractional derivatives, The European Physical Journal Plus, 132 (2017), 439-451. Google Scholar

[4]

K. A. AbroA. A. Memon and A. A. Memon, Functionality of circuit via modern fractional differentiations, Analog Integrated Circuits and Signal Processing, (2018), 1-1. doi: 10.1007/s10470-018-1371-6. Google Scholar

[5]

K. A. AbroI. Khan and J. F. Gomez-Aguilar, A mathematical analysis of a circular pipe in rate type fluid via Hankel transform, Eur. Phys. J. Plus, 133 (2018), 397-407. doi: 10.1140/epjp/i2018-12186-7. Google Scholar

[6]

K. A. AbroM. Hussain and M. M. Baig, Slippage of fractionalized oldroyd-b fluid with magnetic field in porous medium, Progress in Fractional Differentiation and Applications; An international Journal, 3 (2017), 69-80. Google Scholar

[7]

K. A. AbroA. A. Shaikh and S. Dehraj, Exact solutions on the oscillating plate of maxwell fluids, Mehran University Research Journal of Engineering and Technology, 35 (2016), 157-162. Google Scholar

[8]

A. K. AbroA. A. IrfanM. A. Sikandar and K. Ilyas, On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative, Journal of King Saud University-Science, (2018). doi: 10.1016/j.jksus.2018.07.012. Google Scholar

[9]

O. J. Algahtani, Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos, Solitons and Fractals, 89 (2016), 552-559. doi: 10.1016/j.chaos.2016.03.026. Google Scholar

[10]

F. AliS. A. A. JanI. KhanM. Gohar and N. A. Sheikh, Solutions with special functions for time fractional free convection flow of Brinkman-type fluid, The European Physical Journal Plus, 131 (2016), 310-321. Google Scholar

[11]

F. AliM. SaqibI. Khan and N. A. Sheikh, Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters'-B fluid model, The European Physical Journal Plus, 131 (2016), 377-390. Google Scholar

[12]

B. S. T. Alkahtani and A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 539-546. doi: 10.1016/j.chaos.2016.03.012. Google Scholar

[13]

S. AmbreenA. A. Kashif and A. S. Muhammad, Muhammad, Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: Applications to thermal science, Journal of Thermal Analysis and Calorimetry, (2018), 1-10. doi: 10.1007/s10973-018-7897-0. Google Scholar

[14]

A. Atangana and I. Kocab, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467-2480. doi: 10.22436/jnsa.009.05.46. Google Scholar

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A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J Eng Mech, 142 (2016), D4016005. Google Scholar

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A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. Google Scholar

[17]

A. Atangana and B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453. doi: 10.3390/e17064439. Google Scholar

[18]

A. Atangana and S. T. A. Badr, Extension of the RLC electrical circuit to fractional derivative without singular kernel, Adv. Mech. Eng, 7 (2015), 1-6. Google Scholar

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A. Atanganaa, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. Google Scholar

[20]

A. Atanganaa and I. Kocab, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467-2480. doi: 10.22436/jnsa.009.05.46. Google Scholar

[21]

A. Atanganaa and I. Kocab, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012. Google Scholar

[22]

J. F. Gomez-AguilarV. F. Morales-DelgadoM. A. Taneco-HernandezD. BaleanuR. F. Escobar-Jimenez and M. M. Quarashi, Analytical solutions of the electrical RLC circuit via Liouville-Caputo operators with local and non-local kernels, Entropy, 18 (2016), 402-419. doi: 10.3390/e18080402. Google Scholar

[23]

J. Hristov, Derivatives with non-singular kernels from the caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus, 95 (2017), 235-249. Google Scholar

[24]

J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory: Derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey's kernel and analytical solutions, Therm. Sci., 21 (2017), 827-839. Google Scholar

[25]

A. A. KashifA. M. Anwar and M. A. Uqaili, A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives, Eur. Phys. J. Plus, 133 (2018), 113-122. doi: 10.1140/epjp/i2018-11953-8. Google Scholar

[26]

A. A. KashifM. Hussain and M. M. Baig, Influences of magnetic field in viscoelastic fluid, International Journal of Nonlinear Analysis and Applications, 9 (2018), 99-109. doi: 10.22075/ijnaa.2017.1451.1367. Google Scholar

[27]

A. A. KashifH. Mukarrum and M. B. Mirza, A mathematical analysis of magnetohydrodynamic generalized Burger fluid for permeable oscillating plate, Punjab University Journal of Mathematics, 50 (2018), 97-111. Google Scholar

[28]

A. A. Kashif and A. S. Muhammad, Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo-Fabrizoi fractional derivative, Punjab University Journal of Mathematics, 49 (2017), 113-125. Google Scholar

[29]

A. A. KashifM. R. MohammadK. IlyasA. A. Irfan and T. Asifa, Analysis of stokes' second problem for nanofluids using modern fractional derivatives, Journal of Nanofluids, 7 (2018), 738-747. Google Scholar

[30]

A. A. KashifD. C. AliA. A. Irfan and K. Ilyas, Dual thermal analysis of magneto-hydrodynamic flow of nanofluids via modern approaches of Caputo-Fabrizio and Atangana-Baleanu fractional derivatives embedded in porous medium, Journal of Thermal Analysis and Calorimetry, (2018), 1-11. doi: 10.1007/s10973-018-7302-z. Google Scholar

[31]

A. A. Kashif and K. Ilyas, Analysis of Heat and Mass Transfer in MHD Flow of Generalized Casson Fluid in a Porous Space Via Non-Integer Order Derivative without Singular Kernel, Chinese Journal of Physics, 55 (2017), 1583-1595. doi: 10.1016/j.cjph.2017.05.012. Google Scholar

[32]

I. KhanA. Gul and S. Shafie, Effects of magnetic field on molybdenum disulfide nanofluids in mixed convection flow inside a channel filled with a saturated porous medium, Journal of Porous Media, 20 (2017), 77-82. Google Scholar

[33]

I. Khan, Shape effects of MoS 2 nanoparticles on MHD slip flow of molybdenum disulphide nanofluid in a porous medium, Journal of Molecular Liquids, 233 (2017), 442-451. Google Scholar

[34]

I. Koca and A. Atangana, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, Thermal Science, (2017). Google Scholar

[35]

H. L. MuzaffarA. A. Kashif and A. S. Asif, Helical flows of fractional viscoelastic fluid in a circular pipe, International Journal of Advanced and Applied Sciences, 4 (2017), 97-105. Google Scholar

[36]

A. S. NadeemF. AlI. Khan and M. Saqib, A modern approach of Caputo-Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium, Neural Computing and Applications, (2016), 1-11. Google Scholar

[37]

A. M. Qasem, A. A. Kashif and K. Ilyas, Analytical solutions of fractional walter's-B fluid with applications, Complexity, (2018), Article ID 8918541.Google Scholar

[38]

M. SaqibA. FarhadK. IlyasA. S. Nadeem and S. Sharidan, Convection in ethylene glycol based molybdenum disulfide nanofluid: Atangana-Baleanu frictional derivatives approach, J Therm Anal Calorim, (2018), 1-10. doi: 10.1007/s10973-018-7054-9. Google Scholar

[39]

N. A. Shah and I. Khan, Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo-Fabrizio derivatives, Eur Phys J C, 76 (2016), 1-11. Google Scholar

[40]

N. A. SheikhF. AliI. KhanM. Gohar and M. Saqib, On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional models, The European Physical Journal Plus, 132 (2017), 540-558. Google Scholar

[41]

N. A. SheikhF. AliM. SaqibI. KhanS. A. A. JanA. S. Alshomrani and M. S. Alghamdi, Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results in Physics, 7 (2017), 789-800. Google Scholar

[42]

P. SopasakisH. SarimveisP. Macheras and A. Dokoumetzidis, Fractional calculus in pharmacokinetics, J. Pharmacokin. Pharmacodyn, 45 (2018), 107-114. Google Scholar

[43]

A. A. Zafar and C. Fetecau, Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel, Alexandria Engineering Journal, 5 (2016), 2789-2796. Google Scholar

[44]

L. ZhuoL. LiuS. DehghanQ. C. Yang and D. Xue, A review and evaluation of numerical tools for fractional calculus and fractional order controls, International Journal of Control, 90 (2017), 1165-1181. doi: 10.1080/00207179.2015.1124290. Google Scholar

show all references

References:
[1]

A. Abdon and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20 (2016), 763-769. doi: 10.2298/TSCI160111018A. Google Scholar

[2]

K. A. AbroS. H. SaeedN. MustaphaI. Khan and A. Tassadiq, A Mathematical Study of Magnetohydrodynamic Casson Fluid via Special Functions with Heat and Mass Transfer embedded in Porous Plate, Malaysian Journal of Fundamental and Applied Sciences, 14 (2018), 20-38. Google Scholar

[3]

K. A. AbroM. Hussain and M. M. Baig, An analytic study of molybdenum disulfide nanofluids using the modern approach of Atangana-Baleanu fractional derivatives, The European Physical Journal Plus, 132 (2017), 439-451. Google Scholar

[4]

K. A. AbroA. A. Memon and A. A. Memon, Functionality of circuit via modern fractional differentiations, Analog Integrated Circuits and Signal Processing, (2018), 1-1. doi: 10.1007/s10470-018-1371-6. Google Scholar

[5]

K. A. AbroI. Khan and J. F. Gomez-Aguilar, A mathematical analysis of a circular pipe in rate type fluid via Hankel transform, Eur. Phys. J. Plus, 133 (2018), 397-407. doi: 10.1140/epjp/i2018-12186-7. Google Scholar

[6]

K. A. AbroM. Hussain and M. M. Baig, Slippage of fractionalized oldroyd-b fluid with magnetic field in porous medium, Progress in Fractional Differentiation and Applications; An international Journal, 3 (2017), 69-80. Google Scholar

[7]

K. A. AbroA. A. Shaikh and S. Dehraj, Exact solutions on the oscillating plate of maxwell fluids, Mehran University Research Journal of Engineering and Technology, 35 (2016), 157-162. Google Scholar

[8]

A. K. AbroA. A. IrfanM. A. Sikandar and K. Ilyas, On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative, Journal of King Saud University-Science, (2018). doi: 10.1016/j.jksus.2018.07.012. Google Scholar

[9]

O. J. Algahtani, Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos, Solitons and Fractals, 89 (2016), 552-559. doi: 10.1016/j.chaos.2016.03.026. Google Scholar

[10]

F. AliS. A. A. JanI. KhanM. Gohar and N. A. Sheikh, Solutions with special functions for time fractional free convection flow of Brinkman-type fluid, The European Physical Journal Plus, 131 (2016), 310-321. Google Scholar

[11]

F. AliM. SaqibI. Khan and N. A. Sheikh, Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters'-B fluid model, The European Physical Journal Plus, 131 (2016), 377-390. Google Scholar

[12]

B. S. T. Alkahtani and A. Atangana, Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos Solitons Fractals, 89 (2016), 539-546. doi: 10.1016/j.chaos.2016.03.012. Google Scholar

[13]

S. AmbreenA. A. Kashif and A. S. Muhammad, Muhammad, Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: Applications to thermal science, Journal of Thermal Analysis and Calorimetry, (2018), 1-10. doi: 10.1007/s10973-018-7897-0. Google Scholar

[14]

A. Atangana and I. Kocab, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467-2480. doi: 10.22436/jnsa.009.05.46. Google Scholar

[15]

A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J Eng Mech, 142 (2016), D4016005. Google Scholar

[16]

A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. Google Scholar

[17]

A. Atangana and B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453. doi: 10.3390/e17064439. Google Scholar

[18]

A. Atangana and S. T. A. Badr, Extension of the RLC electrical circuit to fractional derivative without singular kernel, Adv. Mech. Eng, 7 (2015), 1-6. Google Scholar

[19]

A. Atanganaa, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. Google Scholar

[20]

A. Atanganaa and I. Kocab, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467-2480. doi: 10.22436/jnsa.009.05.46. Google Scholar

[21]

A. Atanganaa and I. Kocab, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012. Google Scholar

[22]

J. F. Gomez-AguilarV. F. Morales-DelgadoM. A. Taneco-HernandezD. BaleanuR. F. Escobar-Jimenez and M. M. Quarashi, Analytical solutions of the electrical RLC circuit via Liouville-Caputo operators with local and non-local kernels, Entropy, 18 (2016), 402-419. doi: 10.3390/e18080402. Google Scholar

[23]

J. Hristov, Derivatives with non-singular kernels from the caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus, 95 (2017), 235-249. Google Scholar

[24]

J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory: Derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey's kernel and analytical solutions, Therm. Sci., 21 (2017), 827-839. Google Scholar

[25]

A. A. KashifA. M. Anwar and M. A. Uqaili, A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives, Eur. Phys. J. Plus, 133 (2018), 113-122. doi: 10.1140/epjp/i2018-11953-8. Google Scholar

[26]

A. A. KashifM. Hussain and M. M. Baig, Influences of magnetic field in viscoelastic fluid, International Journal of Nonlinear Analysis and Applications, 9 (2018), 99-109. doi: 10.22075/ijnaa.2017.1451.1367. Google Scholar

[27]

A. A. KashifH. Mukarrum and M. B. Mirza, A mathematical analysis of magnetohydrodynamic generalized Burger fluid for permeable oscillating plate, Punjab University Journal of Mathematics, 50 (2018), 97-111. Google Scholar

[28]

A. A. Kashif and A. S. Muhammad, Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo-Fabrizoi fractional derivative, Punjab University Journal of Mathematics, 49 (2017), 113-125. Google Scholar

[29]

A. A. KashifM. R. MohammadK. IlyasA. A. Irfan and T. Asifa, Analysis of stokes' second problem for nanofluids using modern fractional derivatives, Journal of Nanofluids, 7 (2018), 738-747. Google Scholar

[30]

A. A. KashifD. C. AliA. A. Irfan and K. Ilyas, Dual thermal analysis of magneto-hydrodynamic flow of nanofluids via modern approaches of Caputo-Fabrizio and Atangana-Baleanu fractional derivatives embedded in porous medium, Journal of Thermal Analysis and Calorimetry, (2018), 1-11. doi: 10.1007/s10973-018-7302-z. Google Scholar

[31]

A. A. Kashif and K. Ilyas, Analysis of Heat and Mass Transfer in MHD Flow of Generalized Casson Fluid in a Porous Space Via Non-Integer Order Derivative without Singular Kernel, Chinese Journal of Physics, 55 (2017), 1583-1595. doi: 10.1016/j.cjph.2017.05.012. Google Scholar

[32]

I. KhanA. Gul and S. Shafie, Effects of magnetic field on molybdenum disulfide nanofluids in mixed convection flow inside a channel filled with a saturated porous medium, Journal of Porous Media, 20 (2017), 77-82. Google Scholar

[33]

I. Khan, Shape effects of MoS 2 nanoparticles on MHD slip flow of molybdenum disulphide nanofluid in a porous medium, Journal of Molecular Liquids, 233 (2017), 442-451. Google Scholar

[34]

I. Koca and A. Atangana, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, Thermal Science, (2017). Google Scholar

[35]

H. L. MuzaffarA. A. Kashif and A. S. Asif, Helical flows of fractional viscoelastic fluid in a circular pipe, International Journal of Advanced and Applied Sciences, 4 (2017), 97-105. Google Scholar

[36]

A. S. NadeemF. AlI. Khan and M. Saqib, A modern approach of Caputo-Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium, Neural Computing and Applications, (2016), 1-11. Google Scholar

[37]

A. M. Qasem, A. A. Kashif and K. Ilyas, Analytical solutions of fractional walter's-B fluid with applications, Complexity, (2018), Article ID 8918541.Google Scholar

[38]

M. SaqibA. FarhadK. IlyasA. S. Nadeem and S. Sharidan, Convection in ethylene glycol based molybdenum disulfide nanofluid: Atangana-Baleanu frictional derivatives approach, J Therm Anal Calorim, (2018), 1-10. doi: 10.1007/s10973-018-7054-9. Google Scholar

[39]

N. A. Shah and I. Khan, Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo-Fabrizio derivatives, Eur Phys J C, 76 (2016), 1-11. Google Scholar

[40]

N. A. SheikhF. AliI. KhanM. Gohar and M. Saqib, On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional models, The European Physical Journal Plus, 132 (2017), 540-558. Google Scholar

[41]

N. A. SheikhF. AliM. SaqibI. KhanS. A. A. JanA. S. Alshomrani and M. S. Alghamdi, Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results in Physics, 7 (2017), 789-800. Google Scholar

[42]

P. SopasakisH. SarimveisP. Macheras and A. Dokoumetzidis, Fractional calculus in pharmacokinetics, J. Pharmacokin. Pharmacodyn, 45 (2018), 107-114. Google Scholar

[43]

A. A. Zafar and C. Fetecau, Flow over an infinite plate of a viscous fluid with non-integer order derivative without singular kernel, Alexandria Engineering Journal, 5 (2016), 2789-2796. Google Scholar

[44]

L. ZhuoL. LiuS. DehghanQ. C. Yang and D. Xue, A review and evaluation of numerical tools for fractional calculus and fractional order controls, International Journal of Control, 90 (2017), 1165-1181. doi: 10.1080/00207179.2015.1124290. Google Scholar

Figure 1.  Profile of velocity field via Atangana-Baleanu fractional differential operator for fractional parameter
Figure 2.  Profile of velocity field via Atangana-Baleanu fractional differential operator for porous medium
Figure 3.  Profile of velocity field via Atangana-Baleanu fractional differential operator for magnetic field
Figure 4.  Comparative analysis of velocity field via Atangana-Baleanu fractional differential operator verses ordinary differential operator for short time
Figure 5.  Comparative analysis of velocity field via Atangana-Baleanu fractional differential operator verses ordinary differential operator for unit time
Figure 6.  Comparative analysis of velocity field via Atangana-Baleanu fractional differential operator verses ordinary differential operator for larger time
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